Spherical tilings by congruent 4-gons on Archimedean dual skeletons

Transcription

1 Spherical tilings by congruent 4-gons on Archimedean dual skeletons Mathematical Institute, Tohoku University, Sendai Miyagi Japan, June 13, 2016

2 We classify all spherical tilings by congruent 4-gons such that the 4-gon has only 3 equal sides and the skeleton of the tiling is that of an Archimedean dual. Such a tiling induces a perfect face-matching of the skeleton. The classification is by checking the solvability of systems of integer-coefficient linear equations. Such a system is induced by an assignment of inner angles respecting a perfect face-matching to the skeleton. The number of such systems of integer-coefficient linear equations is huge, because the number of the tiles and the number of the perfect face-matchings of the skeleton of the tiling are large. We overcome this difficulty in not only convex 4-gonal tile cases [A.-van Cleemput15] but also concave 4-gonal tile cases, by linear algebraic approach for the 3-valent vertex types of the tilings. We scrutinize how [K. Coolsaet, Spherical quadrangles with three equal sides and rational angles, 2015] reduces the complexity for the convex 4-gonal tile cases.

3 This reverse problem of Grünbaum-Shephard s theorem compares Motivation. Theorem (Grünbaum-Shephard81) The skeleton of a spherical isohedral tiling (i.e., a tiling s.t. the symmetry group acts transitively on the tiles) is exactly the skeleton of a trapezohedron(i.e., the dual of an antiprism), a bipyramid, an Archimedean dual, or a Platonic solid. Proposition [A.13, Hiroshima Math. J.] There is a spherical non-isohedral tiling A by 12 congruent concave 4-gons on a trapezohedron skeleton. Problem[A.15, Japan Conf. on Discr. & Comput. Geom. & Graphs] Is A the only spherical monohedral, non-isohedral tiling with the skeleton listed in Grünbaum-Shephard s Theorem?

4 Symmetry group vs. automorphism group of the skeleton. Deformation. Mani s theorem ([Math. Ann. 1971], a refinement of Steinitz s theorem): For each polyhedral graph G, there exists a 3-dimensional convex polytope P such that the automorphism group of G is the symmetry group of P. Mani s theorem is for polyhedra, and is generalized on any sphere in [Deza et al., Geometric Structure of Chemistry- Relevant Graphs. Zigzags and Central Circuits, Springer, 15]. Other extensions of Steinitz s theorem is found in [Seok-Hee Hong and H. Nagamochi11]. Cauchy s rigidity theorem for convex polyhedra, For deformation of vertex-transitive polyhedra, see [Robertson-Carter-Morton1970]. Angle-assignments to the skeleton (I. Rivin s solution to Steinitz s problem [Ann. of Math1996]).

5 Every sph. tiling by congruent convex 4-gons on the trapezohedron skeleton is isohedral. Case-by-case analysis by finding forbidden substructures.

6 All spherical monohedral polygonal tilings with Platonic or bipyramidal skeleton are isohedral [A.-Yan16]. Their combinatorial types are: F 4 a b c G 8 D G 4n (n 3) C 6 I G 4n+2 (n 1) Figure: For each of them, the degree of freedom of the continuous deformation is the maximum # of change of the symmetry type.

7 Any spherical tiling by congruent 3-gons on the skeleton of an Archimedean dual and any spherical tiling by congr. kites (darts, rhombi) are isohedral [Sakano-A.15]. R 30 T 24 T 60 R 12 Figure: The 4 4-gon-faced ADs R 12, T 24, R 30, T 60 among the 13 ADs (left). The right upper (middle, bottom, resp.) is a spherical tiling by congruent kites (darts, rhombi, resp.) on the trapezohedron skel.

8 Any spherical tiling by congruent type-2 4-gons on an AD skeleton is isohedral. Definition A 4-gon is said to be type-2, if the edge-lengths are aaab (a b). Lemma (Ueno-Agaoka01) A tile of a sph. tiling by congr. 4-gons is a rhomb., a kite, a dart, a type-2, or has cyclic list of edge-lengths aabc (a, b, c distinct). We will prove: Theorem (Main) There is no sph. tiling by congr. type-2 4-gons on R 12 -, R 30 -, or T 60 -skel, but only one sp. ti. by congr. type-2 4-gons on T 24 -skel., It is obtained from the central projection of T 24, by twisting each Y -figure in common irrational multiple of π rad. We will prove this below.

9 A spherical tiling by congruent 4-gons of edge-lengths aaab (a b) induces a solvable angle-assignment of the skeleton S respecting a perfect face-matching of S. T 60 has perfect face-matchings, i.e., 1-regular spanning subgraphs of the dual graph. So T 60 has angle-assgnmnts to the skel. Figure: The left is the 8 perfect face matchings of the cube skel. The right are 4-gons of type 2. A type-2 4-gon has, by definition, 3 equilateral edges (blue) and 1 face-matching edge (red) of different length. α, β, γ, δ stands for inner angles. Each type-2 4-gon has 2 a.-assignments.

10 The ten 3-valent vertex types in spherical tilings by congr. type-2 4-gons. Definition A vertex consisting of i inner angles α, j inner angles β, k inner angles γ, and l inner angles δ (i, j, k, l 0) is said to have type iα + jβ + kγ + lδ. A spherical tiling by congruent 4-gons of type 2 has at most 10 3-valent vertex types: V 3 := {3β, 2β + γ, α + β + δ, 2α + γ, 2α + β, 3γ, 2γ + β, α + γ + δ, 2δ + β, 2δ + γ}. See [A.-van Cleemput, Ars Mathematica Contemporanea, 15].

11 Allowed vertex types Lemma n {12, 24, 30, 60}. a family C n of nonempty sets of vtx typs sph. tiling T by n congr. type-2 4-gons on AD skel. P C n. P T

12 Prf Count face-match edges on the AD bipartites. C 24 := AC(V 4,>0 ). 6-set C 12 := AC(V 3,0, V 3,1, V 4,>0 ). (# = 156) C 30 := AC(V 3,0, V 3,1, V 5,1, V 5,2 ). (# = 1200) C 60 := AC(V 3,1, V 4,>0, V 5,>0 ). (# = 660) AC(T 1, T 2,..., T n ) := {{t 1,..., t n } t i T i }. V d,i := {types of d-val. vertices incident to i face-match edges} V 3,0 := {3β, 2β + γ, β + 2γ, 3γ}. (4-set) V 3,1 := {2α + β, 2α + γ, α + β + δ, α + γ + δ, β + 2δ, γ + 2δ}. V 4,>0. 11-set V 5,i. (10-set) (i = 1, 2). (α, β, γ, δ) (δ, γ, β, α). (6-set)

13 Forbidden vertex types Lemma family F 12, F 24, F 30, F 60 [V 3 ] <4 s.t. sph. tiling T by n > 6 congr. type-2 4-gons. 1 F F 24. F T. 2 n = 12, 30, 60 = F F n. F T.

14 Prf F 24 := {{3β, α + γ + δ}, {α + β + δ, 3γ}} {{2α + γ, 3γ}, {2α + β, 2γ + β}, {3β, β + 2δ}, {2β + γ, 2δ + γ}}. 20-set F 12 := F 30 := F 24 R 62-setF 60 := F 24 {R {S} R R, S V 3,0 }. Here R := {{t 1, t 2 } V 3 t 1 = t 2 Proof. = α = δ or β = γ}. Prop.[A.13] s. ti. T by cngr. type-2 4gns, (1) α γ and β δ, (2) V 3,0 T = T V 3,1 = α δ and β γ. n = 12, 30: V 3,0 T = T V 3,1. n = 60: By T V 3,1, if T V 3,0, then Prop (2) implies α δ and β γ.

15 The characterization of the inner angles (α, β, γ, δ) (Qπ) 4 of convex type-2 sph. 4-gons (1/2). Theorem (Coolsaet15) If the inner angles (α, β, γ, δ) of a conv. type-2 sph. 4-gons are in (Qπ) 4, then (α, β, γ, δ) or (δ, γ, β, α) satisfy one of the following: 1 α = δ, β = γ. (isosceles trapezoid. Does not tile the unit sphere [A.-van Cleemput15]) 2 α = γ 2, δ = β 2, α + δ < π. 3 α = 3γ 2, β = π 3, δ = 2π 3 γ 2., π 2 < γ < 2π 3. 4 α = π 6 + γ 2, β = 2γ, δ = π 2 + γ 2, π 3 < γ < π 2. 5 α = π 6 + γ 2, β = 2γ, δ = π 2 + 3γ 2, 4π 15 < γ < π 3. 6 α = π 6 + γ 2, β = 2π 2γ, δ = 3π 2 3γ 2, π 2 < γ < 5π 6.

16 The characterization of the rational inn. angles of conv. type-2 sph. 4-gons (2/2) sporadic. (cnt d) (4π/n is the area of the 4-gon.) 1 π ( 5 6, 8 21, 5 7, 17 ) ( 42, π 11 15, 7 15, 3, 8 ) ( 15, π 17 π ( 23 30, 1 3, 14 15, 3 ) ( 10, π 31 60, 3 5, 5 6, π ( 19 60, 7 10, 14 15, 13 60). (n = 24). 3 π ( 53 60, 4 15, 7 10, 17 60). (n = 30). 4 π ( 7 10, 4 15, 13 15, , 8 15, 13 15, 11 30), ), π ( 49 60, 3 10, 14 15, 17 ), π ( 49 60, 4 15, 7 10, 17 60). (n = 60). 60). (n = 12). 5 1 tuple for n = 6, 2 tuples for n = 42, 1 tuple for n = 84, 1 tuple for n = 120, 12 tuples for fractional n. ( ) [G. Myerson. Rational products of sines of rational angles. Aequationes mathematicae, 1993].

17 For tiling T, set of vertex types of T induces solvable linear system n {12, 24, 30, 60}, sp. ti. T by n congr. x type-2 4-gons on AD skel., set T of 3-val. vtx typs of T, (ϕ x (n, C n, F n, T )): F F n. F T & P C n s.t. a system {α + β + γ + δ 2 = 4π n, t = c = 2π c P, t T } has indefinite solution, or definite sol. (α, β, γ, δ) in (0, 2π) 4 \ (0, π) 4 (x=cnc) and listed in Coolsaet s Theorerem (x=cnv). Φ(n, i) := { T [V 3 ] <i ϕ x (n, C n, F n, T ) }

18 The sphe. tiling by congr. type-2 4-gons on T 24 -skel.(1/4) Proof. A complete of representative of Φ cnv (24, 3) is {{2β + Y }, {2α + Y }, {α + β + δ} Y = β, γ}, by 2sec computation with MAPLE. There are 1088 perfect face-matchings of the T 24 -skel, and 8 assignments of α, β, γ, δ to the skel. such that the set of 3-val. vertex types is in Φ cnv (24, 3) and the angle-assignments respect perfect face-matchings. 4 assignments among the 8 induce systems of linear equations, which are unsolvable by the coefficient difference. There are only one solvable and canonical assignment (cnt d):

19 0: 1 (α) 2 (α) 3 (α) 4 (α) 16: 8 (β) 23 (δ) 22 (δ) 24 (β) 1: 0 (β) 5 (δ) 6 (δ) 7 (β) 17: 8 (δ) 24 (β) 10 (β) 9 (δ) 2: 0 (δ) 7 (β) 8 (β) 9 (δ) 18: 10 (δ) 24 (β) 22 (β) 25 (δ) 3: 0 (β) 9 (δ) 10 (δ) 11 (β) 19: 10 (β) 25 (δ) 12 (δ) 11 (β) 4: 0 (δ) 11 (β) 12 (β) 5 (δ) 20: 12 (β) 25 (δ) 22 (δ) 21 (β) 5: 1 (γ) 4 (γ) 13 (γ) 21: 13 (γ) 20 (γ) 14 (γ) 6: 1 (α) 13 (α) 14 (α) 15 (α) 22: 14 (α) 20 (α) 18 (α) 16 (α) 7: 1 (γ) 15 (γ) 2 (γ) 23: 14 (γ) 16 (γ) 15 (γ) 8: 2 (α) 15 (α) 16 (α) 17 (α) 24: 16 (γ) 18 (γ) 17 (γ) 9: 2 (γ) 17 (γ) 3 (γ) 25: 18 (γ) 20 (γ) 19 (γ) 10: 3 (α) 17 (α) 18 (α) 19 (α) 11: 3 (γ) 19 (γ) 4 (γ) 12: 4 (α) 19 (α) 20 (α) 13 (α) 13: 5 (δ) 12 (δ) 21 (β) 6 (β) 14: 6 (δ) 21 (β) 22 (β) 23 (δ) 15: 6 (β) 23 (δ) 8 (δ) 7 (β)

20 The sph. tiling by congr. type-2 4-gons on T 24 -skel. (3/4). From the angle-assignment A, the spherical tiling by 24 congruent type-2 quadrangles is constructed from the central projection of T 24 by rotating the central Y-figure of each octant (i.e., right-angled regular triangle), by a fixed radian θ in the direction of the parity of the octant. Here, the octant (+, +, ) has parity and the octant (+,, ) has parity +. The inner angles of the tiles are α = π/2, β = arcsin ( (5 2 3)/(12 6 3) ), γ = 2π/3, and δ = π β. The length( of the edge αδ is b = arccos (4 2 3)/(5 2 ) 3), while the length of the other three edges is a = π/2 b. Note that β is an irrational multiple of π, because no instance of (α, β, γ, δ) is listed in Coolsaet s thm.

21 The sph. tiling by congr. type-2 4-gons on T 24 -skel. (4/4) sph. tiling by congruent concave type-2 4-gons on T 24 -skel. Proof. By 1min computation with MAPLE, Φ cnc (24, 4) = Φ cnv (24, 3) { [{2 α + γ, 2 β + γ}], [{2 β + γ, α + γ + δ}], [{α + β + δ, α + γ + δ}] } assignments of α, β, γ, δ to the skel. such that the set of the 3-val. vtx types is in Φ cnc (24, 4) and the assgnmnts respect perfect face-matchings assgnmnts among the 9704 induce systems of linear equations, but they are unsolvable by the coefficient diff assgnmnts among the 9704 induce systems of linear equations, but they are unsolvable by the linear programming. The other 4 assignment are solvable and correspond to one common canonical solution A of the two previous slides.

22 No sph. tiling by cngr convex 4-gons on T 60 -skel. Lemma For any sp. ti. T by cngr type-2 4gns on T 60 -skel, (1) T V 3,0. (2) F F 12.F T. Proof. (1) Assume otherwise. All 20 3-val. vertices turn out to be incident to match edges. Because T 60 has bipartital skel., the remaining 10 match edges are incident to 5-val. vertices. As # of 5-val. vert is 12, some 5-val. vertex is incident to no matching edge. However, ϕ x (60, AC(V 3,1, V 4,>0, V 5,>0, V 5,0 ),, ). (x {cnv, cnc}). (The total time of computation with MAPLE/Corei7 is 1min.) This is a contradiction.

23 Proof continued (2) F 12 = F 24 {{t 1, t 2 } V 3 t 1 = t 2 = α = δ or β = γ}. We have the conclusion, by T V 3,1, (1), F 24, and Corollary V 3,0 T = T V 3,1 = α δ&β γ. There is no spherical tiling by congruent convex type-2 4-gons on T 60 -skeleton. Proof. 1min computation shows that ϕ cnv (60, AC(V 3,0, V 3,1, V 4,>0, V 5,>0 ),, ). By counting matching edges, we have the conclusion.

24 No sph. tiling by cngr. concave type-2 4-gons on T 60 -skel. Prf. Any 5-val. vertex is adjacent to a 4-val. one in T 60. So, Lemma In a sp. ti. by cng type-2 4gns on T 60 -skel., if every 5-val. vertex does δ (α, resp.) 4 times, then some 4-val. vertex contains α (δ, resp.) at least twice. ϕ cnc (60, AC(V 3,0, V 3,1, V 4,>0, V 5,>0 ),, ) is witnessed by 8 solutions (α, β, γ, δ) = (2π/15, 2π/15, 26π/15, π/15), or(π/30, 28π/15, π/15, π/10) with α δ and/or β γ. Note that AC(...) is symmetric w.r.t such exchange(s) since V s are defined so.

25 No sph. tiling by cngr. concave type-2 4-gons on T 60 -skel. (2/2) If (α, β, γ, δ) = (2π/15, 2π/15, 26π/15, π/15), then a 5-valent vertex type is γ + 4δ and a 4-valent vertex type is α + γ + 2δ or β + γ + 2δ. If (α, β, γ, δ) = (π/30, 28π/15, π/15, π/10), then a 5-valent vertex type is 4α + β and a 4-valent vertex type is 2α + β + γ. These contradict against previous Lemma on # of α and # of δ. The other six solution are obtained by α δ and/or β γ, and contradict similarly.

26 No sph. tiling by congr. type-2 4-gons on R 12 - or R 30 -skel. Proof. Φ cnv (12, 3) = {3 β, 2X + Y }, {3 β, α + β + δ} X = α, δ; Y = β, γ Φ cnc (12, 4) = {3 β, 2X + Y }, {β + 2Y, α + β + δ} X = α, δ; Y = β, Φ cnv (30, 2) =. Φ cnc (30, 4) = {3 β, α + β + δ}, {2 X + γ, 2 β + γ} X = α, δ. 1min, 2min, 10min, 50min computation by (MAPLE/Corei7).

27 No sph. tiling by congr. type-2 4-gons on R 12 - or R 30 -skel. Prf cnt d Proof (cnt d) No member of Φ cnc (n, 4) (n = 12, 30) is suitable, by: Lemma ( 3vs3 ) For a spherical tiling by congr. 4-gons on the R e -skeleton (e = 12, 30), if the set of 3-val. vertex types is {t 1, t 2 }, then a homogeneous system of linear equations it 1 + jt 2 = kα + lβ + lγ + kδ has a non-triv. solution (i, j, k, l) such that i 0, j 0. Proof. The opposite vertex of a 3-val. vertex is 3-val.

28 Summary. Theorem (Main) There is no sph. tiling by congr. type-2 4-gons on R 12 -, R 30 -, or T 60 -skel, but only one spherical tiling by congruent type-2 4-gons on T 24 -skel. It is obtained from the central projection of T 24 by twisting each Y -figure in common irrational multiple of π radian.

29 For R e - or T 60 -skel., 1 Plan a family C of compulsory sets of vertex types, by counting argument # of matching edges in view of the skeleton. 2 Do generate the family Φ of sets of 3-val. vertex types, conjuncting with C, avoiding F, and having indefinite solutions or Coolsaet-rational (concave, resp.) definite ones. 3 Check 3-val. vertex types s.t.: In R e, the opposite of a 3-val. vertex is 3-val. ( 3vs3 Lemma). In T 60, a 4-val. vertex and a 5-val. vertex have types so that they are adjacent (Lemma (T 60 2)). Conclude that no sph. tiling by congruent type-2 4-gons exists if Φ = { }. 4 Act (1-4) for each remaining subcase T Φ (Lemma(T 60 )). A smaller C may save the total computation time. For T 24 -skel., generate Φ from some C. Then construct canonical sph. tilings by cngr. 4-gons from all LP-solvable angle-assignments respecting perfect face-matchings of T 24 s.t. the set of 3-val. vertex types is in Φ. (cf. canonical construction path of McKay)

30 Feature of Coolsaet-rationality for convex type-2 4-gons. Sp. ti. by cng. type-2 s on T 24 -skel. has angle in (R \ Q)π. Efficient. Let A cnv (n) be Φ cnv (n, 3) with listed in Coolsaet s thm replaced by (0, π) 4, that is, {maximalt [V 3 ] 3 F F n. F T & P C n (α + β + γ + δ 2π = 4π n, t T c P. t = c = 2π) has indefinite sol., or definite sol. (α, β, γ, δ) (0, π) 4 }. Φ cnv (12, 3) 2min 8 singl. & 8 pairs A cnv (12) 2min 10 singl. & 18 pairs Φ cnv (24, 3) 20sec 6 singl. A cnv (24) 1min 10 singl. & 18 pairs Φ cnv (30, 2) 10min 0 A cnv (30) 10min 10 singl. & 10 pairs Φ cnv (60, 3) 3min 2 singl. A cnv (60) 15min 10 singl. & 18 pairs

31 The two 5-gon-faced Archi. duals. Amon the 13 ADs, there are two 5-gon-faced ones (figures below). The cyclic list of 5 edges of a tile is aaabb (a b). The #s of faces are 24 and 60. On each of the two 5-gon-faced AD skels., there is a spherical tiling T by congruent equilateral 5-gons [Akama&M. Yan]. The two tilings are a pentagonal subdivision of an octahedron and that of an icosahedron. Hence, any spherical tiling by congruent equilateral 5-gons on an AD skel. is isohedral. Figure: P 24 (left) and P 60 (right). Their skeletons are non-bipartital

32 Prf of Key Lem.(1/4) The 4 4-gonal Archi. duals have bipartital skel. Rhombic e-hedron (R e ), tetragonal 2e-hedron (T 2e ) (e = 12, 30). By the constructions [Conway et al., The Symmetries of Things, 08], the skel. of R e is (L({3, q})) & that of T 2e is (L(L({3, q}))). Here L(G) is the line graph (i.e., link the midpts of adjacent edges) of a graph G. {3, q} (q = 4, 5) is Platonic solid with e edges s.t. q 3-gons share a vtx. So, the skel. of R e is a bipartite btw the set V 1 of f 3-val. vert. & the set of v q-val. vert., where {3, q} has f faces. & v vert. The skel. of T 2e is a bipartite btw V 1 V 3 and a set V 2 of f 4-val. vert. Here V 3 is a set of v q-val. vert.