COMBINATORIAL AND GEOMETRIC RIGIDITY WITH SYMMETRY CONSTRAINTS BY BERND SCHULZE

Transcription

1 COMBINATORIAL AND GEOMETRIC RIGIDITY WITH SYMMETRY CONSTRAINTS BY BERND SCHULZE A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS YORK UNIVERSITY, TORONTO, ONTARIO MAY 2009

2 Abstract In this thesis, we investigate the rigidity and flexibility properties of frameworks consisting of rigid bars and flexible joints that possess non-trivial symmetries Using techniques from group representation theory, we first prove that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form Based on this result, we prove a generalization of the symmetry-extended version of Maxwell s rule given in [25] which can be applied to both injective and non-injective realizations in all dimensions We then use this rule to prove that a symmetric isostatic (ie, minimal infinitesimally rigid) framework must obey some very simply stated restrictions on the number of joints and bars that are fixed by various symmetry operations of the framework In particular, it turns out that a 2-dimensional isostatic framework must belong to one of only six possible point groups For 3-dimensional isostatic frameworks, all point groups are possible, although restrictions on the placement of structural components still apply For three of the five non-trivial symmetry groups in dimension 2 that allow isostatic frameworks, namely for the groups C 2 and C 3 generated by a half-turn and a 3-fold rotation, respectively, and for the group C s generated by a reflection, we establish symmetric versions of Laman s Theorem ([33, 46]) More precisely, we show that the necessary conditions derived from the symmetry-extended version of Maxwell s rule, together with the Laman conditions, are also sufficient for a framework whose joints are positioned as generically as possible subject to the given symmetry conditions to be isostatic Symmetric versions of Henneberg s Theorem ([40, 33]) and Crapo s iv

3 Theorem ([20, 33, 67]) for the groups C 2, C 3, and C s are also established For the remaining two non-trivial symmetry groups in dimension 2 that allow isostatic frameworks, we offer some analogous conjectures Finally, we derive sufficient conditions for the existence of a finite flex of a symmetric framework Finite flexes detected with these results have the nice property that they preserve all of the symmetries of the given framework v

4 To my parents vi

5 Acknowledgements First, I would like to thank my supervisor Prof Walter Whiteley for his invaluable advice, guidance, and support throughout my time as his student His care and enthusiasm for my work as well as his vast knowledge and exceptional insights into mathematics have always been a great source of motivation and inspiration for me The countless enlightening conversations I have had with Prof Walter Whiteley over the last few years have not only played a crucial role in the writing of this thesis, but they have also nourished my intellectual maturity which I will benefit from for a long time to come In addition, Prof Walter Whiteley has always been available for me whenever I faced any sort of trouble or had a question about my research or writing I simply could not have wished for a better or friendlier supervisor Secondly, I would like to express many thanks to Prof Asia Weiss and Prof Mike Zabrocki for taking the time to read my thesis and serve on my supervisory committee in the midst of all their other activities Thanks also to Prof Ada Chan, Prof Andy Mirzaian, and Prof Meera Sitharam for agreeing to be members of my examining committee Further, I would like to thank the organizers and participants of the AIM workshop in Palo Alto in December 2007 and the BIRS workshop in Banff in July 2008, particularly Prof Robert Connelly, Prof Simon Guest, and Prof Brigitte Servatius, for all the interesting and fruitful discussions A special thanks goes to my mother, father, and sister for their continued support and encouragement throughout my studies at home and abroad vii

6 They have always been there for me Last, but not least I would like to thank my partner Krishna Wu for all her care and love vii

7 Contents 1 Introduction 1 11 Background and motivation 1 12 Outline of thesis 7 2 Definitions and preliminaries Graph theory terminology Introduction to rigidity theory Rigidity Infinitesimal rigidity Static rigidity Generic rigidity Basic rigidity results Symmetry in frameworks 38 3 A classification of symmetric frameworks The classification 48 viii

8 32 The notion of (S, Φ)-generic Of what types Φ can a framework be? When is a type Φ of a framework a homomorphism? 74 4 Using group representation theory to analyze symmetric frameworks Block-diagonalization of the rigidity matrix Basic definitions in group representation theory The internal and external representation The block-diagonalization A symmetry-extended version of Maxwell s rule The necessary conditions The rule Example and further remarks Restrictions on the number of fixed joints and bars of symmetric isostatic frameworks Fixed versus geometrically unshifted Isostatic frameworks in dimension Isostatic frameworks in dimension A remark on non-injective realizations Necessary conditions for independence and infinitesimal rigidity150 ix

9 5 Necessary and sufficient conditions for a graph to be (S, Φ)- generically isostatic Preliminary remarks and results Characterizations of (C 3, Φ)-generically isostatic graphs Symmetrized Henneberg moves and 3Tree2 partitions for C The main result for C Characterizations of (C 2, Φ)-generically isostatic graphs Symmetrized Henneberg moves and 3Tree2 partitions for C The main result for C Characterizations of (C s, Φ) generically isostatic graphs Symmetrized Henneberg moves and 3Tree2 partitions for C s The main result for C s Conjectures, algorithms, and further remarks Dimension Dimension Independence and infinitesimal rigidity Symmetry as a sufficient condition for a flex Alternate definitions of rigidity via the edge function 276 x

10 62 Detection of symmetric flexes Examples of flexible frameworks Examples in 2D Examples in 3D Further work Rigidity of other types of symmetric structures Pinned frameworks Body-bar structures Body-hinge and molecular structures Coning, symmetry, and spherical frameworks Symmetric global rigidity 319 A Character tables of selected point groups 322 xi

11 List of Tables 41 Calculations of characters in the 2-dimensional symmetryextended Maxwell s equation Calculations of characters in the 3-dimensional symmetryextended Maxwell s equation 139 xii

12 List of Figures 21 An invariant (b) and a non-invariant subgraph (c) of the graph G under α = (v 1 v 2 v 3 )(v 4 v 5 v 6 ) Aut(G) A rigid (a) and a flexible (b) framework in the plane The flex shown in (c) takes the framework in (b) to the framework in (d) The arrows indicate the non-zero displacement vectors of an infinitesimal rigid motion (a) and infinitesimal flexes (b, c) of frameworks in R (a), (b) The arrows indicate a tension (a) and a compression (b) in a bar (c) An equilibrium load on a non-degenerate triangle This load can be resolved by the triangle as shown in (d) (e) An unresolvable equilibrium load on a degenerate triangle: for any joint of this framework, tensions or compressions in the bars cannot reach an equilibrium with the load vector at this joint 25 xiii

13 25 A Venn diagram showing the relationship between sets of various types of generic configurations and the set of regular points of a graph G (see Definition 2222 in the end of Section 225) The double banana satisfies the counts in Theorem 228 for d = 3, but it is not generically 3-isostatic Illustrations of the Vertex Addition Theorem (a) and the Edge Split Theorem (b) in dimension Illustration of an X-replacement of a graph G A proper (a) and a non-proper (b) 3Tree2 partition Symmetry elements corresponding to symmetry operations in dimension 2: (a) a rotation C m, m 2; (b) a reflection s; (c) the identity Id Symmetry elements corresponding to symmetry operations in dimension 3: (a) an improper rotation S m, m 2; (b) a rotation C m, m 2; (c) a reflection s; (d) the identity Id dimensional realizations of (K 3,3, C s ) of different types dimensional realizations of (G tp, C 2 ) of different types dimensional realizations of (G bp, C s ) of different types By the converse of Pascal s Theorem, the joints of any realization in R (K3,3,C 2 ) or R (K3,3,C s,φ b ) lie on a conic section A 3-dimensional realization of (K 4, C s ) of type Υ a (a) and of type Υ b (b) 64 xiv

14 36 A realization of K 3,3 that is (C 2v, Φ)-generic, but not (C s, Φ a )- generic, where C s is the subgroup of C 2v generated by s v and Φ a = Φ Cs A realization of (G t, C 2 ) of type Θ a and Θ b (a) and a realization of (G bp, C s ) of type Ξ a and Ξ b (b) Non-injective realizations with Aut(G, p) = {id} A graph G (a) and a realization (G, p) R (G,Cs ) (b) for which there does not exist a homomorphism Φ : C s Aut(G) so that (G, p) is of type Φ A graph G (a) and a realization (G, p) R (G,C3 ) (b) for which there does not exist a homomorphism Φ : C 3 Aut(G) so that (G, p) is of type Φ A framework (K 3, p) R (K3,C s,φ) Illustration of the proof of Lemma 411 (i) Illustration of the proof of Lemma 411 (i) in the case where x is a reflection Illustration of the proof of Lemma 411 (ii) (a, b) Vectors of the H e-invariant subspaces V (A ) e V (A ) e (a) and (b) of R 6 ; (c, d) vectors of the H i-invariant subspaces V (A ) i (c) and V (A ) i (d) of R (a) A basis for the subspace V (A ) T ; (b) a basis for the subspace V (A ) T ; (c) a basis for the subspace R = V (A ) R 112 xv

15 47 (a) An infinitesimal flex of (K 3,3, p) R (K3,3,C 2v,Φ) which is symmetric with respect to B 2 (the displacement vector at each joint of (K 3,3, p) remains unchanged under Id and s v and is reversed under C 2 and s h ) (b) An unresolvable equilibrium load on (K 3,3, p) which is symmetric with respect to B 2 (c) A self-stress of (K 3,3, p) which is symmetric with respect to A 1 (the tensions and compressions in the bars of (K 3,3, p) remain unchanged under all symmetry operations in C 2v ) Geometrically unshifted bars in dimension 2: (a) a bar that is geometrically unshifted by a half-turn C 2 ; (b) possible placement of a bar that is geometrically unshifted by a reflection s Possible placement of a bar that is geometrically unshifted by: (a) any rotation C m, m 2 (in dimension 3); (b) a half-turn C 2 (in dimension 3) alone Possible placement of a bar that is geometrically unshifted by a reflection s (in dimension 3): (a) lying in the reflection plane; (b) lying perpendicular to the reflection plane Possible placement of a bar that is geometrically unshifted by: (a) any improper rotation S m, m 2 (in dimension 3); (b) an inversion i = S 2 (in dimension 3) alone 131 xvi

16 412 Examples, for each of the possible point groups, of small 2- dimensional isostatic frameworks: (a) C 1 ; (b) C 2 ; (c) C 3 ; (d) C s ; (e) C 2v ; (f) C 3v For each of C s and C 3v, two examples are given, where in (i) each mirror has a bar centered at and perpendicular to the mirror line, whereas in (ii) each mirror has a bar that lies in the mirror line For C 2v, the bar lying at the center of C 2 must lie in one mirror line and perpendicular to the other A regular octahedron (a), and a convex polyhedron (b) generated by capping every face of the original octahedron with a twisted octahedron The polyhedron in (b) has the rotational but not the reflectional symmetries of the polyhedron in (a) If a framework is constructed from either polyhedron by placing bars along edges, and joints at vertices, the framework will be isostatic An icosahedron (a), and the second stellation of the icosahedron (b) If a framework is constructed from either polyhedron by placing bars along edges, and joints at vertices, the framework will be isostatic The framework (b) could be constructed from the framework (a) by capping each face of the original icosahedron A series of hats added symmetrically along a 3-fold axis of an isostatic framework leaves the framework isostatic 146 xvii

17 416 3-dimensional frameworks with mirror symmetry satisfying Maxwell s original rule (a) A framework which is not isostatic, since b Φa(s) > j Φa(s); (b) a framework which is not isostatic, since b Φb (s) < j Φb (s); (c) a framework which satisfies b Φc (s) = j Φc (s), but is not isostatic, because it contains the frameworks depicted in (a) and (b) (Φ a, Φ b, Φ c are uniquely determined by the injective realizations) dimensional frameworks with half-turn symmetry satisfying Maxwell s original rule (a) An isostatic framework; (b) a framework which is not isostatic, since b Φb (C 2 ) = j Φb (C 2 ) = 0; (c) a framework which satisfies j Φc(C2 ) = 0 and b Φc(C2 ) = 2, but is not isostatic, because it contains the non-isostatic framework depicted in (b) (Φ b and Φ c are uniquely determined by the injective realizations) Independent frameworks in R (G,C2,Φ) with j Φ(C2 ) = b Φ(C2 ) = 0 (a), j Φ(C2 ) = 0, b Φ(C2 ) = 2 (b), and j Φ(C2 ) = 1, b Φ(C2 ) = 0 (c) Infinitesimally rigid frameworks in R (G,C2,Φ) with j Φ(C2 ) = b Φ(C2 ) = 0 (d), j Φ(C2 ) = 0, b Φ(C2 ) = 2 (e), and j Φ(C2 ) = 1, b Φ(C2 ) = 0 (f) Illustration of the proof of Lemma A (C 3, Φ) vertex addition of a graph G, where Φ(C 3 ) = γ and Φ(C3) 2 = γ A (C 3, Φ) edge split of a graph G, where Φ(C 3 ) = γ and Φ(C3) 2 = γ xviii

18 54 A (C 3, Φ) extension of a graph G, where Φ(C 3 ) = γ and Φ(C3) 2 = γ (C 3, Φ) 3Tree2 partitions of graphs, where Φ(C 3 ) = γ and Φ(C3) 2 = γ If a graph G satisfies the conditions in Theorem 521 (ii) and has a vertex v of valence 3, then G is a graph of one of the types depicted above Construction of a (C 3, Φ) 3Tree2 partition of G in the case where G is a (C 3, Φ k 1 ) vertex addition of G k Construction of a (C 3, Φ) 3Tree2 partition of G in the case where G is a (C 3, Φ k 1 ) edge split of G k Construction of a (C 3, Φ) 3Tree2 partition of G in the case where G is a (C 3, Φ k 1 ) extension of G k The frame (G, p, q) The frame (G, p t, q t ) Illustration of the proof of Lemma Illustration of the proof of Lemma Illustration of the proof of Lemma A realization of (G, C 2 ) of type Φ (a) and a realization of (G, C s ) of type Ψ (b) A (C 2, Φ) vertex addition of a graph G, where Φ(C 2 ) = γ A (C 2, Φ) edge split of a graph G, where Φ(C 2 ) = γ 190 xix

19 518 (C 2, Φ) 3Tree2 partitions of graphs, where Φ(C 2 ) = γ The edges in black color represent edges of the invariant trees If a graph G satisfies the conditions in Theorem 531 (ii) and has a vertex v of valence 3, then G is a graph of one of the types depicted above Construction of a (C 2, Φ) 3Tree2 partition of G in the case where G is a (C 2, Φ k 1 ) vertex addition of G k 1 The edges in black color represent edges of the invariant tree T (k) Construction of a (C 2, Φ) 3Tree2 partition of G in the case where G is a (C 2, Φ k 1 ) edge split of G k 1 The edges in black color represent edges of the invariant trees The frame (G, p, q) The frame (G, p t, q t ) A (C s, Φ) single vertex addition of a graph G, where Φ(s) = σ A (C s, Φ) single edge split of a graph G, where Φ(s) = σ A (C s, Φ) double vertex addition of a graph G, where Φ(s) = σ A (C s, Φ) double edge split of a graph G, where Φ(s) = σ A (C s, Φ) X-replacement of a graph G, where Φ(s) = σ A (C s, Φ) 3Tree2 partition of a graph (a) and a (C s, Φ) 3Tree2 partition of a graph (b), where Φ(s) = σ 214 xx

20 530 If a graph G satisfies the conditions in Theorem 541 (ii) and has a vertex v with N G (v) = {v 1, v 2, v 3 } such that σ({v i, v j }) {v i, v j } for all {i, j} {1, 2, 3}, then G is a graph of one of the types depicted above If a graph G satisfies the conditions in Theorem 541 (ii) and has a vertex v with N G (v) = {v 1, v 2, v 3 } such that σ({v i, v j }) = {v i, v j } for exactly one pair {i, j} {1, 2, 3}, then G is a graph of one of the types depicted above If a graph G satisfies the conditions in Theorem 541 (ii), has no vertex of valence two, no vertex of valence three that is fixed by σ, and every 3-valent vertex v of G (except possibly the vertices that are incident with the edge that is fixed by σ) has the property that σ(u) = u for all u N G (v), then there exists v V (G) with N G (v) = {v 1, v 2, v 3 }, σ(v i ) = v i for all i = 1, 2, 3, and val G (v i ) = 4 for some i {1, 2, 3} Construction of a (C s, Φ) or (C s, Φ k 1 ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) single vertex addition of G k Construction of a (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) single edge split of G k 1 The edges in black color represent edges of the invariant trees Construction of a (C s, Φ) or (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) double vertex addition of G k 1 and v 1, v 2 / V ( T (k 1) ) xxi

21 536 Construction of a (C s, Φ) or (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) double vertex addition of G k 1 and at least one of v 1 or v 2 is a vertex of T (k 1) 0 The edges in black color represent edges of the invariant tree Construction of a (C s, Φ) or (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) double edge split of G k 1, {v 1, v 2 }, {σ(v 1 ), σ(v 2 )} E ( T (k 1) ) 0 and either v3 / V ( T (k 1) ) 0 or v 3 V ( T (k 1) ) 0 and σ(v3 ) v 3 The edges in black color represent edges of the invariant trees Construction of a (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) double edge split of G k 1, {v 1, v 2 }, {σ(v 1 ), σ(v 2 )} E ( T (k 1) ) 0, v3 V ( T (k 1) ) 0 and σ(v 3 ) = v 3 The edges in black color represent edges of the invariant trees Construction of a (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) double edge split of G k 1, {v 1, v 2 } E ( T (k 1) ) 1 and {σ(v1 ), σ(v 2 )} E ( T (k 1) ) 2 The edges in black color represent edges of the invariant tree Construction of a (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) X-replacement of G k 1, {v 1, v 2 } E ( T (k 1) ) 1 and {v3, v 4 } E ( T (k 1) ) xxii

22 541 Construction of a (C s, Φ) 3Tree2 partition of G in the case where G is a (C s, Φ k 1 ) X-replacement of G k 1 and {v 1, v 2 }, {v 3, v 4 } E ( T (k 1) ) 0 The edges in black color represent edges of the invariant tree The frame (G, p, q) in Case 1 of the proof of Lemma The frame (G, p t, q t ) in the case where V 1 T 0 is not connected The frame (G, p t, q t ) in the case where V 1 T 2 is not connected The frame (G, ˆp t, ˆq t ) The frame (G, p, q) in Case 2 of the proof of Lemma Two frameworks whose underlying graphs satisfy the conditions of Case B23 in the proof of Lemma 543 with respect to the types Φ that are uniquely determined by the injective realizations Any symmetrized Henneberg s sequence for any of these two graphs needs to include a (C s, Φ i ) X-replacement A (C 3v, Φ) 3Tree2 partition of a graph (a) and a (C 3v, Φ) 3Tree2 partition of a graph (b), where Φ(C 3 ) = γ and Φ(s) = σ A (C s, Φ)-generic realization of the complete bipartite graph K 4, A (C 3, Φ) partial vertex addition of order 0 of a graph G (a), a (C 3, Φ) partial vertex addition of order 1 of a graph G (b), and a (C 3, Φ) addition of a graph G (c) 269 xxiii

23 551 Independent frameworks in R (G,C3,Φ) with j Φ(C3 ) = 1 These frameworks cannot be contained in an isostatic framework that has the same joints and also C 3 symmetry Infinitesimally rigid frameworks in R (G,C3,Φ) with j Φ(C3 ) = 1 These frameworks cannot contain an isostatic framework that has the same joints and also C 3 symmetry Fully (S, Φ)-symmetric infinitesimal motions of frameworks: (a) a fully (C s, Φ)-symmetric infinitesimal rigid motion of (K 3, p) R (K3,C s,φ); (b) a fully (C s, Φ)-symmetric infinitesimal flex of (K 3,3, p) R (K3,3,C s,φ); (c) a fully (C 3, Φ)-symmetric infinitesimal flex of (G tp, p) R (Gtp,C 3,Φ) Since each of the above frameworks is an injective realization, the type Φ is uniquely determined in each case A fully (C s, Φ)-symmetric infinitesimal flex of the independent framework (K 2,2, p) A fully (C 2v, Φ)-symmetric infinitesimal flex of a (C 2v, Φ)- generic realization of K 4, Illustration of the proof that the joints of (K 4,4, p) lie on a conic section A fully (C 2v, Ψ)-symmetric infinitesimal flex of a (C 2v, Ψ)- generic realization of K 4, Flexible octahedra: with point group C 2 (a); with point group C s (b); with point group C 2v (c) 297 xxiv

24 67 An isostatic octahedron in R (G,Cs,Φ d ) A fully (C s, Φ a )-symmetric infinitesimal flex of the framework (G, p) R (G,Cs,Φ a ) A fully (C 2v, Φ b )-symmetric infinitesimal flex of the framework (G, p) R (G,C2v,Φ b ) A fully (C s, Φ)-symmetric infinitesimal flex of a (C s, Φ)-generic realization of K 4, Isostatic pinned frameworks in the plane: (a) with point group C 4 ; (b) with point group C 4v dimensional body-bar frameworks modeling different types of Steward platforms : the non-symmetric body-bar framework in (a) is isostatic; the body-bar framework in (b) is flexible due to the presence of the 6-fold rotational symmetry, as predicted by the necessary counts derived in [36] dimensional body-hinge frameworks whose underlying multigraph is a hexagonal cycle: the non-symmetric body-hinge framework in (a) is isostatic; the body-hinge framework in (b) is flexible due to the half-turn symmetry, as predicted by the counts derived in [36] Illustration of symmetric coning : both the framework (G, p H ) in R d and the coned framework (G {v}, p ) in R d+1 have mirror symmetry 316 xxv

25 75 Frameworks with mirror symmetry which are not globally rigid in R 2 The framework (G, p) in (a) is also not symmetric globally rigid in the sense of problem (2), since the framework in (b) is another non-congruent realization of G in R 2 with the same edge lengths and the same mirror symmetry as (G, p); the framework in (c), however, is symmetric globally rigid within the set of all realizations of G in R 2 with the same mirror symmetry 320 xxvi

26 Chapter 1 Introduction 11 Background and motivation The study of rigidity and flexibility has a rich history in what are currently a number of areas of engineering and mathematics, but historically were connected in the work of many scientists who combined studies of engineering, physics, and mathematics Early work which is now recognized as rigidity theory included the conjecture of L Euler about the rigidity of closed surfaces (eg polyhedra with rigid faces) [23], the static and kinematic analysis of pin-jointed frameworks for engineering structures (from the early 19th century) [49], and the large literature on linkages (frameworks with non-trivial - non-congruent - motions) in the 19th century [52] This wider range of work produced both counting rules (such as JC Maxwell s rules for built structures and M Grübler s counts for linkages) and 1

27 geometric analyses (such as reciprocal diagrams and various other tools for analyzing the resolutions of forces in the structures, and analyzing possible motions) [49, 52, 61, 62, 83] By the end of the 19th century, there were detailed analyses of which frameworks are normally rigid (eg AL Cauchy s famous theorem on the uniqueness of convex triangulated polyhedra with fixed edge lengths [10, 22, 76]), as well as when certain frameworks that were normally rigid became flexible (eg R Bricard s work on flexible frameworks with the bars and joints of an octahedron [9]) The people working on engineering problems developed a number of practical methods for analyzing buildings (eg the Eiffel Tower) as well as numerical and geometric rules of thumb for their design and construction A whole body of work, summarized by the engineer/mathematician L Henneberg, also developed explicit inductive techniques for generating rigid structures [40] Another stream, from AF Möbius through J Plücker and F Klein to L Cremona, investigated the projective geometry of static equilibria and singular forms which lead to building failures, in fields with names such as graphical statics [21, 83] Pockets of work continued through the 20th century, including detailed explorations of the design and analysis of mechanical linkages (mechanisms) and some continuing geometric analyses of built structures such as built trusses However, much of the previous geometric and combinatorial theory was submerged in the numerical analysis permitted by the developing computers and the now standard designs One notable exception to this decline in attention to the theory of rigid- 2

28 ity was the ongoing mathematical work on rigidity of polyhedral surfaces of the Russian school of AD Alexandrov, NV Efimov, and AV Pogorelov, and their analysis of the unique realizability of convex metrics [1] In the past 40 years, there have been two major, parallel developments (A) A strong mathematical theory has flowered, refining old results and techniques, with the splitting of key questions, techniques, and welldeveloped results into combinatorial and geometric aspects These developments have brought in combinatorial methods from graph theory, matroid theory, and associated combinatorial algorithms The work also refined geometric conditions that were sufficient for a shift in the rigidity properties - projective conditions for the static/first-order kinematic theory, and Euclidean and affine geometry for the theory of finite motions The static/first-order kinematic theory is expressed in the linear algebra of the rigidity matrix, whose rank, row dependencies and column dependencies all play key roles in the theory A highlight of these mathematical developments has been the basically complete combinatorial theory of plane frameworks - most notably, the combinatorial characterization of rigid 2-dimensional generic frameworks given by G Laman in 1970 [46] Combined with strong results for certain classes of structures in dimensions d 3, the work clarified the difficulties of characterizing rigid generic frameworks in 3- and higher-dimensional space During the 1970s, remarkable results have also been found in the theory of flexible polyhedra: in 1975 H Gluck was able to refine Euler s famous rigidity conjecture from 1766 [30], and in 1977 R Connelly finally settled the 3

29 conjecture with his celebrated counterexample, the flexible Connelly sphere [11] (B) An expanding web of connections to problems in other fields where the mathematical theory of rigidity makes substantial contributions to clarifying and resolving central questions has arisen Starting with the connections with mechanical and civil engineering (human built structures), there are connections to the general theory of geometric constraints built into Computer Aided Design (CAD) and to computations for robotic motions Less obvious, but very real, are the connections to computer vision/recognition of geometric objects from geometric data In general, a sequence of problems in computational geometry have found rigidity-type results and methods contribute to their understanding and their solutions (as well as sometimes clarifying that the problems are hard ) In turn, computational geometry has contributed new problems and algorithmic insights to the general theory of rigidity One step removed are the connections to scaled down natural structures, such as granular materials which are sometimes modeled with packings of spheres, abstracted as bar and joint frameworks Over the last two decades, there has also been a strong interest in applying rigidity theory to rapid predictions of the rigid and flexible regions of large biomolecules, such as proteins Such predictions, even based on incomplete theories, are implemented on the web because of the importance of flexibility and rigidity to the function of biomolecules and the design of drugs to alter their functioning [82] Overall, both the elegant and expanding mathematical theory and the 4

30 growing network of applications have made rigidity theory a rich area for research, and a source of new questions and new insights into a wide array of pure and applied mathematical theories Symmetry is another central idea in geometry - and appears widely in both natural structures such as crystals and biomolecules, and in structures built by humans One source of such symmetry is the efficiency of forming the shape using multiple copies of a few key components Because the appearance of symmetry is wide-spread, sometimes ubiquitous, there have recently been a series of papers by engineers and chemists, which present criteria for rigidity and flexibility of symmetric frameworks The first of these breakthrough papers is due to the engineers R Kangwai and S Guest: in 2000 they observed that the rigidity matrix of a symmetric framework can be put into a block-diagonalized form using techniques from group representation theory [44] Using this result, some engineers and chemists were able to make some further interesting and useful observations concerning the rigidity of symmetric frameworks (see [25, 43], for example) Many of these observations, however, are incomplete from a mathematical point of view, since they are not presented with a mathematically precise formulation nor with a thorough mathematical foundation or proofs So while this work has resulted in some important heuristics for engineers and chemists to gain further insight into the rigidity properties of a symmetric framework, there has not been a rigorous mathematical investigation of how symmetry impacts the rigidity of frameworks While this thesis was being written, the mathematicians JC Owen and SC Power have also been working on grounding aspects of this theory [53] In Chapter 4 we will describe 5

31 the one key area of overlap between their work and this thesis Again, the focus on classes of frameworks with given symmetries also has both a combinatorial (graph automorphism) level and a geometric aspect (point group, spatial isometries) In our study of these connections, we will see some clearly combinatorial conditions about fixed vertices, edges, etc in the graph automorphisms which must show up in the geometry of the realization When does this necessary geometry of symmetry from the graph automorphisms and the geometry of reflections, rotations, and other isometries force the rank of the rigidity matrix to drop? When does the geometry of symmetry overlap with the geometry of the singular positions which are traditionally expressed in projective form? We will see that under many circumstances, the addition of symmetry does not change the rigidity predicted for more general, asymmetric realizations In other circumstances there are very simply stated added conditions The simplest example is the point group C 3 in dimension 2 (Z 3 as an abstract group) which describes 3-fold rotational symmetry We will see that the combinatorial condition, once there is a group of graph automorphisms associated with the point group C 3, is simply that no vertices are fixed by the automorphism corresponding to the 3-fold rotation (geometrically, no vertices are placed on the center of rotation) This is a necessary condition for any independent and rigid realization of the graph as a framework with C 3 symmetry (Chapter 4) It is also a sufficient condition for the most general C 3 -symmetric realizations to be isostatic (Chapter 5) The result is striking in its simplicity: to test a generic framework with C 3 symmetry for 6

32 isostaticity, we just need to check the number of fixed vertices, as well as the standard conditions for rigidity without symmetry More generally, the techniques in Chapter 4 work with counts of vertices and edges, and counts of vertices and edges fixed by various elements of the group So the necessary conditions will always be of this type: vertices or edges fixed by the automorphisms The examination of when these necessary conditions are also sufficient is the larger theme of Chapter 5, for an array of plane symmetry groups In fact, a collection of rigidity theory methods which can now be called classical are symmetrized in Chapter 5 to establish both necessary and sufficient conditions for realizations which are generic with the given symmetry to be isostatic For general frameworks, an undercount of constraints becomes a prediction of flexibility For symmetric frameworks, we will show that there is an extension that not only predicts finite motions, but predicts motions which preserve the symmetries throughout their path (Chapter 6) The interactions of combinatorics, geometry, and symmetry are rich It is no surprise that at almost every turn, we find not only fascinating and appealing results, but also possible extensions to explore as well as new questions and new generalizations which can be conjectured and anticipated We invite the interested reader to join us in the exploration of these landscapes of possibilities 12 Outline of thesis The thesis is organized as follows 7

33 In Chapter 2, we give a brief introduction to rigidity, its linearized versions infinitesimal and static rigidity, as well as generic (or combinatorial) rigidity We also introduce suitable mathematical definitions for the relevant terms relating to symmetric structures that are frequently used in the chemistry and engineering literature In particular, we give a detailed mathematical description of the Schoenflies notation for point groups in dimensions 2 and 3 We will be using this notation for all the examples throughout this thesis In Chapter 3, we introduce a natural classification of symmetric frameworks This classification is fundamental to all the results of this thesis We then define a symmetry-adapted notion of a generic framework with respect to this classification This symmetrized notion of generic has the two fundamental properties that almost all realizations in a given symmetry class are generic and all generic realizations in this class share the same infinitesimal rigidity properties This classification therefore not only lays the foundation for symmetrizing results in rigidity, infinitesimal rigidity, and static rigidity, but it also allows us to develop a symmetry-adapted version of generic rigidity theory in Chapter 5 In the last two sections of the third chapter, we carefully examine the difficulties that arise in applying techniques from group representation theory to the analysis of symmetric frameworks with non-injective configurations More precisely, in Section 33, we show that a framework with a non-injective configuration can belong to more than one symmetry class, and we examine how many distinct symmetry classes a given framework can possibly belong to In Section 34, we investigate under what conditions techniques from group representation theory can be applied to the frameworks in a given 8

34 symmetry class All the results of the third chapter can be found in the manuscript [55] which has been submitted for review Chapter 4 concerns the application of techniques from group representation theory to the rigidity analysis of symmetric frameworks In Section 41, we first give a complete self-contained mathematical proof that the rigidity matrix of a symmetric framework can be block-diagonalized as described by R Kangwai and S Guest in [44] In Section 42, we use this result to give a detailed proof for the symmetry-extended version of Maxwell s rule given by P Fowler and S Guest in [25] This rule provides further necessary conditions (in addition to Maxwell s original condition from 1864 [49]) for a symmetric framework (G, p) to be isostatic While the rule in [25] is only applicable to 2- or 3-dimensional frameworks with injective configurations, we establish a more general result in this thesis, namely a rule that can be applied to both injective and non-injective realizations in all dimensions The results of Sections 41 and 42 will constitute the main part of [56] An alternate proof for the rule given in [25], as well as various generalizations of this rule to other types of geometric constraint systems, is given by JC Owen and SC Power in [53] In Section 43, we show that the symmetry-extended version of Maxwell s rule can be used to prove that a symmetric isostatic framework must obey some very simply stated restrictions on the number of joints and bars that are fixed by various symmetry operations of the framework In particular, these restrictions imply that the symmetries of a 2-dimensional isostatic framework must belong to one of only six possible point groups For 3-dimensional iso- 9

35 static frameworks, all point groups are possible, although restrictions on the placement of structural components still apply The main part of Section 43 is a mathematically explicit derivation of the results presented without proof in [15] This paper is joint work with R Connelly, P Fowler, S Guest, and W Whiteley Finally, in Section 44, we use the results of the previous sections to also establish necessary conditions for a symmetric framework to be independent or infinitesimally rigid In Chapter 5, we present symmetric versions of some famous results in generic rigidity theory Given a graph G, Laman s Theorem says that Maxwell s condition in 2D, ie, E(G) = 2 V (G) 3, together with the counts E(H) 2 V (H) 3 for all non-trivial subgraphs H of G, are necessary and sufficient for all generic 2-dimensional realizations of G to be isostatic There are well known difficulties in extending this result to higher dimensions (see [32, 33, 46], for example) Using the symmetry-adapted notion of generic introduced in Chapter 3, we establish symmetric versions of Laman s Theorem for three of the five non-trivial symmetry groups in dimension 2 that allow isostatic frameworks, namely for the groups C 2 and C 3 of order 2 and 3 generated by a half-turn and a 3-fold rotation, respectively, and for the group C s of order 2 generated by a reflection More precisely, we show that for each of these groups, the conditions derived from the symmetryextended version of Maxwell s rule, together with the Laman conditions, are necessary and sufficient for realizations of G that are generic within the given symmetry class to be isostatic These results were conjectured in [15] Henneberg s Theorem and Crapo s Tree Covering Theorem are also fa- 10

36 mous combinatorial results that provide characterizations of generically 2- isostatic graphs [20, 40, 33, 67, 68] We show that for each of the symmetry groups C 2, C 3 and C s, there exist symmetric versions of these results as well The other two non-trivial symmetry groups in dimension 2 that allow isostatic frameworks are the dihedral groups of order 4 and 6 For these groups, we offer some analogous conjectures To prove these conjectures with techniques similar to the ones used for the results above, one has to consider an unreasonably large number of cases In the final section of Chapter 5, we briefly discuss symmetric-generically isostatic graphs in dimension 3 The key results of the fifth chapter will be summarized in [57] In Chapter 6, we study finite flexes of symmetric frameworks, ie, flexes that move the joints of a given framework on differentiable displacement paths while holding the lengths of all bars fixed and changing the distance between two unconnected joints We prove that if a framework (G, p) is generic within a given symmetry class and there exists a fully-symmetric infinitesimal flex of (G, p) (ie, the velocity vectors of the infinitesimal flex remain unaltered under all symmetry operations of (G, p)), then (G, p) also possesses a symmetry-preserving finite flex, ie, a flex which displaces the joints of (G, p) in such a way that all the resulting frameworks have the same symmetry as (G, p) (or possibly higher symmetry) This and other related results are obtained by symmetrizing techniques described by L Asimov and B Roth in [3] and by using the fact that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form as shown in Chapter 4 As corollaries of these results, one obtains the results stated (but 11

37 not rigorously proven) in [35] and Proposition 1 in [43] The finite flexes that can be detected with these symmetry-based methods can in general not be found with the analogous non-symmetric methods The work of Chapter 6 will also be presented in [58] Finally, in Chapter 7, we outline how the methods developed in this thesis can be extended to analyze the rigidity and flexibility properties of various other types of symmetric structures Several additional promising directions for future work are also presented in this final chapter 12

38 Chapter 2 Definitions and preliminaries 21 Graph theory terminology We begin by establishing the graph theory vocabulary and notation we will be using throughout this thesis Definition 211 A graph G is a finite nonempty set of objects called vertices together with a (possibly empty) set of unordered pairs of distinct vertices of G called edges The vertex set of G is denoted by V (G) and the edge set of G is denoted by E(G) Definition 212 Two vertices u v of a graph G are adjacent if {u, v} E(G), and independent otherwise A set S of vertices of G is independent if every two vertices of S are independent Definition 213 Let G be a graph The neighborhood N G (v) of a vertex v V (G) is the set of all vertices that are adjacent to v and the elements of 13

39 N G (v) are called the neighbors of v Definition 214 Let G be a graph and e = {u, v} be an edge of G Then we say that u and e are incident, as are v and e The valence val G (v) of a vertex v V (G) is the number of edges of G that are incident with v Equivalently, val G (v) = N G (v) Definition 215 A graph is called complete if every two of its vertices are adjacent We write K n for the complete graph on n vertices A graph G is called bipartite if the vertex set V (G) can be partitioned into two sets X and Y (called partite sets) such that for every edge {x, y} E(G) we have x X and y Y A bipartite graph G with partite sets X and Y is complete if {x, y} E(G) for all x X and y Y We write K m,n for the complete bipartite graph whose partite sets have cardinality m and n Definition 216 A graph H is a subgraph of a graph G if V (H) V (G) and E(H) E(G), in which case we write H G A subgraph H of a graph G is called spanning if V (H) = V (G) The simplest type of subgraph of a graph G is that obtained by deleting a vertex or an edge from G Let v be a vertex and e be an edge of G Then we write G {v} for the subgraph of G that has V (G) \ {v} as its vertex set and whose edges are those of G that are not incident with v Similarly, we write G {e} for the subgraph of G that has V (G) as its vertex set and E(G) \ {e} as its edge set The deletion of a set of vertices or a set of edges from G is defined and denoted analogously If u and v are independent vertices of G, then we write G + { {u, v} } for 14

40 the graph that has V (G) as its vertex set and E(G) { {u, v} } as its edge set The addition of a set of edges is again defined and denoted analogously Definition 217 Let G be a graph and U be a nonempty subset of V (G) Then the subgraph U of G induced by U is the graph having vertex set U and whose edges are those of G that are incident with two elements of U Definition 218 Let G 1 and G 2 be two graphs The intersection G = G 1 G 2 is the graph with V (G) = V (G 1 ) V (G 2 ) and E(G) = E(G 1 ) E(G 2 ) Similarly, the union G = G 1 G 2 is the graph with V (G) = V (G 1 ) V (G 2 ) and E(G) = E(G 1 ) E(G 2 ) Definition 219 An automorphism of a graph G is a permutation α of V (G) such that {u, v} E(G) if and only if {α(u), α(v)} E(G) The automorphisms of a graph G form a group under composition which is denoted by Aut(G) Definition 2110 Let H be a subgraph of a graph G and α Aut(G) We define α(h) to be the subgraph of G that has α ( V (H) ) as its vertex set and α ( E(H) ) as its edge set, where {u, v} α ( E(H) ) if and only if α 1 ({u, v}) = {α 1 (u), α 1 (v)} E(H) We say that H is invariant under α if α ( V (H) ) = V (H) and α ( E(H) ) = E(H), in which case we write α(h) = H Example 211 The graph G in Figure 21 (a) has the automorphism α = (v 1 v 2 v 3 )(v 4 v 5 v 6 ) The subgraph H 1 of G is invariant under α, but the subgraph H 2 of G is not, because α ( E(H 2 ) ) E(H 2 ) 15

41 G: v 1 v 4 v 3 H 1 : v 1 v 3 H 2 : v 1 v 3 v 5 v v6 2 v 2 v 2 (a) (b) (c) Figure 21: An invariant (b) and a non-invariant subgraph (c) of the graph G under α = (v 1 v 2 v 3 )(v 4 v 5 v 6 ) Aut(G) Definition 2111 Let u and v be two (not necessarily distinct) vertices of a graph G A u-v path in G is a finite alternating sequence u = u 0, e 1, u 1, e 2,, u k 1, e k, u k = v of vertices and edges of G in which no vertex is repeated and e i = {u i 1, u i } for i = 1, 2,, k A u-v path is called a cycle if k 3 and u = v Let a u-v path P in G be given by u = u 0, e 1, u 1, e 2,, u k 1, e k, u k = v and let α Aut(G) Then we denote α(p ) to be the α(u)-α(v) path α(u) = α(u 0 ), α(e 1 ), α(u 1 ), α(e 2 ),, α(u k 1 ), α(e k ), α(u k ) = α(v) in G A vertex u is said to be connected to a vertex v in G if there exists a u v path in G A graph G is connected if every two vertices of G are connected A graph with no cycles is called a forest and a connected forest is called a tree A connected subgraph H of a graph G is a component of G if H = H whenever H is a connected subgraph of G containing H 16

42 22 Introduction to rigidity theory We now give a brief introduction to rigidity, its linearized versions infinitesimal and static rigidity, as well as generic rigidity, as we shall symmetrize results from each of these theories The definitions and results listed in this introduction are widely used in the rigidity theory literature so that we will omit the proofs and leave more detailed explanations and illustrations to be found in the references provided 221 Rigidity Definition 221 [32, 33, 81, 83] A framework (in R d ) is a pair (G, p), where G is a graph and p : V (G) R d is a map with the property that p(u) p(v) for all {u, v} E(G) We also say that (G, p) is a d-dimensional realization of the underlying graph G Given the vertex set V (G) = {v 1,, v n } of a graph G and a map p : V (G) R d, it is often useful to identify p with a vector in R dn by using the order on V (G) In this case we also refer to p as a configuration of n points in R d Throughout this thesis we will simplify our notation by not differentiating between an abstract vector and its coordinate vector relative to the canonical basis Definition 222 Let (G, p) be a framework in R d A joint of (G, p) is an ordered pair ( v, p(v) ), where v V (G) A bar of (G, p) is an unordered pair {( u, p(u) ), ( v, p(v) )} of joints of (G, p), where {u, v} E(G) We define 17

43 p(u) p(v) to be the length of the bar {( u, p(u) ), ( v, p(v) )}, where p(u) p(v) is defined by the canonical inner product on R d Note that we allow the map p of a framework (G, p) to be non-injective, that is, two distinct joints ( u, p(u) ) and ( v, p(v) ) of (G, p) may be located at the same point p(u) = p(v) in R d, provided that u and v are independent vertices of G However, if {u, v} E(G), then p(u) p(v), and hence every bar {( u, p(u) ), ( v, p(v) )} of (G, p) has a strictly positive length Definition 223 [32] Let (G, p) be a framework in R d with V (G) = {v 1, v 2,, v n } A motion of (G, p) is an indexed family of functions P i : [0, 1] R d, i = 1, 2,, n, so that (i) P i (0) = p(v i ) for all i; (ii) P i (t) is differentiable on [0, 1] for all i; (iii) P i (t) P j (t) = p(v i ) p(v j ) for all t [0, 1] and {v i, v j } E(G) A motion of a framework (G, p) displaces the joints of (G, p) on differentiable displacement paths while preserving the lengths of all bars of (G, p) Every framework has some trivial motions, namely those that correspond to rigid motions of space (ie, translations, rotations and their combinations) Definition 224 A motion {P i } of a framework (G, p) with V (G) = {v 1, v 2,, v n } is called a rigid motion if it preserves the distances between every pair of joints of (G, p), that is, if P i (t) P j (t) = p(v i ) p(v j ) for all t [0, 1] and all 1 i < j n 18

44 {P i } is called a flex if the distance between at least one pair of joints of (G, p) is changed by {P i }, that is, if P i (t) P j (t) = p(v i ) p(v j ) for all t (0, 1] and some {v i, v j } / E(G) Definition 225 A framework (G, p) is called rigid if every motion of (G, p) is a rigid motion Otherwise (G, p) is called flexible (a) (b) (c) (d) Figure 22: A rigid (a) and a flexible (b) framework in the plane The flex shown in (c) takes the framework in (b) to the framework in (d) Some alternate definitions of a rigid framework are common in the literature [3, 83] all of which are equivalent to Definition 225 We will introduce some of these definitions in Chapter 6, where we examine the motions of symmetric frameworks 222 Infinitesimal rigidity It is in general very difficult to determine whether a given framework is rigid or not since it requires solving a system of quadratic equations It is therefore common to linearize this problem by differentiating the equations in Definition 223 (iii) This gives rise to Definition 226 [32, 33, 81, 83] Let (G, p) be a framework in R d with V (G) = {v 1, v 2,, v n } An infinitesimal motion of (G, p) is a function 19

45 u : V (G) R d such that ( p(vi ) p(v j ) ) (u(v i ) u(v j ) ) = 0 for all {v i, v j } E(G) (21) An infinitesimal motion of a framework (G, p) is a set of displacement vectors u(v i ), one at each joint, that neither stretch nor compress the bars of (G, p) at first order More precisely, condition (21) says that for every edge {v i, v j } E(G), the projections of u(v i ) and u(v j ) onto the line through p(v i ) and p(v j ) have the same direction and the same length (see also Figure 23) Definition 227 An infinitesimal motion u of a framework (G, p) with V (G) = {v 1, v 2,, v n } is called an infinitesimal rigid motion if there exists a rigid motion {P i } of (G, p) such that for i = 1, 2,, n, the vector u(v i ) is the derivative (at t = 0) of P i Otherwise, u is called an infinitesimal flex of (G, p) Remark 221 Let G be a graph with V (G) = {v 1, v 2,, v n } and let u be an infinitesimal motion of a d-dimensional realization (G, p) of G If ( p(vi ) p(v j ) ) (u(v i ) u(v j ) ) 0 for some {v i, v j } / E(G), then u is an infinitesimal flex of (G, p) If the points p(v 1 ),, p(v n ) span all of R d (in an affine sense), then the converse also holds, ie, in this case, u is an infinitesimal flex of (G, p) if and only if ( p(v i ) p(v j ) ) (u(v i ) u(v j ) ) 0 for some {v i, v j } / E(G) or equivalently, u is an infinitesimal rigid motion of (G, p) if and only if ( p(v i ) p(v j ) ) (u(v i ) u(v j ) ) = 0 for all 1 i < j n [32, 33, 81] From now on, when we say that a set of points spans a space, then this will always be in the affine sense 20

46 u 1 p 1 p 2 u 2 (a) u 3 u 1 = 0 u 2 = 0 p 1 p 3 p2 (b) u 1 p 1 p u 4 4 u 3 p 6 u p3 6 u 5 p 2 p 5 u 2 (c) Figure 23: The arrows indicate the non-zero displacement vectors of an infinitesimal rigid motion (a) and infinitesimal flexes (b, c) of frameworks in R 2 Definition 228 [32, 33, 81, 83] A framework (G, p) is infinitesimally rigid if every infinitesimal motion of (G, p) is an infinitesimal rigid motion Otherwise (G, p) is said to be infinitesimally flexible The following theorem gives the main connection between rigidity and infinitesimal rigidity A proof of this result can be found in [3], [17] or [30], for example Theorem 221 If a framework (G, p) is infinitesimally rigid, then (G, p) is rigid Under certain conditions, rigidity and infinitesimal rigidity are equivalent We will give the relevant results in the end of Section 225 after we have established the necessary definitions For a framework (G, p) whose underlying graph G has a vertex set that is indexed from 1 to n, say V (G) = {v 1, v 2,, v n }, we will frequently denote 21

47 p(v i ) by p i for i = 1, 2,, n Similarly, for an infinitesimal motion u of (G, p), we will frequently denote u(v i ) by u i for all i The k th component of a vector x is denoted by (x) k The equations stated in Definition 226 form a system of linear equations whose corresponding matrix is called the rigidity matrix This matrix is fundamental in the study of both infinitesimal and static rigidity Definition 229 [32, 33, 81, 83] Let G be a graph with V (G) = {v 1, v 2,, v n } and let p : V (G) R d The rigidity matrix of (G, p) is the E(G) dn matrix v 1 v i v j v n R(G, p) = 0 0 p i p j 0 0 p j p i 0 0 edge {v i, v j }, that is, for each edge {v i, v j } E(G), R(G, p) has the row with (p i p j ) 1,, (p i p j ) d in the columns d(i 1)+1,, di, (p j p i ) 1,, (p j p i ) d in the columns d(j 1) + 1,, dj, and 0 elsewhere Remark 222 The rigidity matrix is defined for arbitrary pairs (G, p), where G is a graph and p : V (G) R d is a map If (G, p) is not a framework, then there exists a pair of adjacent vertices of G that are mapped to the same point in R d under p and every such edge of G gives rise to a zero-row in R(G, p) If we identify an infinitesimal motion of a d-dimensional framework (G, p) with a column vector in R d V (G) (by using the order on V (G)), then the 22

48 kernel of R(G, p) is the space of infinitesimal motions of (G, p) It is well known that the infinitesimal rigid motions arising from d translations and ( d ) 2 rotations of R d form a basis for the space of infinitesimal rigid motions of (G, p), provided that the points p 1,, p n span an affine subspace of R d of dimension at least d 1 [33, 81] Thus, for such a framework (G, p), we have nullity ( R(G, p) ) d + ( ) ( d 2 = d+1 ) 2 and (G, p) is infinitesimally rigid if and only if nullity ( R(G, p) ) = ( ) ( ) d+1 2 or equivalently, rank R(G, p) = d V (G) ( ) d+1 2 Theorem 222 [3, 30] A framework (G, p) in R d is infinitesimally rigid if and only if either rank ( R(G, p) ) = d V (G) ( ) d+1 2 or G is a complete graph K n and the points p(v), v V (G), are affinely independent Remark 223 Let 1 m d and let (G, p) be a framework in R d If (G, p) has at least m + 1 joints and the points p(v), v V (G), span an affine subspace of R d of dimension less than m, then (G, p) is infinitesimally flexible (recall Figure 23 (b)) In particular, if (G, p) is infinitesimally rigid and V (G) d, then the points p(v), v V (G), span an affine subspace of R d of dimension at least d Static rigidity We now also give a brief introduction to the static approach to rigidity The intuitive test for static rigidity of a framework (G, p) is to apply an external load to (G, p) (ie, a set of forces, one to each joint) and investigate whether there exists a set of tensions and compressions in the bars of (G, p) that reach an equilibrium with this load at the joints (see also Figure 24) 23

49 Of course only loads which do not correspond to a translation or rotation of space can possibly be resolved in this way Definition 2210 [21, 68, 76, 81] Let (G, p) be a framework in R d with V (G) = {v 1, v 2,, v n } A load on (G, p) is a function l : V (G) R d, where for i = 1, 2,, n, the vector l(v i ) represents a force applied to the joint ( vi, p i ) of (G, p) A load l on (G, p) is called an equilibrium load if l satisfies (i) n i=1 l i = 0; (ii) n i=1 ( (li ) j (p i ) k (l i ) k (p i ) j ) = 0 for all 1 j < k d, where l i denotes the vector l(v i ) for each i The physical intuition for conditions (i) and (ii) in Definition 2210 is the following: condition (i) rules out loads that would produce a translation of (G, p) and (ii) says that there is no net rotational twist of (G, p) Definition 2211 [21, 68, 76, 81] Let l be an equilibrium load on a framework (G, p) in R d with V (G) = {v 1, v 2,, v n } A resolution of l by (G, p) is a function ω : E(G) R such that at each joint ( ) v i, p i of (G, p) we have ω ij (p i p j ) + l i = 0, j with {v i,v j } E(G) where ω ij denotes ω({v i, v j }) for all {v i, v j } E(G) The scalars ω ij represent tensions (ω ij < 0) and compressions (ω ij > 0) in the bars of (G, p), so that the bar forces reach an equilibrium with l i at each joint ( ) v i, p i 24

50 (a) (b) (c) (d) (e) Figure 24: (a), (b) The arrows indicate a tension (a) and a compression (b) in a bar (c) An equilibrium load on a non-degenerate triangle This load can be resolved by the triangle as shown in (d) (e) An unresolvable equilibrium load on a degenerate triangle: for any joint of this framework, tensions or compressions in the bars cannot reach an equilibrium with the load vector at this joint Definition 2212 [21, 68, 76, 81] A framework (G, p) is statically rigid if every equilibrium load on (G, p) has a resolution by (G, p) Note that if we identify l and ω with a column vector in R dn and R E(G), respectively, then (after changing the sign of l) the equations in Definition 2211 can be written in a compact form in terms of the rigidity matrix R(G, p) as R(G, p) T ω = l Let (v h, p h ) and (v k, p k ) be two joints of (G, p) Then it is easy to see that the column vector F hk, where (F hk ) T = (0,, 0, p h p k, 0,, 0, p k p h, 0,, 0), is an equilibrium load on (G, p) Further, if {v i, v j } E(G), then (F ij ) T is the row vector of R(G, p) that corresponds to {v i, v j } and F ij is clearly 25

51 resolved by the bar {(v i, p i ), (v j, p j )} of (G, p) Note that if (G, p) is statically rigid, then F hk has a resolution by (G, p) for every pair (v h, p h ), (v k, p k ) of joints of (G, p) (even if {v h, v k } / E(G)) If the points p 1,, p n span all of R d, then the converse also holds, since in this case, the vectors F hk, 1 h < k n, generate the entire space of equilibrium loads on (G, p) (see [76]) This space is a subspace of R dn of dimension dn ( ) d+1 2 (defined by the equations in Definition 2210) Thus, if we want to test such a framework (G, p) for static rigidity, we need to investigate whether the rows of R(G, p) generate a space of dimension dn ( ) d+1 2, that is, the entire space of equilibrium loads on (G, p) In other words, we need to investigate whether rank ( R(G, p) ) ( ) d + 1 T = dn 2 So, the essential information for both infinitesimal and static rigidity of a framework (G, p) is comprised by the rigidity matrix R(G, p) While in infinitesimal rigidity, we investigate the column space and column rank of R(G, p), in static rigidity, we investigate the row space and row rank of R(G, p) In the light of these remarks, the following fundamental facts do not come as a surprise Theorem 223 [54] The load F hk on a framework (G, p) has no resolution by (G, p) if and only if there exists an infinitesimal motion u of (G, p) with (p h p k ) (u h u k ) 0 Theorem 224 [40, 54, 73] A framework (G, p) is infinitesimally rigid if and only if (G, p) is statically rigid 26

52 Theorem 224 allows us to use the terms infinitesimally rigid and statically rigid interchangeably Definition 2213 [21, 76, 81] Given a framework (G, p), a function ω : E(G) R is called a stress of (G, p) Equivalently, a stress of (G, p) is a resolution of a load on (G, p) A resolution of the zero-load is called a selfstress of (G, p) So, if we identify a stress of (G, p) with a vector in R E(G), then a vector ω R E(G) is a self-stress of (G, p) if R(G, p) T ω = 0 The framework (G, p) is said to be independent if the rigidity matrix R(G, p) has linearly independent rows Equivalently, (G, p) is independent if (G, p) has no non-zero self-stress Otherwise, (G, p) is said to be dependent Definition 2214 A framework (G, p) is isostatic if it is infinitesimally (or statically) rigid and independent The rows of the rigidity matrix of an isostatic framework (G, p) form a basis for the space of equilibrium loads on (G, p), provided that the points p(v), v V (G), span all of R d Theorem 225 [33, 83] For a d-dimensional realization (G, p) of a graph G with V (G) d, the following are equivalent: (i) (G, p) is isostatic; (ii) (G, p) is infinitesimally rigid and E(G) = d V (G) ( ) d+1 2 ; (iii) (G, p) is independent and E(G) = d V (G) ( ) d+1 2 ; 27

53 (iv) (G, p) is minimal infinitesimally rigid, ie, (G, p) is infinitesimally rigid and the removal of any bar results in a framework that is not infinitesimally rigid 224 Generic rigidity Generic rigidity is concerned with the infinitesimal (or equivalently, static) rigidity of almost all geometric realizations of a given graph The following standard definition of generic specifies what we mean by almost all Definition 2215 Let K n be the complete graph on n vertices with V (K n ) = {v 1, v 2,, v n } For each i = 1, 2,, n, we introduce a d-tuple p i = ( ) (p i) 1,, (p i) d of variables and let R(n, d) = 0 0 p i p j 0 0 p j p i 0 0 be the matrix that is obtained from the rigidity matrix R(K n, p) of a d- dimensional realization (K n, p) by replacing each (p i ) j R with the variable (p i) j We call R(n, d) the d-dimensional indeterminate rigidity matrix of K n Definition 2216 [32, 33] Let V = {v 1, v 2,, v n } and p : V R d be a map Further, let K n be the complete graph with V (K n ) = V We say that p is generic if the determinant of any submatrix of R(K n, p) is zero only if the determinant of the corresponding submatrix of R(n, d) is (identically) zero A framework (G, p) is said to be generic if p is a generic map 28

54 There are two fundamental facts regarding this definition of generic First, the set of all non-generic maps p of a finite set V = {v 1, v 2,, v n } to R d is a closed set of measure zero [33] To see this, identify p with a vector in R dn and observe that the determinant of every submatrix of R(K n, p) is a polynomial in the variables (p i) j If such a polynomial is not identically zero, then, by general algebraic geometry, it is non-zero for an open dense set of p R dn Since R(K n, p) has only finitely many minors, the set of generic p R dn is still an open dense subset of R dn Secondly, the infinitesimal rigidity properties are the same for all generic realizations of a graph G, as the following result shows: Theorem 226 [32, 33, 83] For a graph G and a fixed dimension d, the following are equivalent: (i) (G, p) is infinitesimally rigid (independent, isostatic) for some map p : V (G) R d ; (ii) every d-dimensional generic realization of G is infinitesimally rigid (independent, isostatic) It follows that for generic frameworks, infinitesimal (and static) rigidity is purely combinatorial, and hence a property of the underlying graph This gives rise to the following definition of infinitesimal rigidity for graphs: Definition 2217 A graph G is called generically rigid (independent, isostatic) in dimension d or generically d-rigid (d-independent, d-isostatic) if d-dimensional generic realizations of G are infinitesimally rigid (independent, isostatic) 29

55 An easy but often useful observation concerning generic frameworks is that if a framework (G, p) in R d is generic, then the joints of (G, p) are in general position, that is, for 1 m d, no m + 1 joints of (G, p) lie in an m 1-dimensional affine subspace of R d [33] In Chapter 3, we introduce a natural classification of symmetric frameworks and introduce a symmetry-adapted notion of generic with respect to this classification Remark 224 There are some other notions of a generic realization of a graph G that are commonly used in rigidity theory (see, for example, [14, 48, 80]) One such notion of generic is obtained by replacing the matrices R(K n, p) and R(n, d) in Definition 2216 with the rigidity matrix R(G, p) and the matrix R (G) (n, d) which is obtained from R(n, d) by deleting those rows that do not correspond to edges of G, respectively [80] We shall refer to this definition of generic as G-generic Clearly, if a map p : V (G) R d is generic, then p is also G-generic Moreover, like the set of all generic realizations of a graph G, the set of all G-generic realizations of G is also a dense open subset of R dn However, the fact that the property of being generic (in the sense of Definition 2216) is invariant under addition or deletion of edges in G makes this definition a more convenient one for our purposes than the definition of G-generic A configuration p R dn is frequently also defined to be generic if the coordinates of p are algebraically independent over Z, ie, if there does not exist a polynomial h(x 1,, x dn ) with integer coefficients such that h ( (p 1 ) 1,, (p n ) d ) = 0 [14, 48] We refer to this definition of generic as A-generic 30

56 The set of all A-generic realizations of G is a dense, but not an open subset of R dn The definition of A-generic is therefore not a very suitable definition of generic for our purposes The relationship between all of these different types of generic realizations of a given graph G is illustrated in Figure 25 by means of a Venn diagram regular points of G G-generic configurations generic configurations A-generic configurations Figure 25: A Venn diagram showing the relationship between sets of various types of generic configurations and the set of regular points of a graph G (see Definition 2222 in the end of Section 225) 225 Basic rigidity results In this section, we give a number of important results (as well as some definitions) in rigidity theory that we will later extend to frameworks that are realized with certain symmetries In 1864, J C Maxwell gave a necessary, but not sufficient condition for a 2- or 3-dimensional framework to be isostatic [49] The following theorem is the d-dimensional version of this condition 31

57 Theorem 227 (Maxwell s rule, 1864) Let (G, p) be a d-dimensional realization of a graph G with V (G) d If (G, p) is isostatic then ( ) d + 1 E(G) = d V (G) 2 Let (G, p) be a framework in R d with the property that the points p(v), v V (G), span an affine subspace of R d of dimension at least d 1, so that the space of infinitesimal rigid motions of (G, p) has dimension ( ) d+1 2 Also, let the vector space of infinitesimal motions of (G, p) be denoted by I(p) and the vector space of self-stresses of (G, p) be denoted by Ω(p) Then the equation in Maxwell s rule can be written in its extended form as E(G) d V (G) = dim ( Ω(p) ) dim ( I(p) ) So, if E(G) ( d V (G) ( )) d+1 2 = k > 0, then we can conclude that (G, p) has at least k linearly independent self-stresses and if E(G) ( d V (G) )) = k < 0, then (G, p) has at least k linearly independent infinitesimal ( d+1 2 flexes (see also [33]) Note that Maxwell s rule is an immediate consequence of Theorem 225 which gives both necessary and sufficient conditions for a framework to be isostatic However, the advantage of Maxwell s rule is that it provides a purely combinatorial necessary condition for (G, p) to be isostatic, and this condition can easily be verified since it only requires a simple count of the edges and vertices of G In Chapter 4, we will use techniques from group representation theory to extend Maxwell s rule to frameworks that possess non-trivial symmetries In addition to the condition in Theorem 227, there exist further necessary conditions for a graph G to be generically d-isostatic The following 32

58 result includes necessary conditions for all non-trivial subgraphs of G Theorem 228 [32, 33] Let G be a graph that is generically d-isostatic with V (G) d Then (i) E(G) = d V (G) ( ) d+1 2 ; (ii) E(H) d V (H) ( ) d+1 2 for all H G with V (H) d Clearly, generic rigidity is a combinatorial concept for all dimensions However, a combinatorial characterization has only been found for dimensions 1 and 2: for d {1, 2}, the counts stated in the previous theorem are also sufficient for a graph to be generically d-isostatic For dimension 1, this says that a graph G is generically 1-isostatic if and only if G is a tree, and G is generically 1-rigid if and only if G is connected [33] For dimension 2, the sufficiency of the counts in Theorem 228 for a graph to be generically 2-isostatic was proven by G Laman in 1970 Theorem 229 (Laman, 1970) [46] A graph G with V (G) 2 is generically 2-isostatic if and only if (i) E(G) = 2 V (G) 3; (ii) E(H) 2 V (H) 3 for all H G with V (H) 2 Various proofs of Laman s Theorem can be found in [32], [33], [67], and [79], for example In Chapter 5, we establish symmetric versions of Laman s Theorem for non-trivial symmetry groups 33

59 Throughout this thesis, we will refer to the conditions (i) and (ii) in Theorem 229 as the Laman conditions For dimension d 3, the counts in Theorem 228 are not sufficient for generic d-rigidity The most famous example to demonstrate this for dimension 3 is the so-called double banana (see Figure 26) [32, 33, 68] Figure 26: The double banana satisfies the counts in Theorem 228 for d = 3, but it is not generically 3-isostatic There are some inductive construction techniques that preserve the generic rigidity properties of a graph These construction techniques can be used to prove theorems such as Laman s Theorem, to analyze graphs for generic rigidity, and to characterize generically 1-isostatic and 2-isostatic graphs For all dimensions d, they provide a tool to generate classes of generically d-isostatic graphs Definition 2218 [68, 81] Let G be a graph, U V (G) with U = d and v / V (G) Then the graph Ĝ with V (Ĝ) = V (G) {v} and E(Ĝ) = E(G) { {v, u} u U } is called a vertex d-addition (by v) of G Theorem 2210 (Vertex Addition Theorem) [32, 33, 68, 81] A vertex d-addition of a generically d-isostatic graph is generically d-isostatic Conversely, deleting a vertex of valence d from a generically d-isostatic graph results in a generically d-isostatic graph 34

60 Definition 2219 [68, 81] Let G be a graph, U V (G) with U = d+1 and {u 1, u 2 } E(G) for some u 1, u 2 U Further, let v / V (G) Then the graph Ĝ with V (Ĝ) = V (G) {v} and E(Ĝ) = ( E(G)\ { {u 1, u 2 } }) { {v, u} u U } is called an edge d-split (on u 1, u 2 ; v) of G Theorem 2211 (Edge Split Theorem) [32, 33, 68, 81] An edge d-split of a generically d-isostatic graph is generically d-isostatic Conversely, if one deletes a vertex v of valence d + 1 from a generically d-isostatic graph, then one may add an edge between one of the pairs of vertices adjacent to v so that the resulting graph is generically d-isostatic (a) for some pair (b) Figure 27: Illustrations of the Vertex Addition Theorem (a) and the Edge Split Theorem (b) in dimension 2 In 1911, L Henneberg gave the following characterization of generically 2-isostatic graphs Theorem 2212 (Henneberg, 1911) [40] A graph is generically 2- isostatic if and only if it may be constructed from a single edge by a sequence of vertex 2-additions and edge 2-splits For a proof of Henneberg s Theorem, see [33] or [68], for example 35

61 There exist a few additional inductive construction techniques that are frequently used in rigidity theory One of these techniques, the X- replacement, will play a pivotal role in proving the symmetrized version of Laman s Theorem for symmetry groups consisting of the identity and a single reflection Definition 2220 [68, 81] Let G be a graph, u 1, u 2, u 3, u 4 be four distinct vertices of G with {u 1, u 2 }, {u 3, u 4 } E(G), and let v / V (G) Then the graph Ĝ with V (Ĝ) = V (G) {v} and E(Ĝ) = ( E(G)\ { {u 1, u 2 }, {u 3, u 4 } }) { {v, ui } i {1, 2, 3, 4} } is called an X-replacement (by v) of G Figure 28: Illustration of an X-replacement of a graph G Theorem 2213 (X-Replacement Theorem) [68, 81] An X-replacement of a generically 2-isostatic graph is generically 2-isostatic The reverse operation of an X-replacement performed on a generically 2-isostatic graph does in general not result in a generically 2-isostatic graph For more details and some additional inductive construction techniques, we refer the reader to [68] Another way of characterizing generically 2-isostatic graphs is due to H Crapo and uses partitions of a graph into edge disjoint trees 36

62 Definition 2221 [20, 47, 67] A 3Tree2 partition of a graph G is a partition of E(G) into the edge sets of three edge disjoint trees T 0, T 1, T 2 such that each vertex of G belongs to exactly two of the trees A 3Tree2 partition is called proper if no non-trivial subtrees of distinct trees T i have the same span (ie, the same vertex sets) (a) (b) Figure 29: A proper (a) and a non-proper (b) 3Tree2 partition Remark 225 If a graph G has a 3Tree2 partition, then it satisfies E(G) = 2 V (G) 3 This follows from the presence of exactly two trees at each vertex of G and the fact that for every tree T we have E(T ) = V (T ) 1 Moreover, note that a 3Tree2 partition of a graph G is proper if and only if every non-trivial subgraph H of G satisfies the count E(H) 2 V (H) 3 [47] Theorem 2214 (Crapo, 1989) [20] A graph G is generically 2-isostatic if and only if G has a proper 3Tree2 partition Symmetrized versions of Crapo s Theorem are discussed in Chapter 5 Finally, as promised, we give some results which assert that under the right conditions, rigidity and infinitesimal rigidity are equivalent In Chapter 6, we establish symmetric analogs to these theorems 37

63 Definition 2222 Let G be a graph with n vertices and let d 1 be an integer A point p R dn is said to be a regular point of G if there exists a neighborhood N p of p in R dn so that rank ( R(G, p) ) rank ( R(G, q) ) for all q N p A framework (G, p) is said to be regular if p is a regular point of G Theorem 2215 [3] Let G be a graph with n vertices and let (G, p) be a d-dimensional framework If p R dn is a regular point of G, then (G, p) is infinitesimally rigid if and only if (G, p) is rigid If a framework (G, p) is generic or independent, then (G, p) is clearly also regular (see also Figure 25), so that we immediately obtain the following results Corollary 2216 If a framework (G, p) is generic, then (G, p) is infinitesimally rigid if and only if (G, p) is rigid Corollary 2217 If a framework (G, p) is independent, then (G, p) is infinitesimally rigid if and only if (G, p) is rigid 23 Symmetry in frameworks In this section we establish the concept of a symmetric framework and give mathematically precise definitions of terms relating to symmetry which might have different meanings in different contexts In the literature about symmetric structures it is common to systematize the notion of symmetry 38

64 by introducing the concept of a symmetry operation and its corresponding symmetry element [6, 19, 37] We begin with our definitions of these terms First, recall that an isometry of R d is a map x : R d R d such that x(a) x(b) = a b for all a, b R d Definition 231 Let (G, p) be a framework in R d A symmetry operation of (G, p) is an isometry x of R d such that for some α Aut(G), we have x ( p(v) ) = p ( α(v) ) for all v V (G) A symmetry operation x of a framework (G, p) carries (G, p) into a framework (G, x p) which is geometrically indistinguishable from (G, p) In other words, up to the labeling of the vertices of the underlying graph G, the frameworks (G, p) and (G, x p) are the same Definition 232 Let x be a symmetry operation of a framework (G, p) in R d The symmetry element corresponding to x is the affine subspace F x of R d which consists of all points a R d such that x(a) = a Since we only consider finite graphs, it follows directly from Definition 231 that a symmetry operation cannot be a translation This implies in particular that a symmetry element is always non-empty In fact, it is easy to see that if x is a symmetry operation of a framework (G, p) with V (G) = {v 1,, v n }, then the point 1 n n i=1 p i must be fixed by x Figures 210 and 211 depict the possible symmetry elements in dimensions 2 and 3 Note that distinct symmetry operations of a framework may have the same corresponding symmetry element For example, distinct rotational 39

65 symmetry operations of a 3-dimensional framework may share the same rotational axis The set of all symmetry operations of a given framework forms a group under composition We adopt the following vocabulary from chemistry and crystallography: Definition 233 Let (G, p) be a framework Then the group which consists of all symmetry operations of (G, p) is called the point group of (G, p) For a systematic method to find the point group of a given framework, see [6, 19, 37], for example If P is the point group of a d-dimensional framework, then, as noted above, there exists a point in R d which is fixed by every symmetry operation in P Note that if the origin of R d is fixed by x P, then x is an orthogonal linear transformation of R d So, if the origin of R d is fixed by every symmetry operation in P, then P is a subgroup of the orthogonal group O(R d ) consisting of all orthogonal linear transformations of R d Definition 234 A subgroup of the orthogonal group O(R d ) is called a symmetry group (in dimension d) Given a d-dimensional framework (G, p), the framework (G, T p), where T is a translation of R d, clearly has the same rigidity properties as (G, p) Therefore, for our purposes we may wlog restrict our attention to frameworks whose point groups are symmetry groups In this thesis, unless otherwise specified, the point group of every framework is assumed to be a symmetry group 40

66 We use the Schoenflies notation to denote symmetry operations and symmetry groups in dimensions 2 and 3, as this is one of the standard notations in the literature about symmetric structures [2, 4, 6, 19, 37] Another motivation for using the Schoenflies notation in this thesis is to be consistent with the notation in the papers [15, 25, 34, 35, 36], for example In the plane, the three kinds of possible symmetry operations are the identity Id, rotations C m about the origin by an angle of 2π, where m 2, m and reflections s in lines through the origin The symmetry elements corresponding to these symmetry operations are shown in Figure 210 (a) (b) (c) Figure 210: Symmetry elements corresponding to symmetry operations in dimension 2: (a) a rotation C m, m 2; (b) a reflection s; (c) the identity Id In the Schoenflies notation we differentiate between the following four types of symmetry groups in dimension 2: C 1, C s, C m and C mv, where m 2 C 1 denotes the trivial group which only contains the identity Id C s denotes any symmetry group in dimension 2 that consists of the identity Id and a single reflection s For m 2, C m denotes any cyclic symmetry group of order m which is generated by a rotation C m, and C mv denotes any symmetry group in dimension 2 that is generated by a pair {C m, s} So, as abstract groups, any group C s is the cyclic group Z 2, any group C m is the cyclic group 41

67 Z m, and any group C mv is the dihedral group of order 2m In 3-space, there are the following symmetry operations: the identity Id, rotations C m about axes through the origin by an angle of 2π, where m 2, m reflections s in planes through the origin, and improper rotations S m fixing the origin, where m 3 An improper rotation S m is a rotation C m followed by the reflection s whose symmetry element is the plane through the origin that is perpendicular to the axis of C m The axis of C m is called the improper rotation axis of S m By convention, S 1 and S 2 are treated separately, since S 1 is simply a reflection s and S 2 is the inversion in the origin which is denoted by i (a) (b) (c) (d) Figure 211: Symmetry elements corresponding to symmetry operations in dimension 3: (a) an improper rotation S m, m 2; (b) a rotation C m, m 2; (c) a reflection s; (d) the identity Id This gives rise to the following families of possible symmetry groups in dimension 3: C 1, C s, C i, C m, C mv, C mh, D m, D mh, D md, S 2m, T, T d, T h, O, O h, I, and I h, where m 2 Analogous to the notation in dimension 2, C 1 again denotes the trivial group that only contains the identity Id, C m denotes any symmetry group in dimension 3 that is generated by a rotation C m, where m 2, and C s denotes any symmetry group in dimension 3 that consists of the identity Id 42

68 and a single reflection s C i is the symmetry group which consists of the identity and the inversion i of R 3 C mv denotes any symmetry group that is generated by a rotation C m and a reflection s whose symmetry element contains the rotational axis of C m Similarly, a symmetry group C mh is generated by a rotation C m and the reflection s whose symmetry element is perpendicular to the axis of C m It follows that every symmetry group C mh contains an improper rotation S m Note that if a symmetry group S contains an improper rotation S m, where m is odd, then S must also contain both the rotation C m whose symmetry element is the improper rotation axis of S m and the reflection s whose mirror plane is perpendicular to the axis of C m Therefore, for odd m, a symmetry group S is of type C mh if and only if S is generated by an improper rotation S m The abstract groups corresponding to C mv and C mh are the dihedral group D m of order 2m and the group Z m Z 2 (which is isomorphic to Z 2m if m is odd), respectively The symbol D m is used to denote a symmetry group in dimension 3 that is generated by a rotation C m and another 2-fold rotation C 2 whose rotational axis is perpendicular to the one of C m As an abstract group, D m is again the dihedral group of order 2m Symmetry groups of the types D mh and D md are generated by the generators C m and C 2 of a group D m and by a reflection s In the case of D mh, the symmetry element of s is the plane that is perpendicular to the C m axis and contains the origin (and hence contains the rotational axis of C 2 ), whereas in the case of D md, the symmetry element of s is a plane that contains the C m 43

69 axis and forms an angle of π with the C m 2 axis (ie, the symmetry element of s bisects the angle between adjacent half-turn axes created by rotating the C 2 axis about the C m axis) As abstract groups, D mh is the group D m Z 2 (which is isomorphic to D 2m if m is odd) and D md is the group D 2m If a symmetry group S in dimension 3 is generated by an improper rotation S k, where k is even, say k = 2m, then S is denoted by S 2m The abstract group that corresponds to S 2m is of course the group Z 2m The remaining seven types of symmetry groups in dimension 3 are related to the Platonic solids and are placed into three divisions: the tetrahedral groups T, T d and T h, the octahedral groups O and O h, and the icosahedral groups I and I h A symmetry group whose elements are all the rotational symmetry operations of a regular tetrahedron is denoted by T, and a symmetry group that consists of all the symmetry operations of a regular tetrahedron is denoted by T h T d denotes a symmetry group that is generated by the elements of a group T and those three reflections whose symmetry elements each contain two of the three axes that correspond to half-turns in T The abstract groups corresponding to T, T d and T h are A 4, S 4 and A 4 Z 2, respectively, where A 4 is the alternating group and S 4 the symmetric group on 4 elements O denotes a symmetry group that consists of all rotational symmetry operations of a regular octahedron (or, equivalently, a regular cube), and O h denotes a symmetry group that consists of all the symmetry operations of a regular octahedron Similarly, a symmetry group whose elements are all rotational symmetry operations of a regular icosahedron (or, equivalently, a regular dodecahedron) is denoted by I, and the symbol I h is used for a 44

70 symmetry group that consists of all the symmetry operations of a regular icosahedron The abstract groups corresponding to O, O h, I and I h are S 4, S 4 Z 2, A 5 and A 5 Z 2, respectively, where A i is the alternating group and S i the symmetric group on i elements for i = 4, 5 45

71 Chapter 3 A classification of symmetric frameworks Recall from Definition 231 that a symmetry operation of a framework (G, p) imposes geometric constraints on (G, p) by taking into account the combinatorial structure of the underlying graph G So, these kinds of symmetry constraints have both a geometric and a combinatorial aspect As the results of this thesis will show, it turns out that both of these aspects of the symmetry constraints play an important role in a symmetry-based rigidity analysis of a symmetric framework The classification of symmetric frameworks we introduce in Section 31 is motivated by this fact The usual starting point in most applications of rigidity of symmetric frameworks is that one is given a symmetric structure with a non-trivial point group (such as a biomolecule, for example) whose rigidity and flexibility properties are to be examined This approach is used in [25, 26, 27, 35, 43, 44, 45], for example The same approach will be used in this thesis 46

72 Alternatively, one could start with a graph G and a subgroup A of Aut(G), and then consider the possible geometric realizations of G that satisfy the symmetry constraints imposed by A This approach is used in [53], for example Note that although these two approaches have a different starting point, they both result in the same geometric and combinatorial conditions for the final theorems As we will see in Sections 33 and 34, if (G, p) is an injective realization of a graph G, and S is a subgroup of the point group of (G, p), then the map Φ : S Aut(G) which turns each isometry x S into a symmetry operation of (G, p) by assigning an appropriate graph automorphism to x is not only uniquely determined, but it is also a group homomorphism In this case, symmetry-based techniques can be applied to the rigidity analysis of (G, p) in a unique way (for a fixed group S) In their studies of symmetric frameworks, engineers and chemists usually restrict their attention to injective realizations While this is a reasonable assumption for most applications (atoms of biomolecules or joints of 3-dimensional physical structures never coincide, for example), there are occasions where we do want to analyze frameworks with non-injective configurations (if we want to model a linkage in the plane with overlapping joints, for example) So, in order to obtain more general mathematical results and a more complete theory, we develop the mathematical foundation for the rigidity of symmetric frameworks in such a way that it also allows us to analyze symmetric frameworks with non-injective configurations We will see in Sections 33 and 34 that if (G, p) is a non-injective real- 47

73 ization of G, then one may have several choices for a graph automorphism to turn an isometry in S into a symmetry operation of (G, p) The subtle difficulties that can occur in such a case, as well as their consequences for the application of symmetry-based techniques to the rigidity analysis of (G, p), are carefully examined in these sections All the key results in this chapter are original and are contained in the manuscript [55] which has been submitted for review 31 The classification In order to symmetrize results in rigidity theory, particularly results in generic rigidity theory, we first of all need an appropriate classification of symmetric frameworks Naturally, we require that frameworks in the same class have the same underlying graph This classification should also be such that almost all frameworks within a given class share the same infinitesimal rigidity properties, so that we can develop a symmetrized version of generic rigidity theory with respect to this classification Definition 311 Let G be a graph and S be a symmetry group in dimension d Then R (G,S) is the set of all d-dimensional realizations of G whose point group is either equal to S or contains S as a subgroup An element of R (G,S) is said to be a realization of the pair (G, S) Theorem 311 Let (G, p) be a d-dimensional realization of a graph G and S be a symmetry group in dimension d Then (G, p) R (G,S) if and only if there exists a map Φ : S Aut(G) such that x ( p(v) ) = p ( Φ(x)(v) ) for all 48

74 v V (G) and all x S Proof It follows immediately from the definitions that (G, p) R (G,S) if and only if S is a subgroup of the point group of (G, p) if and only if every element of S is a symmetry operation of (G, p) if and only if for every x S, there exists an automorphism α x of G that satisfies x ( p(v) ) = p ( α x (v) ) for all v V (G) Remark 311 Note that the set R (G,S) can possibly be empty For example, there clearly exists no realization of (K 2, C 3 ), where C 3 is a symmetry group in dimension 2 Theorem 311 gives rise to the following natural classification of the frameworks within a set R (G,S) Definition 312 Let S be a symmetry group, (G, p) be a framework in R (G,S), and Φ be a map from S to Aut(G) Then (G, p) is said to be of type Φ if the following equations hold: x ( p(v) ) = p ( Φ(x)(v) ) for all v V (G) and all x S The set of all realizations of (G, S) which are of type Φ is denoted by R (G,S,Φ) Given a graph G and a symmetry group S in dimension d, different choices of types Φ : S Aut(G) frequently lead to very different geometric types of realizations of (G, S) This is because a type Φ forces the joints and bars of a framework in R (G,S,Φ) to assume certain geometric positions in R d We give a few examples for small symmetry groups in dimensions 2 and 3 to demonstrate this 49

75 p 4 p 1 p 2 p 5 p6 p 3 p 1 p 6 p 2 p5 p 4 p 3 (a) (b) Figure 31: 2-dimensional realizations of (K 3,3, C s ) of different types Example 311 Figure 31 shows two realizations of (K 3,3, C s ) of different types, where K 3,3 is the complete bipartite graph with partite sets {v 1, v 2, v 3 } and {v 4, v 5, v 6 }, and C s = {Id, s} is a symmetry group in dimension 2 generated by a reflection The framework in Figure 31 (a) is a realization of (K 3,3, C s ) of type Φ a, where Φ a : C s Aut(K 3,3 ) is defined by Φ a (Id) = id Φ a (s) = (v 1 v 2 )(v 5 v 6 )(v 3 )(v 4 ), and the framework in Figure 31 (b) is a realization of (K 3,3, C s ) of type Φ b, where Φ b : C s Aut(K 3,3 ) is defined by Φ b (Id) = id Φ b (s) = (v 1 v 4 )(v 2 v 5 )(v 3 v 6 ) Note that for any framework (K 3,3, p) in the set R (K3,3,C s,φ a), the points p 3 and p 4 must lie in the symmetry element corresponding to s (ie, in the mirror line of s), because s ( p(v i ) ) = p ( Φ a (s)(v i ) ) = p(v i ) for i = 3, 4 This says in particular that for any framework (K 3,3, p) in R (K3,3,C s,φ a ), the entire 50

76 undirected line segment p 3 p 4 which corresponds to the bar { (v 3, p 3 ), (v 4, p 4 ) } of (K 3,3, p) must lie in the mirror line of s We shall immediately become less formal and say that the bar { (v 3, p 3 ), (v 4, p 4 ) } lies in the mirror line of s Similarly, for any framework (K 3,3, p) in R (K3,3,C s,φ b ), the bars { (v1, p 1 ), (v 4, p 4 ) }, { (v 2, p 2 ), (v 5, p 5 ) } and { (v 3, p 3 ), (v 6, p 6 ) } must be perpendicular to and centered at the mirror line of s p 2 p 5 p 1 p 4 p 3 p 6 (a) p 2 p 6 p 1 p 4 p 3 p 5 (b) Figure 32: 2-dimensional realizations of (G tp, C 2 ) of different types Example 312 Figure 32 depicts two realizations of (G tp, C 2 ) of different types, where G tp is the graph of a triangular prism and C 2 = {Id, C 2 } is the half-turn symmetry group in dimension 2 The framework in Figure 32 (a) is a realization of (G tp, C 2 ) of type Ψ a, where Ψ a : C 2 Aut(G tp ) is defined by Ψ a (Id) = id Ψ a (C 2 ) = (v 1 v 4 )(v 2 v 6 )(v 3 v 5 ) and the framework in Figure 32 (b) is a realization of (G tp, C 2 ) of type Ψ b, where Ψ b : C 2 Aut(G tp ) is defined by Ψ b (Id) = id Ψ b (C 2 ) = (v 1 v 4 )(v 2 v 5 )(v 3 v 6 ) 51

77 It follows from the definitions of Ψ a and Ψ b that for any framework (G tp, p) in R (Gtp,C 2,Ψ a ), the bar { (v 1, p 1 ), (v 4, p 4 ) } must be centered at the origin (which is the center of the half-turn C 2 ), whereas for any framework (G tp, p) in R (Gtp,C 2,Ψ b ), all three bars { (v 1, p 1 ), (v 4, p 4 ) }, { (v 2, p 2 ), (v 5, p 5 ) }, and { (v 3, p 3 ), (v 6, p 6 ) } must be centered at the origin p 5 p 3 p 1 p 2 p 4 p 4 p 5 p 3 p 1 p 2 (a) (b) Figure 33: 3-dimensional realizations of (G bp, C s ) of different types Example 313 Finally, Figure 33 depicts two realizations of (G bp, C s ) of different types, where G bp is the graph of a triangular bipyramid and C s = {Id, s} is a symmetry group in dimension 3 The framework in Figure 33 (a) is an element of R (Gbp,C s,ξ a ), where Ξ a : C s Aut(G bp ) is defined by Ξ a (Id) = id Ξ a (s) = (v 1 v 2 )(v 3 )(v 4 )(v 5 ), and the framework in Figure 33 (b) is an element of R (Gbp,C s,ξ b ), where 52

78 Ξ b : C s Aut(G bp ) is defined by Ξ b (Id) = id Ξ b (s) = (v 1 v 2 )(v 4 v 5 )(v 3 ) For any framework (G bp, p) in R (Gbp,C s,ξ a) or R (Gbp,C s,ξ b ), the bar { (v1, p 1 ), (v 2, p 2 ) } must be perpendicular to and centered at the mirror plane of s Further, for any framework (G bp, p) in R (Gbp,C s,ξ a ), the joints (v i, p i ), i = 3, 4, 5, must lie in the mirror plane of s, whereas for a framework (G bp, p) in R (Gbp,C s,ξ b ), only the joint (v 3, p 3 ) must have this property and the joints (v 4, p 4 ) and (v 5, p 5 ) must be mirror images of each other with respect to s Remark 312 Given a non-empty set R (G,S), it is possible that R (G,S,Φ) = for some map Φ : S Aut(G) Consider, for example, the non-empty set R (K2,C 2 ), where C 2 = {Id, C 2 } is the half-turn symmetry group in dimension 2, and let I : C 2 Aut(K 2 ) be the map which sends both Id and C 2 to the identity automorphism of K 2 If (K 2, p) R (K2,C 2,I), then both joints of (K 2, p) must be located at the origin (which is the center of C 2 ) This contradicts Definition 221 of a framework, and hence we have R (K2,C 2,I) = We will see in the next section that almost all frameworks within a set of the form R (G,S,Φ) share the same infinitesimal rigidity properties This will allow us to develop a symmetrized version of generic rigidity theory with respect to the classes R (G,S,Φ) of symmetric frameworks 53

79 32 The notion of (S, Φ)-generic Given a graph G, a non-trivial symmetry group S in dimension d clearly imposes restrictions on the possible geometric positions of realizations of (G, S) in R d For most groups S, these restrictions are in fact so strong that they force the joints of any realization in the set R (G,S) to lie in non-generic positions In some situations, the realizations in R (G,S) are even forced to be non-regular, as the following examples demonstrate Every realization of (K 3, C 2 ), where C 2 is a half-turn symmetry group in dimension 2 or 3, must be a degenerate triangle and is therefore non-regular (recall Figure 23 (b)) For a less trivial example, consider the complete bipartite graph K 3,3 and the symmetry group C 2 in dimension 2 As shown in Figure 34 (a), the joints of any realization (K 3,3, p) in R (K3,3,C 2 ) can be labeled in such a way that for the resulting hexagon p 1 p 2 p 6, there exists a pair of opposite sides which intersect in the origin If all three pairs of opposite sides of this hexagon are extended to their points of intersection, then the half-turn symmetry of (K 3,3, p) guarantees that these three points are collinear Therefore, by the converse of Pascal s Theorem, the joints of (K 3,3, p) must lie on a conic section It is well known that 2-dimensional realizations of K 3,3 whose joints lie on a conic section are in fact non-regular [7, 71, 75] This shows that our notion of generic (without symmetry) is clearly not suitable once we restrict our attention to symmetric frameworks that lie within a set of the form R (G,S) Note also that for a graph G, a symmetry group S, and two distinct maps Φ and Ψ from S to Aut(G), it is possible that all realizations in R (G,S,Φ) are 54

80 infinitesimally flexible, whereas almost all realizations in R (G,S,Ψ) are isostatic For example, consider again the complete bipartite graph K 3,3, a symmetry group C s in dimension 2, and the types Φ a and Φ b from Example 311 K 3,3 is known to be a generically 2-isostatic graph and the pure condition (see [71]) for K 3,3 says that a 2-dimensional realization (K 3,3, p) is infinitesimally flexible if and only if the joints of (K 3,3, p) lie on a conic section It follows (again from the converse of Pascal s Theorem) that every realization in R (K3,3,C s,φ b ) is infinitesimally flexible (see also Figure 34 (b)), whereas almost all realizations in R (K3,3,C s,φ a) are isostatic p 4 p 1 p 5 p 2 p 6 p 1 p 2 p 3 p6 p 4 p 5 p 3 (a) (b) Figure 34: By the converse of Pascal s Theorem, the joints of any realization in R (K3,3,C 2 ) or R (K3,3,C s,φ b ) lie on a conic section Therefore, in order to define a modified, symmetry-adapted notion of generic for a set C R (G,S) of symmetric frameworks in such a way that almost all realizations within C are generic and all generic realizations within C share the same infinitesimal rigidity properties, we need to restrict C to a set of the form R (G,S,Φ) 55

81 Let G be a graph with V (G) = {v 1, v 2,, v n }, S be a symmetry group in dimension d, and Φ be a map from S to Aut(G) We will define a symmetry-adapted notion of generic for the set R (G,S,Φ) in an analogous way as we defined generic in Definition 2216 This requires the definition of a symmetry-adapted indeterminate rigidity matrix for R (G,S,Φ) The following observations lay the foundation for the definition of such a matrix Recall that for every framework (G, p) in the set R (G,S,Φ), the equations stated in Definition 312 are satisfied, that is, we have x ( p(v i ) ) = p ( Φ(x)(v i ) ) for all i = 1, 2,, n and all x S Since every element of S is an orthogonal linear transformation, we may identify each x S with its corresponding orthogonal matrix M x that represents x with respect to the canonical basis of R d Therefore, for each x S, the equations in Definition 312 corresponding to x form a system of linear equations which can be written as M (x) p 1 p 2 p n = P Φ(x) p 1 p 2 p n, where M (x) = M x M x 0, 0 0 M x and P Φ(x) is the dn dn matrix which is obtained from the permutation matrix corresponding to Φ(x) by replacing each 1 by a d d identity matrix 56

82 and each 0 by a d d zero matrix Equivalently, we have ) (M (x) P Φ(x) p 1 p 2 p n = 0 We denote L x,φ = ker ( ) M (x) P Φ(x) and U = x S L x,φ Then U is a subspace of R dn which may be interpreted as the space of all those (possibly non-injective) configurations of n points in R d that possess the symmetry imposed by S and Φ In particular, if (G, p) is a framework in R (G,S,Φ), then the configuration p is an element of U Therefore, if we fix a basis B U = {u 1, u 2,, u k } of U, then every framework (G, p) R (G,S,Φ) can be represented uniquely by the k 1 coordinate vector of p relative to B U We are now ready to define the symmetry-adapted indeterminate rigidity matrix for R (G,S,Φ) Definition 321 Let G be a graph with V (G) = {v 1, v 2,, v n }, K n be the complete graph with V (K n ) = V (G), S be a symmetry group in dimension d, and Φ be a map from S to Aut(G) Further, let B U = {u 1, u 2,, u k } be a basis of U = x S L x,φ The symmetry-adapted indeterminate rigidity matrix for R (G,S,Φ) (corresponding to B U ) is the matrix R BU (n, d) which is obtained from the indeterminate rigidity matrix R(n, d) by introducing a k-tuple (t 1, t 2,, t k ) of variables and replacing the dn variables (p i) j of R(n, d) as follows For each i = 1, 2,, n and each j = 1,, d, we replace the variable (p i) j in R(n, d) by the linear combination t 1(u 1 ) ij + t 2(u 2 ) ij + + t k (u k) ij 57

83 Remark 321 Let (G, p) R (G,S,Φ) and B U = {u 1, u 2,, u k } be a basis of U = x S L x,φ Then p 1 p 2 p n = t 1 u t k u k, for some t 1,, t k R So, if for i = 1,, k, the variable t i in R BU (n, d) is replaced by t i then we obtain the rigidity matrix R(K n, p) of the framework (K n, p) With the help of Definition 321 we can now also give the formal definition of our symmetry-adapted notion of generic for a set R (G,S,Φ) Definition 322 Let G be a graph with V (G) = {v 1, v 2,, v n }, K n be the complete graph with V (K n ) = V (G), S be a symmetry group in dimension d, Φ be a map from S to Aut(G), and B U be a basis of U = x S L x,φ A map p : V (G) R d is said to be (S, Φ, B U )-generic if the following holds: If the determinant of any submatrix of R(K n, p) is equal to zero, then the determinant of the corresponding submatrix of R BU (n, d) is (identically) zero The map p is said to be (S, Φ)-generic if p is (S, Φ, B U )-generic for some basis B U of U A framework (G, p) R (G,S,Φ) is (S, Φ, B U )-generic if p is an (S, Φ, B U )-generic map, and (G, p) is (S, Φ)-generic if (G, p) is (S, Φ, B U )- generic for some basis B U of U 58

84 Theorem 321 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) If (G, p) R (G,S,Φ) is (S, Φ)-generic, then (G, p) is (S, Φ, B U )-generic for every basis B U of U = x S L x,φ Proof Suppose S is a symmetry group in dimension d and the vertex set of G is V (G) = {v 1, v 2,, v n } Let (G, p) R (G,S,Φ) be (S, Φ)-generic, say (G, p) is (S, Φ, B U )-generic, where B U = {u 1,, u k } is a basis of U Let B U = {u 1,, u k } be another basis of U Then we need to show that (G, p) is (S, Φ, B U)-generic Let p 1 p 2 p n = t 1 u t k u k = t 1u t ku k, where t i, t i R for all i = 1,, k Then there exists an invertible matrix of real numbers (s ij ) such that t 1 = s 11 t s 1k t k (31) t k = s k1 t s kk t k Let R BU (n, d) be the symmetry-adapted indeterminate rigidity matrix corresponding to B U with variables t 1,, t k, and R B (n, d) be the symmetry- U adapted indeterminate rigidity matrix corresponding to B U with variables t 1,, t k Then note that if for i = 1,, k, we replace the variable t i in R BU (n, d) analogously to (31) by t i = s i1 t s ik t k, (32) 59

85 then we obtain the matrix R B U (n, d) If each t i in R B U (n, d) is replaced by t i, then, by Remark 321, we obtain the rigidity matrix R(K n, p) Consider the determinant of a submatrix of R(K n, p) which is equal to zero The determinant of the corresponding submatrix of R BU (n, d) is a polynomial in t 1,, t k, say a(a1,,a k )t a 1 1 t a k k, where a (a 1,,a k ) R (33) Since (G, p) is (S, Φ, B U )-generic, the polynomial in (33) is the zero polynomial If in (33) we replace the variables t i as in (32), then we again obtain the zero polynomial On the other hand, this polynomial is the determinant of the corresponding submatrix of R B U (n, d) This says that (G, p) is (S, Φ, B U)-generic and the proof is complete Note that it follows directly from Definition 322 that the set of (S, Φ)- generic realizations of a graph G is an open dense subset of the set R (G,S,Φ) Moreover, as we will show next, the infinitesimal rigidity properties are the same for all (S, Φ)-generic realizations of G Lemma 322 Let G be a graph with V (G) = {v 1, v 2,, v n }, S be a symmetry group in dimension d, and Φ be a map from S to Aut(G) If for some framework (G, p) R (G,S,Φ), the points p 1,, p n span an affine subspace of R d of dimension k, then for any (S, Φ)-generic realization (G, q) of G, the points q 1,, q n span an affine subspace of R d of dimension at least k Proof Let (G, p) R (G,S,Φ) be a framework for which the points p 1,, p n span an affine subspace of R d of dimension k Then there are k + 1 affinely independent points among p 1,, p n, say wlog p 1,, p k+1 Let A be the 60

86 k d matrix defined by (p 1 p 2 ) 1 (p 1 p 2 ) 2 (p 1 p 2 ) d (p 1 p 3 ) 1 (p 1 p 3 ) 2 (p 1 p 3 ) d A = (p 1 p k+1 ) 1 (p 1 p k+1 ) 2 (p 1 p k+1 ) d Then the rows of A are linearly independent and hence there exists a k k submatrix B of A whose determinant is non-zero Fix a basis B U of U = x S L x,φ and let R BU (n, d) be the symmetry-adapted indeterminate rigidity matrix for R (G,S,Φ) corresponding to B U Then the determinant of the submatrix B of R BU (n, d) which corresponds to B is not identically zero Now, let (G, q) be an (S, Φ)-generic realization of G and suppose the points q 1,, q n span an affine subspace of R d of dimension m < k Then the matrix  which is obtained from A by replacing each (p i) j by (q i ) j has a non-trivial row dependency, which says that the determinant of every k k submatrix of  is equal to zero This contradicts the fact that (G, q) is (S, Φ)-generic and that the determinant of B is not identically zero Theorem 323 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) such that R (G,S,Φ) The following are equivalent (i) There exists a framework (G, p) R (G,S,Φ) that is infinitesimally rigid (independent, isostatic); (ii) every (S, Φ)-generic realization of G is infinitesimally rigid (independent, isostatic) 61

87 Proof Suppose S is a symmetry group in dimension d Let (G, p) R (G,S,Φ) be infinitesimally rigid and let (G, q) be an (S, Φ)-generic realization of G Suppose first that V (G) d Then, by Remark 223, the points p(v), v V (G), span an affine subspace of R d of dimension at least d 1 Therefore, the infinitesimal rigid motions arising from d translations and ( d 2) rotations of R d form a basis for the space of infinitesimal rigid motions of (G, p) (see [3, 33] for details), and hence we have rank ( R(G, p) ) ( ) d + 1 = d V (G) 2 By the definition of (S, Φ)-generic, rank ( R(G, q) ) rank ( R(G, p) ) By Lemma 322, the points q(v), v V (G), also span an affine subspace of R d of dimension at least d 1, which says that nullity ( R(G, q) ) ( ) d+1 2 Therefore, It follows that rank ( R(G, q) ) ( ) d + 1 d V (G) 2 rank ( R(G, q) ) ( ) d + 1 = d V (G), 2 and hence (G, q) is infinitesimally rigid Suppose now that V (G) d 1 Then the dimension of the space of infinitesimal rigid motions of (G, p) is strictly smaller than ( ) d+1 2 (see again [3, 33] for details) Therefore, we have nullity ( R(G, p) ) < ( ) d+1 2, and hence rank ( R(G, p) ) > d V (G) ( ) d+1 2 It follows from Theorem 222 that G is a complete graph and the points p(v), v V (G), are affinely independent By 62

88 Lemma 322, the points q(v), v V (G), must also be affinely independent, and hence (G, q) is infinitesimally rigid If (G, p) is independent, then it follows from the definition of (S, Φ)- generic that (G, q) is also independent Therefore, if (G, p) is isostatic, so is (G, q) So, being infinitesimally rigid (independent, isostatic) is an (S, Φ)-generic property This gives rise to Definition 323 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) G is said to be (S, Φ)-generically infinitesimally rigid (independent, isostatic) if all realizations of G which are (S, Φ)-generic are infinitesimally rigid (independent, isostatic) Examples 311 and 312 show that a graph G which is (S, Φ)-generically isostatic is not necessarily (S, Ψ)-generically isostatic, where Φ and Ψ are two distinct maps from S to Aut(G) In Example 311, (C s, Φ a )-generic realizations in R (K3,3,C s,φ a ) are isostatic, whereas all the realizations in R (K3,3,C s,φ b ) are infinitesimally flexible, because the joints of any realization in R (K3,3,C s,φ b ) lie on a conic section, as we already observed in the beginning of this section In Example 312, the graph G tp is (C 2, Ψ a )-generically isostatic, but none of the realizations in R (Gtp,C2,Ψ b ) is isostatic This follows from the pure condition for G tp, which says that a 2-dimensional realization of G tp is not isostatic if and only if the triangles p 1 p 2 p 3 and p 4 p 5 p 6 are perspective from a line [71] Equivalently, by Desargues Theorem, a 2-dimensional realization of G tp is not isostatic if and only if the triangles p 1 p 2 p 3 and p 4 p 5 p 6 are perspective 63

89 from a point or at least one of those triangles is degenerate For an example in 3-space, consider the complete graph K 4 with V (K 4 ) = {v 1, v 2, v 3, v 4 }, a symmetry group C s = {Id, s} in dimension 3, and the maps Υ a and Υ b from C s to Aut(K 4 ), where Υ a maps Id to the identity automorphism id of K 4 and s to (v 1 v 2 )(v 3 )(v 4 ), and Υ b maps both Id and s to id Then K 4 is (C s, Υ a )-generically isostatic, but all realizations in R (K4,C s,υ b ) are infinitesimally flexible, because all the joints of a realization in R (K4,C s,υ b ) must lie in the mirror plane corresponding to s and are therefore coplanar p 4 p 4 p 3 p 1 p 2 p 3 p 2 p 1 (a) (b) Figure 35: A 3-dimensional realization of (K 4, C s ) of type Υ a (a) and of type Υ b (b) Remark 322 Let G be a graph, S be a symmetry group in dimension d, and Φ : S Aut(G) be a homomorphism Frameworks in the set R (G,S,Φ), particularly (S, Φ)-generic realizations of G, can then be visualized in a very intuitive way via the following approach The map of S V (G) onto V (G) that sends (x, v) to Φ(x)(v) defines a group action on V (G) and the orbits Sv = {Φ(x)(v) x S} form a partition 64

90 of V (G) Let {v 1,, v r } be a subset of V (G) obtained by choosing one representative from each of these orbits, and recall from Definition 232 that for every x S, F x denotes the symmetry element corresponding to x If for i = 1,, r, we define F (v i ) = x S with Φ(x)(v i )=v i F x, then for every framework (G, p) R (G,S,Φ) and for every i {1,, r}, the point p(v i ) must be contained in the subspace F (v i ) of R d Note that the positions of all joints of a framework (G, p) R (G,S,Φ) are uniquely determined by the positions p(v 1 ),, p(v r ) of the joints ( v1, p(v 1 ) ),, ( v r, p(v r ) ) and the symmetry constraints imposed by S and Φ In other words, we may construct frameworks in R (G,S,Φ) by first choosing a point p(v i ) F (v i ) for each i = 1,, r and then letting S and Φ determine the positions of the remaining joints In particular, note that we obtain an (S, Φ)-generic framework (G, p) in this way for almost all choices of points p(v i ) that satisfy p(v i ) F (v i ) for i = 1,, r Consider, for example, the set R (K4,C s,υ a ) an element of which is shown in Figure 35 (a) The orbits for the group action from C s V (K 4 ) onto V (K 4 ) are given by {v 1, v 2 }, {v 3 }, and {v 4 } If (K 4, p) is a framework in R (K4,C s,υ a), then both p 3 and p 4 must be contained in the mirror plane F s of s, because F (v 3 ) = F (v 4 ) = F Id F s = F s Furthermore, since v 1 and v 2 are vertices of the same orbit, the position of the point p 2 is uniquely determined by the position of p 1 and the symmetry constraints imposed by C s and Υ a Since F (v 1 ) = F Id = R 3, the point p 1 may be chosen to be any point in R 3 ; however, if p 1 lies in the mirror plane of s, then p 1 = p 2, in which case (K 4, p) is not a framework 65

91 We will return to this approach in Chapter 6 We conclude this section by giving a few more interesting properties of (S, Φ)-generic frameworks Theorem 324 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) Further, let (G, p) R (G,S,Φ), S be a subgroup of S, and Φ = Φ S If (G, p) is (S, Φ )-generic, then (G, p) is also (S, Φ)-generic Proof Suppose S is a symmetry group in dimension d and G is a graph with n vertices Let (G, p) R (G,S,Φ) be (S, Φ )-generic We fix a basis B U = {u 1,, u k } of U = x S L x,φ and extend it to a basis B U = {u 1,, u k, u k+1,, u m } of U = x S L x,φ Consider a submatrix of the rigidity matrix R(K n, p) whose determinant is equal to zero We need to show that the determinant of the corresponding submatrix of the symmetry-adapted indeterminate rigidity matrix R BU (n, d) is identically zero Let = a (a1,,a m )t a 1 1 t a m m, where a (a1,,a m ) R, be the determinant of the corresponding submatrix of R BU (n, d) Then is the zero polynomial since (G, p) is (S, Φ )-generic But note that is a polynomial that is obtained from by deleting all those terms in that have one or more variables in {t k+1,, t m} Thus, is also the zero polynomial shows The converse of Theorem 324 does not hold, as the following example Example 321 The realization (K 3,3, p) in Figure 36 is (C 2v, Φ)-generic, where C 2v = {Id, C 2, s h, s v } is a symmetry group in dimension 2 and Φ : 66

92 C 2v Aut(K 3,3 ) is defined by Φ(Id) = id Φ(C 2 ) = (v 1 v 6 )(v 2 v 5 )(v 3 v 4 ) Φ(s h ) = (v 1 v 5 )(v 2 v 6 )(v 3 v 4 ) Φ(s v ) = (v 1 v 2 )(v 5 v 6 )(v 3 )(v 4 ) However, (K 3,3, p) is not (C s, Φ a )-generic, where C s is the subgroup of C 2v generated by s v and Φ a = Φ Cs is the map we defined in Example 311 p 4 p 1 p 2 p 5 p 6 p 3 s h s v Figure 36: A realization of K 3,3 that is (C 2v, Φ)-generic, but not (C s, Φ a )- generic, where C s is the subgroup of C 2v generated by s v and Φ a = Φ Cs Corollary 325 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) If (G, p) R (G,S,Φ) is generic (in the sense of Definition 2216) then (G, p) is also (S, Φ)-generic Proof Suppose S is a symmetry group in dimension d and G is a graph with n vertices Let (G, p) R (G,S,Φ) be generic Then Φ maps the symmetry operation Id S to the identity automorphism id of G, for otherwise the map q of every realization (G, q) in R (G,S,Φ) is non-injective, contradicting the fact 67

93 that (G, p) R (G,S,Φ) is generic So, Φ C1 = I, where I : C 1 Aut(G) maps Id to id, and we have x C 1 L x,i = L Id,I = R dn Now, observe that the indeterminate rigidity matrix R(n, d) is equal to the symmetry-adapted indeterminate rigidity matrix R BR (n, d), where dn B R dn is the canonical basis of R dn Therefore, (G, p) is generic if and only if (G, p) is (C 1, I)-generic The result now follows immediately from Theorem 324 The converse of Corollary 325 is of course false A (C s, Φ b )-generic realization of K 3,3, for example, where C s and Φ b are as in Example 311, has all of its joints on a conic section and is therefore not generic (in fact, such a framework is even non-regular) 33 Of what types Φ can a framework be? Let (G, p) be a d-dimensional framework with point group symmetry P Then (G, p) R (G,S) for every subgroup S of P Fix a subgroup S of P Then it follows from Theorem 311 that there exists a map Φ : S Aut(G) such that (G, p) R (G,S,Φ) The following examples show that it is possible for (G, p) R (G,S) to be of more than just one such type Φ Note that each of these examples is a non-injective realization Example 331 Let G t be the graph of two triangles sharing an edge and C 2 = {Id, C 2 } be the half-turn symmetry group in dimension 2 Figure 37 (a) shows a realization (G t, p) of (G t, C 2 ) that is of type Θ a as well as Θ b, 68

94 where Θ a : C 2 Aut(G t ) is defined by Θ a (Id) = id Θ a (C 2 ) = (v 1 v 2 )(v 3 )(v 4 ), and Θ b : C 2 Aut(G t ) is defined by Θ b (Id) = id Θ b (C 2 ) = (v 1 v 2 )(v 3 v 4 ) p 2 p 4 = p 5 p 3 = p 4 p 3 p 1 p 1 p 2 (a) (b) Figure 37: A realization of (G t, C 2 ) of type Θ a and Θ b (a) and a realization of (G bp, C s ) of type Ξ a and Ξ b (b) Example 332 Consider the graph G bp of a triangular bipyramid and a symmetry group C s = {Id, s} in dimension 3 The framework (G bp, p) in Figure 37 (b) is a realization of (G bp, C s ) that is of type Ξ a as well as Ξ b, where Ξ a : C s Aut(G bp ) and Ξ b : C s Aut(G bp ) are defined as in Example

95 Since for a given framework (G, p) in a set of the form R (G,S), the specification of a type Φ : S Aut(G) plays a key role in a symmetry-based rigidity analysis of (G, p), it is natural to ask how we can find all the types Φ of (G, p), how these types are related to each other, and under what conditions (G, p) is of a unique type The following definition is essential to answer all of these questions Definition 331 Let (G, p) be a framework Then we denote Aut(G, p) to be the set of all α Aut(G) which satisfy p(v) = p ( α(v) ) for all v V (G) Given a framework (G, p) and an automorphism α Aut(G, p), it is easy to see that only vertices of G that have the same image under p can possibly belong to the same permutation cycle of α In particular, for every framework (G, p) with an injective map p, we have Aut(G, p) = {id}, as we will see in the proof of Corollary 333 For the framework (G t, p) in Figure 37 (a), we have Aut(G t, p) = {id, (v 3 v 4 )(v 1 )(v 2 )} and for the framework (G bp, p) in Figure 37 (b), we have Aut(G bp, p) = {id, (v 4 v 5 )(v 1 )(v 2 )(v 3 )} Clearly, Aut(G, p) is a subgroup of Aut(G) Theorem 331 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) Further, let (G, p) R (G,S,Φ) and x S Then Φ(x)Aut(G, p) = Aut(G, p)φ(x), and an automorphism α of G satisfies x ( p(v) ) = p ( α(v) ) for all v V (G) if and only if α is an element of Φ(x)Aut(G, p) Proof First, we show that Φ(x)Aut(G, p) = Aut(G, p)φ(x) Since the cosets Φ(x)Aut(G, p) and Aut(G, p)φ(x) have the same cardinality, it suffices 70

96 to show that Φ(x)Aut(G, p) Aut(G, p)φ(x) Let α Φ(x)Aut(G, p), say α = Φ(x) β, where β Aut(G, p) Then for v V (G), we have x ( p(v) ) ( = x p ( β(v) )) ( = p Φ(x) ( β(v) )) = p ( α(v) ) Since we also have x ( p(v) ) = p ( Φ(x)(v) ), it follows that p ( α(v) ) = p ( Φ(x)(v) ) for all v V (G) Therefore, p (α ( Φ(x) ) ) 1 (v) = p(v) for all v V (G), and hence α (Φ(x)) 1 Aut(G, p) Thus, α Aut(G, p)φ(x) Now, α Aut(G) satisfies x ( p(v) ) = p ( α(v) ) for all v V (G) if and only if if and only if p ( α(v) ) = p ( Φ(x)(v) ) for all v V (G) p (α ( Φ(x) ) ) 1 (v) = p(v) for all v V (G) if and only if if and only if α ( Φ(x) ) 1 Aut(G, p) α Aut(G, p)φ(x) = Φ(x)Aut(G, p) Corollary 332 Let G be a graph, S be a symmetry group, Φ be a map from S to Aut(G), and (G, p) R (G,S,Φ) Then for every Ψ : S Aut(G) distinct from Φ, we have (G, p) / R (G,S,Ψ) if and only if Aut(G, p) = {id} 71

97 Proof It follows directly from Theorem 331 that Aut(G, p) = {id} if and only if for every x S, the automorphism Φ(x) is the only automorphism of G that satisfies x ( p(v) ) = p ( Φ(x)(v) ) for all v V (G) Corollary 332 asserts that the type Φ : S Aut(G) of a framework (G, p) R (G,S) is unique if and only if Aut(G, p) only contains the identity automorphism of G In particular, we have the following result Corollary 333 Let G be a graph, S be a symmetry group, and Φ be a map from S to Aut(G) If the map p of a framework (G, p) R (G,S,Φ) is injective, then (G, p) / R (G,S,Ψ) for every Ψ : S Aut(G) distinct from Φ Proof Let α be an element of Aut(G, p) Then we have p(v) = p ( α(v) ) for all v V (G), and since p is injective it follows that v = α(v) for all v V (G) Thus, α is the identity automorphism of G and the result follows from Corollary 332 The following examples show that the converse of Corollary 333 does not hold, that is, a framework (G, p) R (G,S) that is of a unique type Φ : S Aut(G) can possibly have a non-injective map p Example 333 The framework (G, p) in Figure 38 (a) is a non-injective realization of (G, C 2 ) (since p 5 = p 6 ) with Aut(G, p) = {id} So, (G, p) R (G,C2 ) is of the unique type Φ : C 2 Aut(G), where Φ(Id) = id and Φ(C 2 ) = (v 1 v 2 )(v 3 v 4 )(v 5 v 6 ) Example 334 The framework (G, p) in Figure 38 (b) is a non-injective realization of (G, C 3 ) (since p 4 = p 5 = p 6 ) with Aut(G, p) = {id} So, 72

98 (G, p) R (G,C3 ) is of the unique type Φ : C 3 Aut(G), where Φ is the homomorphism defined by Φ(C 3 ) = (v 1 v 2 v 3 )(v 4 v 5 v 6 ) p 2 p 3 p 6 p5 p 4 p 1 (a) p 3 p 6 p 5 p 1 p 4 p 2 (b) Figure 38: Non-injective realizations with Aut(G, p) = {id} Remark 331 Let (G, p) R (G,S,Φ) be a framework with Aut(G, p) = {id} and let (G, q) R (G,S,Φ) be an (S, Φ)-generic framework It follows immediately from the definition of (S, Φ)-generic (Definition 322) that two joints (v i, q i ) and (v j, q j ) of (G, q) can only satisfy q i = q j if p i = p j This says that (G, q) also satisfies Aut(G, q) = {id} Therefore, by Corollary 332, being of a unique type is an (S, Φ)-generic property Remark 332 If a framework (G, p) R (G,S) is of distinct types Φ 1, Φ k, where k 2, then (G, p) is not (S, Φ t )-generic for some t {1,, k}, as the following argument shows Suppose to the contrary that (G, p) is (S, Φ i )-generic for all i = 1,, k and let l {1,, k} Since Aut(G, p) {id}, there exist vertices v w of G such that p(v) = p(w) and α(v) = w for some α Aut(G, p) Since (G, p) is (S, Φ l )-generic, there must exist non-trivial symmetry operations x, y S such that Φ l (x)(v) = v and Φ l (y)(w) = w, and the symmetry 73

99 elements corresponding to x and y must be the origin 0 = p(v) = p(w) If for each x S with Φ l (x)(v) = v, we replace Φ l (x) by α Φ l (x), then we obtain a map Φ t, t l, with the property that for all x S, Φ t (x)(v) v Thus, (G, p) is not (S, Φ t )-generic, a contradiction As an example, consider the framework (G t, p) in Figure 37 (a) (G t, p) is (C 2, Θ a )-generic, but not (C 2, Θ b )-generic, because p 3 = p 4 and Θ b (v 3 ) = v 4 (see Example 331) The framework in Figure 37 (b) is a realization of (G bp, C s ) of type Ξ a and Ξ b which is neither (C s, Ξ a )-generic nor (C s, Ξ b )-generic, because p 4 = p 5 (see Example 332) 34 When is a type Φ of a framework a homomorphism? We will see in the next chapter that in order to use techniques from group representation theory to analyze the rigidity properties of a symmetric framework (G, p) R (G,S,Φ), we need Φ to be a homomorphism In this section, we therefore investigate the natural question of whether a type Φ : S Aut(G) of a given framework (G, p) R (G,S) is in fact a homomorphism (rather than just a map) Theorem 341 Let S be a symmetry group and (G, p) be a framework in R (G,S) with Aut(G, p) = {id} Then the unique map Φ : S Aut(G) for which (G, p) R (G,S,Φ) is a homomorphism 74

100 Proof Let x and y be any two elements of S Then Φ(y) Φ(x) Aut(G) satisfies ( )( ) ( y x p(v) = y p ( Φ(x)(v) )) ( (Φ(y) ) ) = p Φ(x) (v) for all v V (G) and, by Corollary 332, Φ(y) Φ(x) is the only automorphism of G with this property Thus, Φ(y x) = Φ(y) Φ(x) In particular, it follows from Corollary 333 and Theorem 341 that if the map p of (G, p) R (G,S) is injective, then the unique type Φ of (G, p) is a group homomorphism Theorem 342 Let S be a symmetry group, Φ : S Aut(G) be a map, and (G, p) be a framework in R (G,S,Φ) (i) If Φ is a homomorphism, then Φ(S) is a subgroup of Aut(G); (ii) if Φ(S) is a subgroup of Aut(G) and Φ(x) = Φ(y) whenever Φ(y) Φ(x)Aut(G, p), then Φ is a homomorphism Proof (i) It is a standard result in algebra that the homomorphic image of a group is again a group (ii) Let x and y be any two elements of S By the same argument as in the proof of Theorem 341, we have ( )( ) ( (Φ(y) ) ) y x p(v) = p Φ(x) (v) for all v V (G) It follows from Theorem 331 that Φ(y x) ( Φ(y) Φ(x) ) Aut(G, p) By assumption, Φ(S) contains at most one element of each of the cosets of Aut(G, p) Since Φ(S) is a group, the element of the coset ( Φ(y) 75

101 Φ(x) ) Aut(G, p) that lies in Φ(S) must be Φ(y) Φ(x) It follows that Φ(y x) = Φ(y) Φ(x) and the proof is complete For a framework (G, p) R (G,S) with Aut(G, p) {id}, there does not necessarily exist any homomorphism Φ : S Aut(G) for which (G, p) R (G,S,Φ), as the following examples illustrate v 1 v 4 v 2 v 3 (a) p 1, p 3 p 2, p 4 (b) Figure 39: A graph G (a) and a realization (G, p) R (G,Cs) (b) for which there does not exist a homomorphism Φ : C s Aut(G) so that (G, p) is of type Φ Example 341 Consider the graph G and the 2-dimensional realization (G, p) of G shown in Figure 39 (a) and (b), respectively Let s be the reflection whose mirror line is shown in Figure 39 (b) All vertices of G that are illustrated with the same color in Figure 39 have the same image under p Observe that the 1 -turn-automorphism σ of G that permutes 4 the vertices v 1, v 2, v 3, and v 4 according to the cycle (v 1 v 2 v 3 v 4 ) satisfies s ( p(v i ) ) = p ( σ(v i ) ) for all v i V (G) Thus, s is a symmetry operation of (G, p), and hence (G, p) is an element of R (G,Cs ), where C s = {Id, s} Note that Aut(G, p) = {id, σ 2 } Therefore, by Theorem 331, id and σ 76

102 are the two automorphisms of G that can turn Id C s into a symmetry operation of (G, p) Similarly, either one of the elements of σaut(g, p) = {σ, σ 3 } can turn s C s into a symmetry operation of (G, p) It now follows from Theorem 342 (i) that there does not exist any homomorphism Φ : C s Aut(G) such that (G, p) R (G,Cs ) is of type Φ, because we cannot choose two elements, one from each of the cosets Aut(G, p) and σaut(g, p), that form a subgroup of Aut(G) v 2 v 1 v 3 v 4 v 5 v 6 (a) v 9 v 8 v7 p 1, p 4, p 7 p 2, p 5, p 8 p 3, p 6, p 9 (b) Figure 310: A graph G (a) and a realization (G, p) R (G,C3 ) (b) for which there does not exist a homomorphism Φ : C 3 Aut(G) so that (G, p) is of type Φ Example 342 Consider the graph G and the 2-dimensional realization (G, p) of G shown in Figure 310 (a) and (b), respectively As in the previous example, all vertices of G that are illustrated with the same color in Figure 310 have the same image under p Note that (G, p) is an element of R (G,C3 ), where C 3 = {Id, C 3, C3} 2 is a symmetry group in dimension 2, because the ( automorphism γ = (v 1 v 2 v 9 ) of G satisfies C 3 p(vi ) ) = p ( γ(v i ) ) for all v i V (G) and γ 2 satisfies C3( 2 p(vi ) ) = p ( γ 2 (v i ) ) for all v i V (G) We have Aut(G, p) = {id, γ 3, γ 6 }, and hence γaut(g, p) = {γ, γ 4, γ 7 } and 77

103 γ 2 Aut(G, p) = {γ 2, γ 5, γ 8 } Since C 3 C 3 has order 3 and each element in γaut(g, p) has order 9 it follows that there does not exist any homomorphism Φ : C 3 Aut(G) such that (G, p) is of type Φ Note that Examples 341 and 342 can easily be extended to obtain further examples of frameworks (G, p) and symmetry groups S with the property that there exists no homomorphism Φ : S Aut(G) for which (G, p) R (G,S,Φ) 78

104 Chapter 4 Using group representation theory to analyze symmetric frameworks It is a common method in engineering, physics, and chemistry to apply techniques from group representation theory to the analysis of symmetric structures (see, for example, [26, 27, 34, 35, 44, 45]) In particular, some recent papers have used these techniques to gain insight into the rigidity properties of symmetric frameworks consisting of rigid bars and flexible joints [15, 25, 43, 44, 53] One of the fundamental observations resulting from this approach for studying the rigidity of symmetric frameworks is due to R Kangwai and S Guest ([44]): given a symmetric framework (G, p) and a non-trivial subgroup S of its point group, there are techniques to block-diagonalize the rigidity matrix of (G, p) into submatrix blocks in such a way that each block corre- 79

105 sponds to an irreducible representation of S A number of interesting and useful results concerning the rigidity of symmetric frameworks are based on this block-diagonalization of the rigidity matrix [15, 25, 43] However, since the main focus of the work in [44], as well as in [15], [25], and [43], lies on applications in engineering and chemistry, many of these results are not presented with a mathematically precise formulation nor with a complete mathematical verification In this chapter, we establish several major results First, in Section 41, we use the mathematical foundation we established in the previous chapter to give a complete proof for the fact that the rigidity matrix of a symmetric framework can be block-diagonalized in the way described above Fundamental to this proof are our mathematically explicit definitions for the external and internal representation which were introduced in [25] and [44] only by means of an example, and Lemma 411 which establishes the key connection between these two representations Secondly, in Section 42, we apply the results of Section 41 to give a detailed mathematical proof for the symmetry-extended version of Maxwell s rule given in [25] This rule provides further necessary conditions (in addition to Maxwell s original condition given in Theorem 227) for a symmetric framework to be isostatic While the symmetry-extended version of Maxwell s rule, as formulated in [25], is only applicable to 2- or 3-dimensional frameworks with injective configurations, we establish a more general result in Section 42, namely a rule that can be applied to both injective and noninjective realizations in all dimensions The proof of this result is based on Theorem 422 which in turn relies on the fact that the rigidity matrix of a 80

106 symmetric framework can be block-diagonalized as described in Section 41 An alternate approach to proving the symmetry-extended version of Maxwell s rule in [25], as well as various generalizations of this rule to other types of geometric constraint systems, is presented in [53] (see also Chapter 7) In order to apply the symmetry-extended version of Maxwell s rule to a given framework (G, p), it is necessary to determine the dimensions of the subspaces of infinitesimal rigid motions of (G, p) that are invariant under the external representation While in [25], the question of how to find the dimensions of these subspaces is only briefly addressed and not answered completely from a mathematical point of view (in particular, for all frameworks in dimensions higher than 3, this question is not addressed at all), in Section 42, we describe in detail how to determine the dimensions of these subspaces for an arbitrary-dimensional framework The results of Sections 41 and 42 will also be presented in the paper [56] Since in [25] and [44], the rigidity properties of a symmetric framework are studied from both the kinematic and static point of view simultaneously, we develop the corresponding mathematical theory in this chapter in the same manner In Section 43, we use the symmetry-extended version of Maxwell s rule to show that a symmetric isostatic framework in 2D or 3D must obey some very simply stated restrictions on the number of structural elements that are fixed by various symmetry operations of the framework In particular, it turns out that a 2-dimensional isostatic framework must belong to one 81

107 of only six point groups For 3-dimensional isostatic frameworks, all point groups are possible However, there still exist restrictions on the placement of structural components While analogous restrictions on the number of fixed structural components can be established for symmetric frameworks in an arbitrary dimension using the results of Section 42, we focus our attention on frameworks in dimensions 2 and 3, since they are of special interest for current applications Most of the results in Section 43 appeared in the joint paper [15] The derivations also appeared there, and Sections 41 and 42 now provide a proof that these methods are correct Finally, in Section 44 we use the results of Sections 41 and 42 to establish necessary conditions for a symmetric framework to be independent or infinitesimally rigid 41 Block-diagonalization of the rigidity matrix 411 Basic definitions in group representation theory We need the following notions from group representation theory Definition 411 Let S be a group and V be an n-dimensional vector space over the field F A linear representation of S with representation space V is a group homomorphism H from S to GL(V ), where GL(V ) denotes the group of all automorphisms of V The dimension n of V is called the degree of H 82

108 Two linear representations H 1 : S GL(V 1 ) and H 2 : S GL(V 2 ) are said to be equivalent if there exists an isomorphism h : V 1 V 2 such that h H 1 (x) h 1 = H 2 (x) for all x S Definition 412 Let S be a group, V be a vector space over the field F and H : S GL(V ) be a linear representation of S A subspace U of V is said to be H-invariant (or simply invariant if H is clear from the context) if H(x)(U) U for all x S H is called irreducible if V and {0} are the only H-invariant subspaces of V Note that the property of irreducibility depends on the field F Since we only consider frameworks in the real vector space R d, the representation space of any linear representation in this thesis is assumed to be a real vector space In the examples throughout this thesis we use the Mulliken symbols (see Appendix A or [6, 19, 37]) to denote the irreducible representations of a given group This is one of the standard notations in group representation theory and its applications Definition 413 A linear representation H : S GL(V ) is said to be unitary with respect to a given inner product v, w if H(x)(v), H(x)(w) = v, w for all v, w V and all x S Remark 411 A unitary representation has the property that the orthogonal complement of an invariant subspace is again invariant [60] Definition 414 Let H : S GL(V ) be a linear representation of a group S and let U be an invariant subspace of V If for all x S, we restrict the 83

109 automorphism H(x) of V to the subspace U, then we obtain a new linear representation H (U) of S with representation space U H (U) is said to be a subrepresentation of H Definition 415 Let H 1 : S GL(V 1 ) and H 2 : S GL(V 2 ) be two linear representations of a group S Then H 1 H 2 : S GL(V 1 V 2 ) is the representation of S which sends x S to H 1 H 2 (x), where H 1 H 2 (x) ( (v 1, v 2 ) ) = ( H 1 (x)(v 1 ), H 2 (x)(v 2 ) ) for all v 1 V 1 and v 2 V 2 Definition 416 Let S be a group and F be a field A matrix representation of S is a homomorphism H from S to GL(n, F ), where GL(n, F ) denotes the group of all invertible n n matrices with entries in F Two matrix representations H 1 : S GL(n, F ) and H 2 : S GL(n, F ) are said to be equivalent if there exists an invertible matrix M such that MH 1 (x)m 1 = H 2 (x) for all x S, in which case we write H 1 H 2 Let S be a group, V be an n-dimensional vector space over the field F, and H : S GL(V ) be a linear representation of S Given a basis B of V, we may associate a matrix representation H B : S GL(n, F ) to H by defining H B (x) to be the matrix that represents the automorphism H(x) with respect to the basis B for all x S H B is then said to correspond to H with respect to B Note that two matrix representations H 1 and H 2 correspond to equivalent linear representations if and only if H 1 H 2 84

110 412 The internal and external representation Given a graph G, a symmetry group S, and a homomorphism Φ : S Aut(G), we define two particular matrix representations of S, the external and the internal representation, both of which depend on G and Φ These two representations play the key role in a symmetry-based rigidity analysis of a framework (G, p) R (G,S,Φ) Note that our definitions of these representations are mathematically explicit definitions of the external and internal representation introduced in [25] and [44] Definition 417 Let G be a graph with V (G) = {v 1, v 2,, v n } and E(G) = {e 1, e 2,, e m }, S be a symmetry group in dimension d, and Φ be a homomorphism from S to Aut(G) For x S, let M x denote the orthogonal d d matrix which represents x with respect to the canonical basis of R d The external representation of S (with respect to G and Φ) is the matrix representation H e : S GL(dn, R) that sends x S to the matrix H e (x) which is obtained from the transpose of the n n permutation matrix corresponding to Φ(x) (with respect to the enumeration V (G) = {v 1, v 2,, v n }) by replacing each 1 with the matrix M x and each 0 with a d d zero-matrix The internal representation of S (with respect to G and Φ) is the matrix representation H i : S GL(m, R) that sends x S to the transpose of the permutation matrix corresponding to the permutation of E(G) (with respect to the enumeration E(G) = {e 1, e 2,, e m }) which is induced by Φ(x) Remark 412 It is easy to verify that both the external representation H e and the internal representation H i of S (with respect to G and Φ) are in fact 85

111 matrix representations of the group S, provided that Φ is a homomorphism If, however, Φ is not a homomorphism, then H e and H i are also not homomorphisms, in which case neither H e nor H i is a matrix representation of the group S Example 411 To illustrate the previous definition, let K 3 be the complete graph with V (K 3 ) = {v 1, v 2, v 3 } and E(K 3 ) = {e 1, e 2, e 3 }, where e 1 = {v 1, v 2 }, e 2 = {v 1, v 3 } and e 3 = {v 2, v 3 } Further, let C s = {Id, s} be the symmetry group in dimension 2 with M Id = 1 0 and M s = , and let Φ : C s Aut(K 3 ) be the homomorphism defined by Φ(s) = (v 1 v 2 )(v 3 ) Then we have H e (Id) = , H e (s) = , H i (Id) = , H i(s) = For further examples, see [44] or [45] 86

112 p 3 e 2 e 3 p 1 e 1 p2 Figure 41: A framework (K 3, p) R (K3,C s,φ) 413 The block-diagonalization In this section, we use the mathematically explicit definitions of the external and internal representation from the previous section to prove that the rigidity matrix of a symmetric framework can be transformed into a blockdiagonalized form Basic to this proof is Lemma 411 which discloses the essential mathematical connection between the external and internal representation Recall from Section 22 that in the study of infinitesimal rigidity, we consider the equation R(G, p)u = z, where R(G, p) is the rigidity matrix of a framework (G, p), u R d V (G) is a column vector that represents an assignment of d-dimensional displacement vectors to the joints of (G, p), and z R E(G) is the column vector that represents the distortions in the bars of (G, p) that are induced by u The component of z that corresponds to the edge {v i, v j } of G is also known as the strain induced on the bar {(v i, p i ), (v j, p j )} by u 87

113 Similarly, in the study of static rigidity, we consider the equation R(G, p) T ω = l, where the column vector ω R E(G) is a stress of (G, p) and the column vector l R d V (G) is the load on (G, p) which is resolved by ω Now, suppose (G, p) is a symmetric framework in the set R (G,S,Φ), where S is a symmetry group in dimension d and Φ : S Aut(G) is a homomorphism Then, using the notation of Definition 417, and assuming that the ith row of the rigidity matrix R(G, p) of (G, p) corresponds to the edge e i of G, we have the following fundamental property of the external and internal representation of S (with respect to G and Φ) Lemma 411 Let G be a graph, S be a symmetry group, Φ be a homomorphism from S to Aut(G), and p x S L x,φ (i) If R(G, p)u = z, then for all x S, we have R(G, p)h e (x)u = H i (x)z; (ii) if R(G, p) T ω = l, then for all x S, we have R(G, p) T H i (x)ω = H e (x)l Proof (i) Suppose R(G, p)u = z Fix x S and let M x be the orthogonal matrix representing x with respect to the canonical basis of R d Also, let Φ(x)(v i ) = v k and Φ(x)(v j ) = v l, and let e f = {v i, v j } and e h = {v k, v l } Then, since p x S L x,φ, we have M x p i = p k and M x p j = p l By the definition of H i (x), we have ( Hi (x)z ) = (z) h f 88

114 p j M x M x p j = p l e f e h p i M x M x p i = p k H e (x) u l M x u j H i (x) (z) h (z) f H u k e (x) M x u i Figure 42: Illustration of the proof of Lemma 411 (i) Similarly, it follows from the definition of H e (x) that if u R dn is replaced by H e (x)u, then u k R d is replaced by M x u i and u l R d by M x u j By the definition of R(G, p), we have ( ) R(G, p)u = (z) h h = (p k p l ) u k + (p l p k ) u l Therefore, ( R(G, p)he (x)u ) h = (p k p l ) M x u i + (p l p k ) M x u j = ( M x p i M x p j ) Mx u i + ( M x p j M x p i ) Mx u j = ( M x (p i p j ) ) M x u i + ( M x (p j p i ) ) M x u j = (p i p j ) u i + (p j p i ) u j = (z) f The penultimate equality sign is valid because the canonical inner product on R d is invariant under the orthogonal transformation x S This proves (i) (ii) Suppose R(G, p) T ω = l Fix x S and let Φ(x)(v i ) = v k Then, since p x S L x,φ, we have M x p i = p k Let v i1, v i2,, v ij be the vertices in V (G) that are adjacent to v i, and let e ft = {v i, v it } for t = 1, 2,, j Further, choose an enumeration of the j 89

115 (z) f p j u j u l M xp j = p l (z) h (z) h p j M x u j p l M x u l (z) f p i u i x M x p i = p k u k p i M x u k M x u i x pk Figure 43: Illustration of the proof of Lemma 411 (i) in the case where x is a reflection vertices that are adjacent to v k in such a way that M x p it = p kt, and let e ht = {v k, v kt } for t = 1, 2,, j For the vertex v k, the equation R(G, p) T ω = l yields the vector-equation (p k p k1 )(ω) h1 + + (p k p kj )(ω) hj = l k (41) e f1 p i1 M x M x p i1 = p k1 e h1 (ω) h1 p i M x M x p i = p k l k e fj M x e hj (ω) hj p ij M x p ij = p kj H i (x) (ω) f1 H e (x) H i (x) (ω) fj Mx l i Figure 44: Illustration of the proof of Lemma 411 (ii) If l R dn is replaced by H e (x)l, then on the right-hand side of equation (41), l k R d is replaced by M x l i and if ω is replaced by H i (x)ω, then the 90

116 left-hand side of equation (41) is replaced by (p k p k1 )(ω) f1 + + (p k p kj )(ω) fj = ( M x p i M x p i1 ) (ω)f1 + + ( M x p i M x p ij ) (ω)fj = M x ( (pi p i1 )(ω) f1 + + (p i p ij )(ω) fj ) = M x l i This completes the proof In the following, we again let G be a graph with V (G) = {v 1, v 2,, v n } and E(G) = {e 1, e 2,, e m }, S be a symmetry group in dimension d, and Φ be a homomorphism from S to Aut(G) Let H e be the external and H i be the internal representation of S (with respect to G and Φ) Then we let H e : S GL(R dn ) be the linear representation of S that sends x S to the automorphism H e(x) which is represented by the matrix H e (x) with respect to the canonical basis of the R-vector space R dn Similarly, we let H i : S GL(R m ) be the linear representation of S that sends x S to the automorphism H i(x) which is represented by the matrix H i (x) with respect to the canonical basis of the R-vector space R m So, the external representation H e corresponds to the linear representation H e with respect to the canonical basis of R dn and the internal representation H i corresponds to the linear representation H i with respect to the canonical basis of R m From group representation theory we know that every finite group has, up to equivalency, only finitely many irreducible linear representations and that every linear representation of such a group can be written uniquely, up to equivalency of the direct summands, as a direct sum of the irreducible linear 91

117 representations of this group [42, 60] So, let S have r pairwise non-equivalent irreducible linear representations I 1, I 2,, I r and let H e = λ 1 I 1 λ r I r, where λ 1,, λ r N {0} (42) For each t = 1,, r, there exist λ t subspaces ( V (I t) e )1,, ( V (I ) t) e λ t of the R-vector space R dn which correspond to the λ t direct summands in (42), so that R dn = V (I 1) e V (I r) e, (43) where V (I t) e = ( V (I t) e )1 ( ) V (I t) e λ t (44) Let ( B (I t) e )1,, ( B (I ) t) e be bases of the subspaces in (44) Then λ t B (I t) e = ( B (I t) e )1 ( ) B (I t) e λ t is a basis of V (It) e and B e = B (I 1) e is a basis of the R-vector space R dn B (Ir) e (45) Consider now the matrix representation H e that corresponds to the linear representation H e with respect to the basis B e For x S, we have H e (x) = T 1 e H e (x)t e, where the ith column of T e is the coordinate vector of the ith basis vector of B e relative to the canonical basis, that is, T e is the matrix of the basis transformation from the canonical basis of the R-vector space R dn to the basis B e The column vectors of H e (x) are the coordinates of the images of 92

118 the basis vectors in B e under H e(x) relative to the basis B e So, for each x S, the matrix H e (x) has the same block form, namely ( (I A 1 )) e (x) 1 0 ( (I A 1 )) e λ 1 (x) H e (x) = ( A (I ) r) e (x) 1 0 ( The block-matrix ( A e (It) ) j (x) represents the restriction of the linear trans- )j with respect to the basis ( B (I ) t) formation H e(x) to the subspace ( V (I t) e Since for a given t, each of the subspaces ( V (I t) e A (I ) r) e λ r (x) ) e j j, j = 1,, λ t, corresponds to the same irreducible linear representation I t, we can choose the bases of the subspaces ( V e (It) ) in such a way that j ( A (I t) e )1 (x) = = ( A (It) e )λ t (x) =: A (It) e (x) In the following we assume that the basis B e is chosen in this way The above observations about the linear representation H e of S can be transferred analogously to the linear representation H i of S Let the direct sum decomposition of H i be given by H i = µ 1 I 1 µ r I r, where µ 1,, µ r N {0} (46) For each t = 1,, r, there exist µ t subspaces ( V (It) i )1,, ( V (It) ) i µ t of the R-vector space R m which correspond to the µ t direct summands in (46), so that R m = V (I 1) i V (Ir) i, (47) 93

119 where Let ( B (It) i )1,, ( B (It) i V (I t) i = ( V (I t) i )1 ( V (I ) t) i µ t (48) ) µ t be bases of the subspaces in (48) Then B (It) i = ( B (It) i )1 ( B (It) ) i µ t is a basis of V (I t) i and B i = B (I 1) i B (I r) i is a basis of the R-vector space R m Consider now the matrix representation H i that corresponds to the linear representation H i with respect to the basis B i Let T i be the matrix of the basis transformation from the canonical basis of the R-vector space R m to the basis B i Then for x S, we have H i (x) = T 1 i H i (x)t i So, the matrix H i (x) has the same block form for each x S, namely ( (I A 1 )) i (x) 1 0 ( (I A 1 )) i µ 1 (x) H i (x) =, ( A (I ) r) i (x) 1 0 ( A (Ir) ) i µ r (x) and for each t = 1, 2,, r, we can choose the bases of the subspaces ( V (It) ) i in such a way that ( A (I t ) i )1 (x) = = ( A (I t) i )µ t (x) =: A (I t) i 94 (x) = A (It) e (x) j

120 In the following we assume that B i is chosen in this way Definition 418 With the notation above, we say that a vector v R dn is symmetric with respect to the irreducible linear representation I t of S if v V (I t) e Similarly, we say that a vector w R m is symmetric with respect to the irreducible linear representation I t of S if w V (I t) i We are now in the position to state the fundamental theorem for analyzing the rigidity properties of a symmetric framework using group representation theory Theorem 412 Let G be a graph, S be a symmetry group with pairwise nonequivalent irreducible linear representations I 1,, I r, Φ be a homomorphism from S to Aut(G), and p x S L x,φ (i) If R(G, p)u = z and u is symmetric with respect to I t, then z is also symmetric with respect to I t ; (ii) if R(G, p) T ω = l and ω is symmetric with respect to I t, then l is also symmetric with respect to I t Proof (i) Suppose S is a symmetry group in dimension d and G is a graph with n vertices Let u ( V (I t) e )j By the direct sum decomposition of V (I t) e in (44), the result follows if we can show that z = R(G, p)u V (I t) i By the decomposition of R E(G) into direct summands in (48), z has a unique decomposition of the form z = r µ α α=1 β=1 z α,β, where z α,β ( V (I ) α) i β 95

121 We now interpret R(G, p) : R dn R E(G) as a linear transformation and for given m and k, we define the projection map R m,k R(G, p) ( ) V (I t e ) j by R m,k : ( V (I t ) e )j ( V (I m) i k u z m,k ) corresponding to We need to show that for all m t, R m,k is the zero map So, let m t Clearly, R m,k is a linear transformation The image of R m,k is an H i-invariant subspace of ( V (I m) i ) k, as the following argument shows Fix x S and let z be in the image of R m,k, say z = R m,k (u ) Then, by assumption, H e(x)(u ) ( V e (It) ) and, by Lemma 411 (i), H i(x)(z ) is the image of H e(x)(u ) under R m,k Since I m is an irreducible linear representation of S, ( V (I ) m) i and {0} k are the only H i-invariant subspaces of ( V (Im) ) i If the image of R k m,k is the null-space, then we are done, otherwise R m,k is surjective Next, we show that the kernel of R m,k is an H e-invariant subspace of ( (I V t )) e Fix x S and let j u be in the kernel of R m,k, that is, R m,k (u ) = 0 Then, again by Lemma 411 (i), the image of H e(x)(u ) under R m,k H i(x)(0) = 0, and hence H e(x)(u ) is also in the kernel of R m,k Since I t is an irreducible linear representation of S, we either have ker (R m,k ) = ( V e (It) ), in which case we are done, or ker (R j m,k) = {0}, in which case R m,k is injective So, assume R m,k is bijective Let the matrix that represents R m,k with respect to the bases ( B e (It) )j and ( B (Im) ) i be denoted by R k m,k Then R m,k is an invertible matrix Let ũ be the coordinate vector of an element in ( V e (It) ) j relative to the basis ( B (I ) t) e and let z be the coordinate vector of the image j 96 j is

122 of ũ under R m,k relative to the basis ( B (Im) ) i Then, by Lemma 411 (i), k for any x S, we have R m,k ( A (I t ) e )j (x)ũ = ( A (Im) i )k (x) z = ( A (Im) i ) k (x) R m,k ũ, and hence also R m,k ( A (I t ) e )j (x) = ( A (I ) m) i (x) R k m,k Therefore, ( R m,k A (I t) 1 e (x) R )j m,k = ( A (I m) i )k (x) = ( ) A (Im) e (x) for all x S, k which says that I t and I m are equivalent representations, a contradiction This completes the proof of part (i) With the help of Lemma 411 (ii), part (ii) can be proved completely analogously to part (i) Theorem 412 (i) says that if u R dn is an assignment of displacement vectors to the joints of a framework (G, p) R (G,S,Φ) and u is symmetric with respect to I t, then the strains induced on the bars of (G, p) by u must also be symmetric with respect to I t Similarly, Theorem 412 (ii) says that if ω is a resolution of an equilibrium load l on (G, p) R (G,S,Φ) and ω is symmetric with respect to I t, then l must also be symmetric with respect to I t An immediate consequence of Theorem 412 is that the matrices R(G, p) and R(G, p) T can be block-diagonalized in such a way that the original rigidity problems R(G, p)u = z and R(G, p) T ω = l are decomposed into subproblems, where each subproblem considers, respectively, the relationship between vectors u and z and vectors ω and l that are symmetric with respect to the same irreducible linear representation I t This is specified in 97

123 Corollary 413 Let G be a graph, S be a symmetry group with pairwise non-equivalent irreducible linear representations I 1,, I r, Φ be a homomorphism from S to Aut(G), and p x S L x,φ Then the matrices T 1 i R(G, p)t e and Te 1 R(G, p) T T i are block-diagonalized in such a way that there exists (at most) one submatrix block for each irreducible linear representation I t of S Proof Suppose R(G, p)u = z, and let ũ be the coordinate vector of u relative to the basis B e and z be the coordinate vector of z relative to the basis B i Further, let R(G, p) be the matrix that represents the linear transformation R(G, p) with respect to the bases B e and B i, that is, R(G, p) = T 1 i R(G, p)t e Then, by changing coordinates relative to the canonical bases of R dn and R m into coordinates relative to the bases B e and B i, the equation R(G, p)u = z is converted into the equation R(G, p)ũ = z By Theorem 412 (i), the matrix R(G, p) is block-diagonalized in such a way that there exists (at most) one submatrix block for each irreducible linear representation I t of S and the submatrix block corresponding to I t is a matrix of the size dim ( V (I ) ( t) (I i dim V t )) e In particular, a submatrix block can possibly be an empty matrix which has rows but no columns or alternatively columns but no rows 98

124 Similarly, if we denote ω to be the coordinate vector of ω relative to the basis B i, l to be the coordinate vector of l relative to the basis B e, and R(G, p) T = T 1 e R(G, p) T T i, then we may carry out the same changes of coordinates as above to convert the equation R(G, p) T ω = l into the equation R(G, p) T ω = l By Theorem 412 (ii), the matrix R(G, p) T is again block-diagonalized in such a way that there exists (at most) one block for each I t Remark 413 Note that the matrix R(G, p) T is equal to the transpose of the matrix R(G, p) if and only if both of the matrices T e and T i are orthogonal matrices (ie, T 1 e = T T e and T 1 i = T T i ) if and only if both B e and B i are orthonormal bases Since the external and internal representation are both unitary representations (for all x S, H e (x) and H i (x) are orthogonal matrices), the invariant subspaces in (43) and (47) are mutually orthogonal (see [24, 60], for example) Thus, B e and B i can always be chosen to be orthonormal Example 412 Let K 3, C s = {Id, s}, and Φ be as in Example 411 and consider the framework (K 3, p) R (K3,C s,φ) shown in Figures 41 and 45, where p 1 = 1 0, p 2 = 1 0, and p 3 =

125 The rigidity matrix of (K 3, p) is given by (p 1 p 2 ) 1 (p 1 p 2 ) 2 (p 2 p 1 ) 1 (p 2 p 1 ) R(K 3, p) = (p 1 p 3 ) 1 (p 1 p 3 ) (p 3 p 1 ) 1 (p 3 p 1 ) (p 2 p 3 ) 1 (p 2 p 3 ) 2 (p 3 p 2 ) 1 (p 3 p 2 ) = The symmetry group C s has two non-equivalent irreducible linear representations both of which are of degree 1 In the Mulliken notation, they are denoted by A and A A maps both Id and s to the identity transformation, whereas A maps Id to the identity transformation and s to the linear transformation A (s) which is defined by A (s)(x) = x for all x R We have and R 6 = V (A ) e V (A ) e R 3 = V (A ) i V (A ) i It is easy to see that the elements of the subspace V (A ) e u 1 u 2 u 1, where u 1, u 2, u 3 R, u 2 0 u of R 6 are of the form

126 (see Figure 45 (a)), so that an orthonormal basis B (A ) e of V (A ) e is given by B (A ) e = , , Similarly, the elements of the subspace V (A ) e of R 6 are of the form u 1 u 2 u 1 u 2 u 3 0, where u 1, u 2, u 3 R, (see Figure 45 (b)), so that an orthonormal basis B (A ) e of V (A ) e is given by B (A ) e = , , Orthonormal bases B (A ) i and B (A ) i for the subspaces V (A ) i and V (A ) i of R 3 101

127 can be found analogously (see Figure 45 (c), (d)) We let B (A ) i = 1 0 0, and B (A ) i = Therefore, we have p 3 ( ) 0 u 3 ( u3 0 ) p 3 ( u1 u 2 ) e 2 e 3 p 1 e 1 p2 ( ) u1 u 2 ( u1 u 2 ) e 2 e 3 p 1 e 1 p2 ( u1 u 2 ) (a) (b) p 3 p 3 e 2 e 3 z 2 z 2 p 1 e 1 z 1 p2 (c) e 2 e 3 z 1 z 1 p 1 e 1 0 p 2 (d) Figure 45: (a, b) Vectors of the H e-invariant subspaces V (A ) e (a) and V (A ) e (b) of R 6 ; (c, d) vectors of the H i-invariant subspaces V (A ) i (c) and V (A ) i (d) of R 3 102

128 T e = and Thus, and R(K 3, p) = T 1 i R(K 3, p)t e = T i = R(K 3, p) T = T 1 e R(K 3, p) T T i = Remark 414 In the previous example, we were able to find the invariant subspaces V (A ) e, V (A ) e of R 6 and V (A ) i, V (A ) i of R 3 by inspection because C s is a small symmetry group with only two elements This is of course generally not possible There are, however, some standard methods and algorithms for finding the symmetry adapted bases B e and B i for any given symmetry group Good sources for these methods are [24, 50], for example As we will see in Section 42, knowledge of only the sizes of the submatrix blocks that appear in the block-diagonalized rigidity matrices of a given symmetric framework allows us to gain significant insight into the rigidity properties of the framework Since, with the aid of character theory, the 103

129 sizes of these submatrix blocks can be determined very easily without explicitly finding the bases B e and B i, there exist a number of applications of Corollary 413 (such as the symmetry-extended version of Maxwell s rule we will discuss in the following sections) that do not require finding the blockdiagonalized rigidity matrices explicitly Remark 415 The matrices R(G, p) T R(G, p) and R(G, p)r(g, p) T are also of interest in some areas of rigidity theory [17, 44] In structural engineering, these matrices are called the stiffness matrix and the flexibility matrix, respectively It follows immediately from Corollary 413 that if p x S L x,φ, then these matrices can also be block-diagonalized in such a way that there exists (at most) one block for each irreducible representation I t of S In fact, it is easy to see that the matrices Te 1 R(G, p) T R(G, p)t e and T 1 i R(G, p)r(g, p) T T i have the desired block-form In the following sections of this chapter, as well as in Chapter 6, we will discuss some interesting applications of the fact that the rigidity matrix of a symmetric framework can be block-diagonalized in the way described in Corollary A symmetry-extended version of Maxwell s rule Recall from Section 225 that Maxwell s rule (Theorem 227) gives a necessary condition for a d-dimensional framework (G, p) to be isostatic If 104

130 (G, p) is a 2- or 3-dimensional symmetric framework with an injective configuration, then the symmetry-extended version of Maxwell s rule given in [25] provides further necessary conditions for (G, p) to be isostatic Though the rule in [25] is a useful tool for engineers and chemists to analyze the rigidity properties of symmetric structures in 2D and 3D, it is unsatisfactory from a mathematical point of view since it cannot be applied to frameworks in dimensions higher than 3 and since a complete mathematical proof of this result has not been provided In the following sections, we aim to give a mathematical proof not only for the rule in [25], but also for an extended rule that can be applied to a symmetric framework with a possibly non-injective configuration in an arbitrary dimension In this section, we first develop all the necessary mathematical background that was omitted in [25] This background consists of three major parts First, we show that the subspaces R and T of all rotational and translational infinitesimal rigid motions of a given symmetric framework (G, p) are invariant under the external representation H e (see Lemma 421) This allows us to define subrepresentations of H e for the subspaces R and T We then prove that the block-diagonalized form of the rigidity matrix of (G, p) gives rise to additional necessary conditions for (G, p) to be isostatic (see Theorem 422) The symmetry-extended version of Maxwell s rule is based on these conditions Finally, we describe in detail how to determine the dimensions of the H e-invariant subspaces of R and T This is essential in applying the symmetry-extended version of Maxwell s rule to a given symmetric framework Using some basic techniques from character theory, all the results of this 105

131 section combined will allow us to formulate the symmetry-extended version of Maxwell s rule given in [25] (as well as its extension to higher dimensions) as a mathematical theorem in Section 422 For the remainder of this chapter, we will continue to use the notation of the previous section 421 The necessary conditions As before, we let G be a graph, S be a symmetry group in dimension d with pairwise non-equivalent irreducible linear representations I 1,, I r, Φ be a homomorphism from S to Aut(G), and (G, p) be a framework in R (G,S,Φ) In this section, we make the additional assumption that the points p(v), v V (G), span all of R d Recall from the previous section that we have the decomposition R dn = V (I 1) e V (I r) e (49) with V (It) e = ( V (It) e )1 ( ) V e (It) λ t (410) of R dn into H e-invariant subspaces While the scalars λ t (as well as the subspaces that appear as direct summands in (49)) are uniquely determined in this decomposition, the subspaces that appear as direct summands in (410) are not [60] In order to derive the symmetry-extended version of Maxwell s rule, the subspaces in (410) shall now be chosen appropriately Since the points p(v), v V (G), span all of R d, the subspace N = 106

132 ker ( R(K n, p) ) of R dn, where K n is the complete graph on V (G), is the space consisting of all infinitesimal rigid motions of (G, p) This space can be written as the direct sum N = T R, where T is the space of all translational and R is the space of all rotational infinitesimal rigid motions of (G, p) More precisely, a basis of T is given by {T j j = 1,, d}, where for j = 1,, d, T j : V (G) R d is the map that sends each v V (G) to the jth canonical basis vector e j of R d, and a basis of R is given by {R ij 1 i < j d}, where for 1 i < j d, R ij : V (G) R d is the map defined by R ij (v k ) = (p k ) i e j (p k ) j e i for all k = 1,, n [81] Each of the maps T j and R ij is of course identified with a vector in R dn (by using the order on V (G)) Note that in the context of static rigidity, T is the space of all translational loads and R is the space of all rotational loads on (G, p) Using the notation of the previous paragraph we have the following result Lemma 421 For every dimension d, the subspaces T, R, and N of R dn are H e-invariant Proof Fix a dimension d We show first that N = ker ( R(K n, p) ) is H e-invariant Since p x S L x,φ, it follows from Lemma 411 that if R(K n, p)u = z, then for all x S, we have R(K n, p)h e (x)u = Ĥi(x)z, (411) where Ĥi is the internal representation of S with respect to K n and Φ Let u N, ie, R(K n, p)u = 0 Then for any x S, we have Ĥ i (x)r(k n, p)u = Ĥi(x)0 = 0 107

133 By (411), we have Ĥi(x)R(K n, p)u = R(K n, p)h e (x)u, and hence R(K n, p)h e (x)u = 0 Thus, for all x S, H e (x)u ker ( R(K n, p) ), which says that N is H e- invariant Next, we show that T is also H e-invariant Let x S and let, as usual, M x denote the orthogonal matrix that represents x with respect to the canonical basis of R d Then for j = 1,, d, we have M x e j H e (x)t j = = (M x) 1j T (M x ) dj T d M x e j It follows that T is H e-invariant It remains to show that R is H e-invariant Since for all x S, H e (x) is an orthogonal matrix, H e is a unitary representation (with respect to the canonical inner product on R dn ) Therefore, the subrepresentation H (N) e of H e with representation space N is also unitary (with respect to the inner product obtained by restricting the canonical inner product on R dn to N) So, by Remark 411, it suffices to show that R is the orthogonal complement of T in N Let t be any element of T and r be any element of R Then w t = for some w Rd w 108

134 and r = V p 1 for some skew-symmetric matrix V V p n Since the point n i=1 p i must be fixed by every symmetry operation x S, we may wlog define an origin so that n i=1 p i = 0 Then the inner product of t and r is given by This gives the result t r = n w T V p i i=1 = w T V n p i = 0 Since, by Lemma 421, N is an H e-invariant subspace of R dn, it follows from Maschke s Theorem (see [42, 51, 60], for example) that N has an i=1 H e-invariant complement Q in R dn We may therefore form the subrepresentation H (Q) e of H e with representation space Q Since H (Q) e of irreducible linear representations of S, say is a direct sum H (Q) e = κ 1 I 1 κ r I r, where κ 1,, κ r N {0}, (412) we obtain, analogously to (410), a decomposition of Q of the form Q = V (I 1) Q V (Ir) Q, where V (I t) Q = ( V (I t) Q )1 ( V (I ) t) Q κ t (413) Similarly, since both T and R are also H e-invariant subspaces of R dn, we may form the subrepresentations H (T ) e and H (R) e of H e with respective representation spaces T and R This gives rise to a decomposition of T of the 109

135 form T = V (I 1) T V (I r) T, where V (I t) T = ( V (I t) T )1 ( V (I ) t) T θ t, and to a decomposition of R of the form R = V (I 1) R V (I r) R, where V (It) R = ( V (It) R )1 ( V (It) ) R ρ t We can now choose the decomposition in (410) in such a way that V (It) e = V (I t) Q V (I t) T V (I t) R (414) In the following we assume that the subspaces ( V (I ) t) e are chosen in this way j We are now in the position to derive the necessary conditions for (G, p) R (G,S,Φ) to be isostatic upon which the symmetry-extended version of Maxwell s rule is based Theorem 422 Let G be a graph, S be a symmetry group in dimension d with pairwise non-equivalent irreducible linear representations I 1,, I r, and Φ : S Aut(G) be a homomorphism If (G, p) is an isostatic framework in R (G,S,Φ) with the property that the points p(v), v V (G), span all of R d, then for t = 1, 2,, r, we have dim ( V (It) Q ) ( (I = dim V t) ) (415) i 110

136 Proof Suppose first that dim ( V (It) Q ) ( (I > dim V t) ) for some t In this case we give two separate arguments to show that (G, p) is not isostatic, one that is based on infinitesimal rigidity and another one that is based on static rigidity This will later allow us to obtain information about both kinematic and static rigidity properties of symmetric frameworks with the symmetryextended version of Maxwell s rule Since dim ( V (I ) ( t) (I Q > dim V t )) i, it follows from Corollary 413 that there exists an element u 0 in V (It) Q i that lies in the kernel of the linear transformation which is represented by the matrix R(G, p) with respect to the bases B e and B i In other words, u is an infinitesimal flex of (G, p) (which is symmetric with respect to I t ), and hence (G, p) is not isostatic Alternatively, it follows from Corollary 413 that there exists an element l in V (I t) Q that does not lie in the image of the linear transformation which is represented by the matrix R(G, p) T with respect to the bases B e and B i This says that l is an unresolvable equilibrium load on (G, p) (which is symmetric with respect to I t ), so that we may again conclude that (G, p) is not isostatic Suppose now that dim ( V (It) Q ) ( (I < dim V t) ) for some t Then, analogously as above, there exists an element ω 0 in V (I t) i i that lies in the kernel of the linear transformation which is represented by the matrix R(G, p) T with respect to the bases B e and B i This says that ω is a non-zero self-stress of (G, p) (which is symmetric with respect to I t ) So, it again follows that (G, p) is not isostatic Example 421 Recall from Example 412 that for the framework (K 3, p) 111

137 R (K3,C s,φ) shown in Figure 46, we have dim ( ) V (A ) e = 3 dim ( V (A )) i = 2 dim ( V (A ) e dim ( V (A ) i ) = 3 ) = 1 It is easy to see that the 2-dimensional space T of all translational infinitesimal rigid motions of (K 3, p) can be written as the direct sum where V (A ) T T = V (A ) T V (A ) T, is the space of dimension 1 generated by the infinitesimal rigid motion shown in Figure 46 (a), and V (A ) T is the space of dimension 1 generated by the infinitesimal rigid motion shown in Figure 46 (b) Moreover, the 1-dimensional space R of rotational infinitesimal rigid motions of (K 3, p) is clearly generated by the infinitesimal rigid motion shown in Figure 46 (c), so that R = V (A ) R and dim ( V (A )) R = 0 It follows from equation (414) that p 3 p 3 p 3 p 1 p2 (a) p 1 p2 (b) p 1 p2 (c) Figure 46: (a) A basis for the subspace V (A ) T ; (b) a basis for the subspace V (A ) T ; (c) a basis for the subspace R = V (A ) R 112

138 dim ( V (A ) Q and dim ( V (A ) Q ) = dim ( V (A ) e ) = dim ( V (A ) e ) dim ( V (A ) T ) dim ( V (A ) T ) dim ( V (A ) R ) dim ( V (A ) R ) = dim ( V (A ) i ) = dim ( V (A ) i ) = 2 ) = 1, so that the conditions (415) in Theorem 422 are satisfied for the isostatic framework (K 3, p) In general, finding the dimensions of the subspaces V (I t) Q and V (I t) i inspection is not as easy as in the previous example In the following, we therefore describe a systematic method, based on techniques from character theory, for determining the dimensions of these subspaces, so that we can apply Theorem 422 to a symmetric framework with an arbitrary point group in any dimension We begin by introducing the necessary vocabulary by Definition 421 Let A = (a ij ) be an n n square matrix The trace of A is defined to be T r(a) = n i=1 a ii It is an important and well-known fact that the trace of a matrix is invariant under a similarity transformation [19, 37] This gives rise to Definition 422 Let H : S GL(V ) be a linear representation of a group S, B be a basis of V, and H B be the matrix representation that corresponds to H with respect to B The character χ(h) of H is the function χ(h) : S R that sends x S to T r ( H B (x) ) For a fixed enumeration {x 1,, x k } of the elements of the group S, we ( will frequently also refer to the vector T r ( H B (x 1 ) ),, T r ( H B (x k ) )) as the character of H 113

139 In the following we need some well-known results from character theory which we summarize in Theorem 423 [19, 37, 42, 60] Let S be a group with r pairwise nonequivalent irreducible linear representations I 1,, I r and let H : S GL(V ) be a linear representation of S with H = α 1 I 1 α r I r, where α t N {0} for all t = 1,, r (i) If H = H 1 H 2 for some linear representations H 1 and H 2 of S, then χ(h) = χ(h 1 ) + χ(h 2 ); (ii) χ(h) can be written uniquely as a linear combination of the characters χ(i 1 ),, χ(i r ) as χ(h) = α 1 χ(i 1 ) + + α r χ(i r ); (iii) For every t = 1,, r, we have α t = 1 χ(i t ) 2 ( χ(h) χ(it ) ) We first explain how we can determine the dimensions of the subspaces V (It) i for all t = 1,, r It follows from the direct sum decomposition of H i in (46) that for t = 1,, r, the dimension of V (I t) i is the degree of I t multiplied by µ t Since the degree of each irreducible linear representation I t can be read off from the character tables in Appendix A (or, if a more complete list of character tables is required, from the tables in [2, 4, 37]), we only need to determine the values of the µ t This can easily be done by means of the formula given in Theorem 423 (iii), because the characters of the irreducible representations 114

140 I t can simply be read off from the above-mentioned character tables and the character of H i can be found by setting up the internal representation matrices H i (x), x S Finding the dimensions of the subspaces V (It) Q for all t = 1,, r requires a little more work It follows from (414) that for t = 1,, r, we have dim ( V (I t) Q ) = dim ( V (I t) The dimensions of the subspaces V (I t) e e ) dim ( V (I t ) T ) dim ( V (I t ) R can be determined in the analogous way as the dimensions of the subspaces V (It) i : for t = 1,, r, the dimension of the subspace V (I t) e is equal to the degree of I t multiplied by λ t Note that the values of the λ t in (410) can again easily be computed with the help of Theorem 423 (iii) since the character of H e can be found by setting up the external representation matrices H e (x), x S For t = 1,, r, the dimension of the subspace V (I t) T multiplied by θ t and the dimension of the subspace V (I t) R ) is the degree of I t is the degree of I t multiplied by ρ t So, in order to determine the dimensions of the subspaces V (I t) T and V (I t) R with the formula in Theorem 423 (iii), it only remains to determine the characters χ(h e (T ) ) and χ(h e (R) ) We first show how to compute the character χ(h e (T ) ) It follows directly from the proof of Lemma 421 that if S is a symmetry group in dimension d and x S, then the matrix that represents the linear transformation H e (T ) (x) with respect to the basis {T 1,, T d } is the orthogonal matrix M x that represents x with respect to the canonical basis of R d This says that for a fixed enumeration {x 1,, x k } of the elements of S, we have χ(h (T ) e ) = ( T r(m x1 ),, T r(m xk ) ) 115

141 For example, if S is a symmetry group in dimension 2, then the component of χ(h e (T ) ) that corresponds to the identity Id S is equal to 2, each component of χ(h (T ) e C m S is equal to 2 cos ( 2π m ) that corresponds to a rotational symmetry operation ) (T ), and each component of χ(h e ) that corresponds to a reflection s S is equal to 0 (see also Table 41 in Section 432) For a symmetry group S in dimension 3, the explicit values of the components of χ(h (T ) e ) are summarized in Table 42 in Section 433 The character χ(h e (R) ) can be determined similarly As an example, we compute the character χ(h e (R) ) in the case where S is a symmetry group in dimension 2 Every element of S is then either the identity Id, a rotation C m about the origin by an angle of 2π, or a reflection s in a line through the m origin Note that R is a one-dimensional subspace of R 2n a basis of which is given by {R 12 } Let C m be a rotational symmetry operation of (G, p) with M Cm = cos ( ) ( 2π m sin 2π ) m sin ( ) 2π cos ( ) 2π m m Then, by using the definition of the external representation H e of S (with respect to G and Φ) and the fact that (G, p) R (G,S,Φ), it is easy to verify that H e (C m )R 12 = R 12 Similarly, if s is a reflectional symmetry operation of (G, p) with cos (θ) M s = sin (θ) then it is again easy to verify that sin (θ) cos (θ), H e (s)r 12 = R

142 It follows that the matrices which represent the linear transformations H (R) e (Id), H (R) (C m ), and H (R) (s) with respect to the basis {R 12 } are the e e 1 1 matrices (ie, scalars) 1, 1, and 1, respectively Therefore, if d = 2, the character χ(h e (R) ) is the vector defined as follows: each component of χ(h e (R) ) that corresponds to the identity Id S or a rotational symmetry operation C m S is equal to 1, and each component of χ(h e (R) ) that corresponds to a reflection s S is equal to 1 (see also Table 41 in Section 432) Note that analogous calculations as above can easily be carried out for any symmetry group in dimension d > 2 as well For a symmetry group S in dimension 3, the values of the components of χ(h e (R) ) are again summarized in Table 42 in Section 433 Example 422 Let us apply the methods described above to the framework (K 3, p) R (K3,C s,φ) from Example 421 From the representation matrices in Example 411 we immediately deduce that χ(h e) = (6, 0) and χ(h i) = (3, 1) Therefore, if we let H e = λ 1 A λ 2 A H i = µ 1 A µ 2 A, then, by the formula in Theorem 423 (iii), we have λ 1 = 1 ) = 3 2( λ 2 = 1 ) ( 1) = 3 2( µ 1 = 1 ) = 2 2( µ 2 = 1 ) ( 1) = 1 2( 117

143 Further, for the characters χ(h e (T ) ) and χ(h e (R) ), we have, as shown above, χ(h (T ) e ) = (2, 0) and χ(h (R) ) = (1, 1) So, if we let e H (T ) e = θ 1 A θ 2 A H (R) e = ρ 1 A ρ 2 A, then, again by the formula in Theorem 423 (iii), we obtain θ 1 = 1, θ 2 = 1, ρ 1 = 0, and ρ 2 = 1 Since both A and A are linear representations of degree 1, it follows that dim ( V (A ) Q dim ( V (A ) Q ) = dim ( V (A ) e = = 2 ) = dim ( V (A ) e = = 1 ) dim ( V (A ) T ) dim ( V (A ) T ) dim ( V (A ) R ) ) dim ( V (A ) R ) and dim ( V (A )) i = 2 dim ( V (A )) i = The rule Using the mathematical background established in the previous section, we can now prove a symmetry-extended version of Maxwell s rule that can be applied to both injective and non-injective symmetric realizations in any dimension Note that for dimensions 2 and 3, Theorem 424 is a rigorous 118

144 mathematical formulation of the rule given in [25] The condition (416) in Theorem 424 is obtained by combining all of the conditions in (415) into a single equation using some basic techniques from character theory This enables us to check the conditions in (415) with very little computational effort, so that the symmetry-extended version of Maxwell s rule is in the same spirit as Maxwell s original rule in the sense that it only requires a simple count of joints and bars that are fixed by various symmetry operations From now on we will simplify our notation of the previous section by denoting the characters χ(h e), χ(h i), χ(h e (Q) ), χ(h e (T ) ), and χ(h e (R) ) by X e, X i, X Q, X T, and X R, respectively Theorem 424 (Symmetry-extended version of Maxwell s rule) Let G be a graph, S be a symmetry group in dimension d with pairwise nonequivalent irreducible linear representations I 1,, I r, and Φ : S Aut(G) be a homomorphism If (G, p) is an isostatic framework in R (G,S,Φ) with the property that the points p(v), v V (G), span all of R d, then we have X Q = X i (416) Proof Suppose X Q X i Then, by Theorem 423 (ii) and equations (46) and (412), we have κ 1 χ(i 1 ) + + κ r χ(i r ) µ 1 χ(i 1 ) + + µ r χ(i r ), which implies that κ t µ t for some t {1,, r} Therefore, dim ( V (I ) t) Q dim ( V (I ) t) i The result now follows from Theorem

145 So, by checking the condition (416) in Theorem 424, we implicitly check all the conditions in (415) Since we have it follows from Theorem 423 (i) that H e = H e (Q) H e (T ) H e (R), X Q = X e X T X R Note that we have already shown how to compute each of the above characters in the previous section In fact, for dimensions 2 and 3, the characters X T and X R can be read off from Tables 41 and 42 in Section 43 So, in order to check condition (416) for d = 2 or d = 3, it only remains to compute the characters X e and X i So far, our method of determining X e and X i has been to set up the external and internal representation matrices H e (x) and H i (x) for all x S, and then to determine the traces of these matrices In the following, we generalize the method of P Fowler and S Guest presented in [25] to determine the characters X e and X i without explicitly finding the external and internal representation of S This will simplify significantly the calculations required to apply the symmetry-extended version of Maxwell s rule to a given framework Recall from Section 33 that for a framework (G, p) R (G,S), there exists a unique map Φ : S Aut(G) such that (G, p) R (G,S,Φ) if and only if Aut(G, p) = {id} Definition 423 Let G be a graph with V (G) = {v 1,, v n }, S be a symmetry group, Φ be a map from S to Aut(G), (G, p) be a framework in R (G,S,Φ), 120

146 and x S A joint (v i, p i ) of (G, p) is said to be fixed by x with respect to Φ (or simply fixed by x if Aut(G, p) = {id}) if Φ(x)(v i ) = v i Similarly, a bar {(v i, p i ), (v j, p j )} of (G, p) is said to be fixed by x with respect to Φ (or simply fixed by x if Aut(G, p) = {id}) if Φ(x) ( {v i, v j } ) = {v i, v j } The number of joints of (G, p) that are fixed by x (with respect to Φ) is denoted by j Φ(x) and the number of bars of (G, p) that are fixed by x (with respect to Φ) is denoted by b Φ(x) Recall from Definition 417 that for x S, the external representation matrix H e (x) is obtained from the transpose of the permutation matrix corresponding to Φ(x) by replacing each 1 with the d d orthogonal matrix M x and each 0 with a d d zero-matrix Note that the transpose of the permutation matrix corresponding to Φ(x) has a 1 in the diagonal if and only if the corresponding joint of (G, p) is fixed by x with respect to Φ Therefore, a joint of (G, p) can make a contribution to the trace of H e (x) only if it is fixed by x with respect to Φ So, for a fixed enumeration {x 1,, x k } of the elements of S, we have X e = (T r ( H e (x 1 ) ),, T r ( H e (x k ) )) = ( j Φ(x1 )T r(m x1 ),, j Φ(xk )T r(m xk ) ) = X J X T, where X J = (j Φ(x1 ),, j Φ(xk )) and denotes componentwise multiplication Similarly, for x S, the internal representation matrix H i (x) has a 1 in the diagonal if and only if the corresponding bar of (G, p) is fixed by x with 121

147 respect to Φ Thus, X i = (b Φ(x1 ),, b Φ(xk )) So, condition (416) in the symmetry-extended version of Maxwell s rule can be written as X J X T X T X R = X i, (417) and each of the characters in (417) can be determined with very little computational effort In the following we refer to (417) as the symmetry-extended Maxwell s equation Example 423 The symmetry-extended version of Maxwell s rule, applied to the framework (K 3, p) R (K3,C s,φ) from Example 422, yields the counts X J = (j Φ(Id), j Φ(s) ) = (3, 1) X T = (2, 0) X R = (1, 1) X Q = X J X T X T X R = (3, 1) X i = (b Φ(Id), b Φ(s) ) = (3, 1) Thus, condition (416) in Theorem 424 is satisfied for the isostatic framework (K 3, p) Remark 421 Suppose the symmetry-extended version of Maxwell s rule detects that a framework (G, p) R (G,S,Φ) is not isostatic Then we may use Theorem 423 (iii) and the proof of Theorem 422 to obtain information on the symmetry properties of self-stresses of (G, p), infinitesimal flexes of (G, p), and unresolvable equilibrium loads on (G, p) in the following way 122

148 Suppose for the framework (G, p), we have X Q X i Using the formula in Theorem 423 (iii), we may then determine the values of the κ t and µ t in (412) and (46) for all t = 1,, r By the proof of Theorem 424, there must exist t {1,, r} such that κ t µ t It follows from the proof of Theorem 422 that if κ t > µ t, say κ t µ t = k > 0, then there exist k linearly independent infinitesimal flexes of (G, p) which are symmetric with respect to I t, as well as k unresolvable equilibrium loads on (G, p) which are symmetric with respect to I t Similarly, if κ t < µ t, say µ t κ t = k > 0, then there exist k linearly independent self-stresses of (G, p) which are symmetric with respect to I t 423 Example and further remarks To illustrate how the symmetry-extended version of Maxwell s rule can give a significantly improved insight into the rigidity properties of a symmetric framework in comparison to Maxwell s original rule, we consider the framework (K 3,3, p) R (K3,3,C 2v,Φ) from Example 321 The symmetry group C 2v = {Id, C 2, s h, s v } has four non-equivalent irreducible linear representations A 1, A 2, B 1, and B 2 each of which is of degree C 2v Id C 2 s h s v A A B B

149 1 The characters of these representations are shown in the table above (see also Appendix A) We have X J = (j Φ(Id), j Φ(C2 ), j Φ(sh ), j Φ(sv )) = (6, 0, 0, 2) X T = (2, 2, 0, 0) X R = (1, 1, 1, 1) X Q = X J X T X T X R = (9, 1, 1, 1) X i = (b Φ(Id), b Φ(C2 ), b Φ(sh ), b Φ(sv )) = (9, 3, 3, 1) Since X Q X i, we may conclude that (K 3,3, p) is not isostatic Note that Maxwell s original rule would not have detected this because E(K 3,3 ) = 2 V (K 3,3 ) 3 = 9 With the help of the formula in Theorem 423 (iii) we obtain X Q = 3A 1 + 2A 2 + 2B 1 + 2B 2 and X i = 4A 1 + 2A 2 + 2B 1 + B 2, which implies that (K 3,3, p) has a non-zero self-stress which is symmetric with respect to A 1 and an infinitesimal flex (as well as an unresolvable equilibrium load) which is symmetric with respect to B 2 (see Figure 47) Remark 422 Given a framework (G, p) R (G,S), we need to specify a type Φ : S Aut(G) in order to apply the symmetry-extended version of Maxwell s rule (Theorem 424) to (G, p) and S, because Φ determines the characters X e and X i Of course, we also need to make sure that Φ is a homomorphism, for otherwise the external and internal representation (with respect to G and Φ) are not matrix representations of S (see Remark 412) 124

150 p 4 p 1 p 2 p 5 p 6 p 3 s h p 4 p 1 p 2 p 5 p 6 p 3 s h s v (a) s v (b) p 4 p 1 p 2 p 5 p6 p 3 s h s v (c) Figure 47: (a) An infinitesimal flex of (K 3,3, p) R (K3,3,C 2v,Φ) which is symmetric with respect to B 2 (the displacement vector at each joint of (K 3,3, p) remains unchanged under Id and s v and is reversed under C 2 and s h ) (b) An unresolvable equilibrium load on (K 3,3, p) which is symmetric with respect to B 2 (c) A self-stress of (K 3,3, p) which is symmetric with respect to A 1 (the tensions and compressions in the bars of (K 3,3, p) remain unchanged under all symmetry operations in C 2v ) 125

151 Recall that if Aut(G, p) = {id}, then, by Corollary 332, (G, p) R (G,S) is of a unique type Φ and, by Theorem 341, Φ is a homomorphism, so that the external and internal representation are uniquely determined in this case and Theorem 424 can be applied to (G, p) and S in a unique way By Corollary 333, this is in particular the case if (G, p) is a framework whose map p is injective In the following section, we will see that if p is injective, then the characters X e and X i can be found in a particularly easy way (without determining the type Φ) by simply examining the geometric positions of the joints and bars of (G, p) Since in [25] only injective realizations in R 2 and R 3 are considered, Theorem 424 includes the symmetrized version of Maxwell s rule given in [25] as a special case If Aut(G, p) {id}, then, by Theorem 331, (G, p) R (G,S) is not of a unique type, and hence we may apply Theorem 424 to (G, p) and S by using any homomorphism Φ for which (G, p) R (G,S,Φ) Note that Examples 341 and 342 show that there exist frameworks (G, p) R (G,S) for which we cannot apply the symmetry-extended version of Maxwell s rule to (G, p) and S at all, because there does not exist any homomorphism Φ so that (G, p) R (G,S) is of type Φ Remark 423 Let G be a graph, S be a symmetry group in dimension d, and Φ : S Aut(G) be a homomorphism, so that the set R (G,S,Φ) contains a framework (G, p) with the property that the points p(v), v V (G), span all of R d Then it follows from Lemma 322 and Theorem 323 that the condition (416) in the symmetry-extended version of Maxwell s rule (Theorem 424) is also a necessary condition for G to be (S, Φ)-generically isostatic 126

152 In particular, if the condition (416) does not hold (ie, if X Q X i ) then we may conclude that every framework in the set R (G,S,Φ) is not isostatic Remark 424 There exist a number of further classical counting rules, similar to Maxwell s rule, that can predict the rigidity and flexibility properties of various other types of structures, such as pinned frameworks (ie, frameworks that have some of their joints firmly anchored to the ground), bodybar frameworks, and body-hinge frameworks, for example For each of these rules, symmetry extensions can be derived using techniques from group representation theory (see, for example, [25, 34, 36, 53, 56]) We will discuss some of these rules in Chapter 7 43 Restrictions on the number of fixed joints and bars of symmetric isostatic frameworks In this section, we show that the necessary conditions given in the symmetry-extended version of Maxwell s rule for a 2- or 3-dimensional symmetric framework (G, p) R (G,S,Φ) to be isostatic are equivalent to some very simply stated restrictions on the number of joints and bars of (G, p) that are fixed by various symmetry operations in S The basic results in this section are from the joint paper [15] 127

153 431 Fixed versus geometrically unshifted We begin by summarizing some simple observations regarding the geometric position of a joint or a bar of a framework (G, p) R (G,S,Φ) that is fixed by an element in S (with respect to Φ) Definition 431 Let x be a symmetry operation of a framework (G, p) Then a joint ( v, p(v) ) of (G, p) is said to be geometrically unshifted by x if x ( p(v) ) = p(v) or equivalently, if p(v) is contained in the symmetry element F x corresponding to x Similarly, a bar {( v, p(v) ), ( w, p(w) )} of (G, p) is said to be geometrically unshifted by x if x ( {p(v), p(w)} ) = {p(v), p(w)} or equivalently, if the undirected line segment p(v)p(w) is equal to the undirected line segment x ( p(v) ) x ( p(w) ) C 2 s (a) (b) Figure 48: Geometrically unshifted bars in dimension 2: (a) a bar that is geometrically unshifted by a half-turn C 2 ; (b) possible placement of a bar that is geometrically unshifted by a reflection s Theorem 431 Let G be a graph, S be a symmetry group, Φ : S Aut(G) be a map, (G, p) be a framework in R (G,S,Φ), and x be an element in S (i) If a joint j of (G, p) is fixed by x (with respect to Φ), then j is geometrically unshifted by x; 128

154 (ii) if a bar b of (G, p) is fixed by x (with respect to Φ), then b is geometrically unshifted by x; (iii) if p is injective and a joint j of (G, p) is geometrically unshifted by x, then j is fixed by x; (iv) if p is injective and a bar b of (G, p) is geometrically unshifted by x, then b is fixed by x Proof (i) Let j = ( v, p(v) ) be a joint of (G, p) that is fixed by x (with respect to Φ) Then we have x ( p(v) ) = p ( Φ(x)(v) ) = p(v), which says that j is geometrically unshifted by x (ii) Let b = {( v, p(v) ), ( w, p(w) )} be a bar of (G, p) that is fixed by x with respect to Φ Then we have x ( {p(v), p(w)} ) = { x ( p(v) ), x ( p(w) )} = { p ( Φ(x)(v) ), p ( Φ(x)(w) )} = {p(v), p(w)}, which says that b is geometrically unshifted by x (iii) Let j = ( v, p(v) ) be geometrically unshifted by x Then we have p(v) = x ( p(v) ) = p ( Φ(x)(v) ) Since p is injective, it follows that v = Φ(x)(v) Thus, j is fixed by x (iv) Let b = {( v, p(v) ), ( w, p(w) )} be geometrically unshifted by x Then we have {p(v), p(w)} = x ( {p(v), p(w)} ) = { x ( p(v) ), x ( p(w) )} = { p ( Φ(x)(v) ), p ( Φ(x)(w) )} Since p is injective, it follows that {v, w} = {Φ(x)(v), Φ(x)(w)} = Φ(x) ( {v, w} ) Thus, b is fixed by x It follows from Theorem 431 that if the map p of a framework (G, p) R (G,S,Φ) is injective, then a joint or a bar of (G, p) is geometrically unshifted by x S if and only if it is fixed by x Therefore, if (G, p) is a framework whose map p is injective, then the determination of whether a joint j or a 129

155 C m (a) (b) Figure 49: Possible placement of a bar that is geometrically unshifted by: (a) any rotation C m, m 2 (in dimension 3); (b) a half-turn C 2 (in dimension 3) alone (a) (b) s Figure 410: Possible placement of a bar that is geometrically unshifted by a reflection s (in dimension 3): (a) lying in the reflection plane; (b) lying perpendicular to the reflection plane 130

156 S m i = S 2 (a) (b) Figure 411: Possible placement of a bar that is geometrically unshifted by: (a) any improper rotation S m, m 2 (in dimension 3); (b) an inversion i = S 2 (in dimension 3) alone bar b of (G, p) is fixed by x S only requires a simple examination of the geometric positions of j and b with respect to the symmetry element corresponding to x Figures 48, 49, 410 and 411 show the possible geometric positions of a bar that is geometrically unshifted by the relevant symmetry operations in dimensions 2 and 3 If the map p of (G, p) R (G,S,Φ) is not injective, then a joint or a bar that is geometrically unshifted by x S is not necessarily fixed by x (with respect to Φ) For example, the joints (v 3, p 3 ) and (v 4, p 4 ) of the framework (G t, p) R (Gt,C 2,Θ b ) in Example 331 are both geometrically unshifted by C 2, since both p 3 and p 4 lie on the center of rotation of C 2, but they are not fixed by C 2 with respect to Θ b, since Θ b (C 2 )(v 3 ) = v 4 Similarly, the bar { (v3, p 3 ), (v 4, p 4 ) } of the framework (G bp, p) R (Gbp,C s,ξ b ) in Example 332 is geometrically unshifted by s, but not fixed by s with respect to Ξ b, since Ξ b ({v 3, v 4 }) = {v 3, v 5 } = {v 3, v 4 } So, if the map p of a framework (G, p) R (G,S,Φ) is not injective, then 131

157 we can only find the joints and bars of (G, p) that are fixed by x S (with respect to Φ) by considering the graph automorphism Φ(x) 432 Isostatic frameworks in dimension 2 Suppose (G, p) is an isostatic framework in R (G,S,Φ), where S is a symmetry group in dimension 2, Φ : S Aut(G) is a homomorphism, and the points p(v), v V (G), span all of R 2 Since S is a symmetry group in dimension 2, every element of S is of one of the following three types: the identity Id, a rotation C m, where m 2, or a reflection s This allows us to calculate the (2-dimensional) symmetryextended Maxwell s equation (417) for (G, p) componentwise, as shown in Table 41 In this table we distinguish a half-turn C 2 from a rotation C m, where m > 2, because there may exist bars of (G, p) that are fixed by a half-turn, but there cannot be any bars of (G, p) that are fixed by a rotation C m, where m > 2 By Table 41, the symmetry-extended Maxwell s equation for the isostatic framework (G, p) R (G,S,Φ) reduces to the following four equations: Id: E(G) = 2 V (G) 3 (418) C 2 : 2j Φ(C2 ) + b Φ(C2 ) = 1 (419) C m, m > 2: (j Φ(Cm ) 1) cos ( ) 2π = 1 m 2 (420) s: b Φ(s) = 1, (421) 132

158 Id C 2 C m, m > 2 s X J V (G) j Φ(C2 ) j Φ(Cm ) j Φ(s) X T cos ( ) 2π 0 m X R X Q 2 V (G) 3 2j Φ(C2 ) + 1 2(j Φ(Cm) 1) cos ( 2π m ) 1 1 X i E(G) b Φ(C2 ) 0 b Φ(s) Table 41: Calculations of characters in the 2-dimensional symmetryextended Maxwell s equation where a given equation applies when the corresponding symmetry operation is present in S Some observations arising from this set of equations are: (i) Since every symmetry group contains the identity Id, equation (418) holds and simply restates the condition in Maxwell s original rule (Theorem 227) (ii) If S contains a half-turn C 2, then it follows from (419) and the fact that both j Φ(C2 ) and b Φ(C2 ) must be non-negative integers that j Φ(C2 ) = 0 and b Φ(C2 ) = 1 In particular, since all bars of (G, p) (except the one that is fixed by C 2 with respect to Φ) and all joints of (G, p) occur in pairs, it follows that V (G) is even and E(G) is odd (iii) If S contains a rotation C m, m > 2, then it follows from (420) that either j Φ(Cm ) = 0 and m = 3 or j Φ(Cm ) = 2 and m = 6 However, if j Φ(Cm) = 2 and m = 6, then S also contains a half-turn C 2 = C6 3 with j Φ(C2 ) = 2, contradicting (419) Therefore, S cannot contain a 133

159 rotational symmetry operation C m with m > 3 and when either C 2 or C 3 is present in S, then j Φ(C2 ) = 0 and j Φ(C3 ) = 0 Note that if S contains a rotation C 3, then all joints and bars of (G, p) occur in sets of 3 (iv) Finally, equation (421) says that if S contains a reflection s, then we must have b Φ(s) = 1 However, (421) does not impose a restriction on the number of joints that are fixed by s (with respect to Φ) An immediate consequence of these observations is that the point group of (G, p) must be one of the following six symmetry groups in dimension 2: C 1, C 2, C 3, C s, C 2v or C 3v Figure 548 depicts examples of small 2-dimensional isostatic frameworks for each of these symmetry groups Group by group, the above results can be summarized as follows Theorem 432 Let G be a graph, S be a symmetry group in dimension 2, Φ : S Aut(G) be a homomorphism, and (G, p) be an isostatic framework in R (G,S,Φ) with the property that the points p(v), v V (G), span all of R 2 (i) If S = C 1, then E(G) = 2 V (G) 3; (ii) if S = C 2, then E(G) = 2 V (G) 3, j Φ(C2 ) = 0 and b Φ(C2 ) = 1; (iii) if S = C 3, then E(G) = 2 V (G) 3 and j Φ(C3 ) = 0; (iv) if S = C s, then E(G) = 2 V (G) 3 and b Φ(s) = 1; (v) if S = C 2v, then E(G) = 2 V (G) 3, j Φ(C2 ) = 0 and b Φ(C2 ) = b Φ(s) = 1 for all reflections s C 2v ; 134

160 (a) (b) (c) (di) (dii) (e) (fi) (fii) Figure 412: Examples, for each of the possible point groups, of small 2- dimensional isostatic frameworks: (a) C 1 ; (b) C 2 ; (c) C 3 ; (d) C s ; (e) C 2v ; (f) C 3v For each of C s and C 3v, two examples are given, where in (i) each mirror has a bar centered at and perpendicular to the mirror line, whereas in (ii) each mirror has a bar that lies in the mirror line For C 2v, the bar lying at the center of C 2 must lie in one mirror line and perpendicular to the other 135

161 (vi) if S = C 3v, then E(G) = 2 V (G) 3, j Φ(C3 ) = 0 and b Φ(s) = 1 for all reflections s C 3v ; (vii) S must be one of the above symmetry groups Remark 431 By the above results, an isostatic framework in the plane cannot possess any rotational symmetry operation C m with m > 3 These kinds of restrictions for isostatic symmetric frameworks in the plane become intuitively plausible if one looks at some simple examples of frameworks with m-fold rotational symmetries Suppose, for instance, that (G, p) is a 2-dimensional framework with point group C m, where m is a multiple of 4 Then E(G) must be an even number, because every bar of (G, p) belongs to a symmetry orbit of size m or m 2 Since 2 V (G) 3 is an odd number, however, it follows that the Maxwell count E(G) = 2 V (G) 3 cannot be attained by (G, p) Similarly, if m > 3 is odd, then every bar of (G, p) belongs to a symmetry orbit of size m So, (G, p) can only satisfy the Maxwell count if at least one of the joints of (G, p) is fixed by the m-fold rotation C m, and we have already seen in the previous sections that this kind of restriction has the potential for affecting the rigidity properties of (G, p) Symmetric subgraphs Let S, Φ, and (G, p) R (G,S,Φ) be as in Theorem 432 Then, besides the conditions for G, S, and Φ given in Theorem 432, the Laman conditions for all nontrivial subgraphs of G are also necessary conditions for (G, p) to be isostatic Further, if H is a subgraph of G with the prop- 136

162 erty that E(H) = 2 V (H) 3 and (H, p V (H) ) is a symmetric framework, say (H, p V (H) ) R (G,S,Φ ), then the corresponding conditions in Theorem 432 for S and Φ are clearly also necessary conditions for (G, p) to be isostatic (provided that the points p(v), v V (H), span all of R 2 and Φ : S Aut(H) is a homomorphism) However, if S is a subgroup of S and Φ : S Aut(H) is defined by Φ (x) = Φ(x) V (H), then these conditions are implicitly contained in the overall conditions for S and Φ This is obvious for the conditions concerning the number of joints that are fixed by any symmetry operation, since this number is either 0 or unrestricted, and for the conditions concerning the number of bars that are fixed by a rotation C m, where m > 2, since this number is 0 If x is a half-turn or a reflection, then we must have b Φ (x) = 1 for (G, p) to be isostatic But if b Φ(x) = 1, then we have b Φ (x) 1, and b Φ (x) cannot be zero, for otherwise E(H) is an even number, contradicting the count E(H) = 2 V (H) 3 Thus, b Φ(x) = 1 implies b Φ (x) = 1 In particular, it follows that while the Laman conditions, as well as the conditions in Theorem 432 concerning G, S, and Φ, are necessary conditions for the graph G to be (S, Φ)-generically isostatic (see also Remark 423), there are no additional necessary conditions concerning symmetric subgraphs for G to be (S, Φ)-generically isostatic In Chapter 5, we consider whether the conditions in Theorem 432 for G, S, and Φ, together with the Laman conditions, are also sufficient for G to be (S, Φ)-generically isostatic 137

163 433 Isostatic frameworks in dimension 3 Suppose that (G, p) is an isostatic framework in R (G,S,Φ), where S is a symmetry group in dimension 3, Φ : S Aut(G) is a homomorphism, and the points p(v), v V (G), span all of R 3 Recall from Section 23 that since S is a symmetry group in dimension 3, every element of S is of one of the following types: the identity Id, a rotation C m, where m 2, a reflection s, or an improper rotation S m This gives rise to the componentwise calculations for the 3-dimensional symmetry-extended Maxwell s equation (417) for (G, p) shown in Table 42 In this table we again distinguish a half-turn C 2 from a rotation C m, where m > 2, and an inversion i = S 2 from an improper rotation S m, where m > 2 By Table 42, the symmetry-extended Maxwell s equation for the isostatic framework (G, p) R (G,S,Φ) reduces to the following six equations: Id: E(G) = 3 V (G) 6 (422) C 2 : j Φ(C2 ) + b Φ(C2 ) = 2 (423) ( ( ) ) 2π C m, m > 2: (j Φ(Cm) 2) 2 cos + 1 = b Φ(Cm) (424) m s: j Φ(s) = b Φ(s) (425) i: 3j Φ(i) + b Φ(i) = 0 (426) S m, m > 2: j Φ(Sm) ( ( ) ) 2π 2 cos 1 = b Φ(Sm), (427) m 138

164 Id C2 Cm, m > 2 s i Sm, m > 2 XJ V (G) jφ(c2) jφ(cm) jφ(s) jφ(i) jφ(sm) XT cos ( ) ( 2π m cos 2π ) m 1 XR cos ( ) ( 2π m cos 2π ) m + 1 ( ( XQ 3 V (G) 6 jφ(c2) cos 2π ) ) ( ( m + 1 (j Φ(Cm) 2) jφ(s) 3jΦ(i) 2 cos 2π ) ) m 1 j Φ(Sm) Xi E(G) bφ(c2) bφ(cm) bφ(s) bφ(i) bφ(sm) Table 42: Calculations of characters in the 3-dimensional symmetry-extended Maxwell s equation 139

165 where a given equation applies when the corresponding symmetry operation is present in S Some observations arising from this set of equations are: (i) By equation (422), (G, p) must satisfy the condition in Maxwell s original rule (Theorem 227) (ii) If S contains a half-turn C 2, then equation (423) holds The solutions of (423) are (j Φ(C2 ), b Φ(C2 )) = (2, 0), (1, 1), (0, 2) A bar {( v, p(v) ), ( w, p(w) )} of (G, p) contributes to b Φ(C2 ) if and only if Φ(C 2 ) either fixes both v and w or interchanges v and w However, if (G, p) has a fixed bar of the first kind (ie, a bar whose corresponding joints are both fixed by C 2 with respect to Φ), then this bar contributes 2 to j Φ(C2 ) and 1 to b Φ(C2 ), contradicting (423) Thus, the vertices corresponding to the joints of any bar that is fixed by C 2 with respect to Φ must be images of each other under Φ(C 2 ) This says in particular that all bars included in b Φ(C2 ) must lie perpendicular to the C 2 axis (iii) If S contains a rotation C m, m > 2, then equation (424) holds The non-negative integer solution j Φ(Cm) = 2 and b Φ(Cm) = 0 is possible for all m For m > 2, the factor 2 cos ( 2π m ) + 1 is rational at m = 3, 4, 6, but generates a further distinct solution only for m = 3: m = 3: 0(j Φ(C3 ) 2) = b Φ(C3 ), and hence b Φ(C3 ) = 0, but j Φ(C3 ) is unrestricted 140

166 m = 4: j Φ(C4 ) 2 = b Φ(C4 ) If S contains C 4, then S also contains C 2 = C4 2 Therefore, we must have j Φ(C4 ) = j Φ(C2 ) = 2 and b Φ(C4 ) = b Φ(C2 ) = 0 m = 6: 2(j Φ(C6 ) 2) = b Φ(C6 ) If S contains C 6, then S also contains C 2 = C6 3 and C 3 = C6, 2 and hence we must have j Φ(C6 ) = j Φ(C3 ) = j Φ(C2 ) = 2 and b Φ(C6 ) = b Φ(C3 ) = b Φ(C2 ) = 0 Thus, b Φ(Cm ) must be 0 for all m > 2 and only in the case m = 3 may j Φ(Cm ) depart from 2 (iv) Suppose S has the group I of all rotational symmetries of a regular icosahedron as a subgroup Then it follows from (iii) that for every C 5 S, we have j Φ(C5 ) = 2 Thus, all the natural vertices of the regular icosahedron must be present (as joints) in (G, p) Similarly, if S has the group O of all rotational symmetries of a regular octahedron as a subgroup, then for every C 4 S, we have j Φ(C4 ) = 2, and hence all the natural vertices of the regular octahedron must be present (as joints) in (G, p) (v) If S contains a reflection s, then equation (425) says that the number of joints of (G, p) that are fixed by s with respect to Φ is equal to the number of bars of (G, p) that are fixed by s with respect to Φ (vi) If S contains an inversion i, then it follows from (426) that (G, p) has neither a joint nor a bar that is fixed by i with respect to Φ If p is 141

167 injective, this says that there is no joint and no bar that is located at the center of the inversion i (vii) If S contains an improper rotation S m, m > 2, then equation (427) holds The non-negative integer solution j Φ(Cm ) = 0 and b Φ(Cm ) = 0 is possible for all m For m > 2, the factor 2 cos ( 2π m ) 1 is rational at m = 3, 4, 6, but generates no further solutions: m = 3: and hence j Φ(S3 ) = b Φ(S3 ) = 0 m = 4: and hence j Φ(S4 ) = b Φ(S4 ) = 0 m = 6: 2j Φ(S3 ) = b Φ(S3 ), j Φ(S4 ) = b Φ(S4 ), 0j Φ(S6 ) = b Φ(S6 ), and hence b Φ(S6 ) = 0 But if S contains S 6, then S also contains i = S 3 6, so that j Φ(S6 ) = 0 Symmetry operation by symmetry operation, the above results can be summarized as follows Theorem 433 Let G be a graph, S be a symmetry group in dimension 3, Φ : S Aut(G) be a homomorphism, and (G, p) be an isostatic framework in R (G,S,Φ) with the property that the points p(v), v V (G), span all of R 3 Then 142

168 (i) E(G) = 2 V (G) 3; (ii) if C 2 S, then (j Φ(C2 ), b Φ(C2 )) = (2, 0), (1, 1), (0, 2); (iii) if C 3 S, then b Φ(C3 ) = 0; (iv) if C m S, where m > 3, then j Φ(Cm ) = 2 and b Φ(Cm ) = 0; (v) if s S, then j Φ(s) = b Φ(s) ; (vi) if i S, then j Φ(i) = b Φ(i) = 0; (vii) if S m S, where m > 2, then j Φ(Sm ) = b Φ(Sm ) = 0 In contrast to the 2-dimensional case, the conditions we derived from the 3-dimensional symmetry-extended Maxwell s equation do not exclude any point group For every symmetry group S in dimension 3, we can construct a fully triangulated convex polyhedron that has S as its point group and is isostatic by the Theorem of Cauchy and Dehn [10, 22, 76] One possible approach to construct such frameworks is to begin with the regular triangulated convex polyhedra (the tetrahedron, octahedron and icosahedron), and to expand them by using operations of truncation and capping In fact, for every symmetry group S in dimension 3, an infinite family of isostatic frameworks with point group symmetry S can be created in this way For example, to generate isostatic frameworks with only the rotational symmetries of a given triangulated polyhedron, we can cap each face with a twisted octahedron, consistent with the rotational symmetries of the underlying polyhedron The resultant polyhedron will then be an isostatic framework with the rotational symmetries of the underlying polyhedron, but 143

169 none of the reflectional symmetries An example of the capping of a regular octahedron is shown in Figure 413 (a) (a) (b) (b) Figure 413: A regular octahedron (a), and a convex polyhedron (b) generated by capping every face of the original octahedron with a twisted octahedron The polyhedron in (b) has the rotational but not the reflectional symmetries of the polyhedron in (a) If a framework is constructed from either polyhedron by placing bars along edges, and joints at vertices, the framework will be isostatic One interesting possibility arises from consideration of symmetry groups that contain 3-fold rotational symmetry operations Equation (424) allows an unlimited number of joints, though not bars, that are fixed by a 3-fold rotation Thus, if we start with an isostatic framework and add joints symmetrically along a 3-fold axis using vertex 3-additions (see Definition 2218), the resultant frameworks will remain isostatic as long as each of the new joints is added so that it is not coplanar with the three joints it is linked to So, for instance, we can cap every face of an icosahedron to give the compound icosahedron-plus-dodecahedron (the second stellation of the icosa- 144

170 hedron), as illustrated in Figure 414, and this process can be continued ad infinitum adding a pile of hats consisting of a new joint, linked to all three joints of an original icosahedral face (see Figure 415) (a) (a) (b) (b) Figure 414: An icosahedron (a), and the second stellation of the icosahedron (b) If a framework is constructed from either polyhedron by placing bars along edges, and joints at vertices, the framework will be isostatic The framework (b) could be constructed from the framework (a) by capping each face of the original icosahedron Similar constructions starting from octahedral and tetrahedral symmetric isostatic frameworks can be envisaged Symmetric subgraphs Let S be a symmetry group in dimension 3, Φ : S Aut(G) be a homomorphism, and (G, p) be a framework in R (G,S,Φ) with the property that the points p(v), v V (G), span all of R 3 Then, besides the conditions in Theorem 433 for the symmetry operations in S (with Φ as the underlying type), the conditions in Theorem 228 for all nontrivial subgraphs of G are also necessary conditions for (G, p) to be isostatic Further, if H is a 145