Coxeter Groups as Automorphism Groups of Solid Transitive 3-simplex Tilings

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1 Filomat 8:3 4, DOI 98/FIL43557S Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Coxeter Groups as Automorphism Groups of Solid ransitive 3-simplex ilings Milica Stojanović a a Faculty of Organizational Sciences, Jove Ilića 54, 4 Belgrade, Serbia Abstract In the papers of IK Zhuk, then more completely of E Molnár, I Prok, J Szirmai all simplicial 3-tilings have been classified, where a symmetry group acts transitively on the simplex tiles he involved spaces depends on some rotational order parameters When a vertex of a such simplex lies out of the absolute, eg in hyperbolic space H 3, then truncation with its polar plane gives a truncated simplex or simply, trunc-simplex Looking for symmetries of these tilings by simplex or trunc-simplex domains, with their side face pairings, it is possible to find all their group extensions, especially Coxeter s reflection groups, if they exist So here, connections between isometry groups and their supergroups is given by expressing the generators and the corresponding parameters here are investigated simplices in families F3, F4, F6 and appropriate series of trunc-simplices In all cases the Coxeter groups are the maximal ones Introduction he isometry groups, acting discontinuously on the hyperbolic 3-space with compact fundamental domain, are called hyperbolic space groups One possibility to describe them is to look for their fundamental domains Face pairing identifications of a given polyhedron give us generators and relations for a space group by Poincaré heorem [], [3], [7] he simplest fundamental domains are simplices and truncated simplices by polar planes of vertices when they lie out of the absolute here are 64 combinatorially different face pairings of fundamental simplices [7], [6], furthermore 35 solid transitive non-fundamental simplex identifications [6] I K Zhuk [7] has classified Euclidean and hyperbolic fundamental simplices of finite volume up to congruence Some completing cases are discussed in [], [5], [], [], [3], [4], [5], [6] An algorithmic procedure is given by E Molnár and I Prok [5] In [6], [8] and [9] the authors summarize all these results, arranging identified simplices into 3 families Each of them is characterized by the so-called maximal series of simplex tilings Besides spherical, Euclidean, hyperbolic realizations there exist also other metric realizations in 3-dimensional simply connected homogeneous Riemannian spaces, moreover, metrically non-realizable topological simplex tilings occur as well [4] his paper is a continuation of [4] and the presented results are as follows: Main result Investigating symmetries of the series of simplices in families F3, F4, F6 and corresponding series of trunc-simplices, it is given relationship between their isometry groups and possible supergroups he maximal Mathematics Subject Classification Primary 5M; Secondary 5C, H5, F55 Keywords hyperbolic space group, fundamental domain by simplex and trunc-simplex, Coxeter groups as supergroups Received: 3 May ; Accepted: 5 September 3 Communicated by Vladimir Dragović address: milicas@fonrs Milica Stojanović

2 M Stojanović / Filomat 8:3 4, supergroups are always the Coxeter reflection groups, whose realizations can be described by the usual machinery, illustrated here only for family F6 Moreover, there are also constructed some new series of fundamental domains, by truncating simplices and half-simplices he results will be given in Sections 3, 4 and in tables Notations used here for simplices and families of simplices are introduced in [6], [8], [9], while the investigated trunc-simplices are introduced in [4] Maximal tilings and principles of truncating simplices Let, Γ denote a face-to-face tiling by topological simplices,, as a usual incidence structure of -, -, -, 3-dimensional constituents in a simply connected topological 3-space X 3, G with a group Γ < G that acts on by topological mapping of X 3, preserving the incidences Let Γ be tile-transitive on, ie for any two 3-tiles, there exists at least one γ Γ such that γ :, := γ wo tilings, Γ and, Γ are called combinatorially topologically equivariant, and lie in the same equivariance class, iff there is a bijective incidence preserving topological mapping φ :, := φ such that Γ = φ Γφ, Γ is a maximal tiling iff Γ = Aut is a maximal group of automorphisms of A maximal tiling =, Γ = Aut represents a family of tilings, Γ such that there is a bijection topological mapping above φ :, with φ Γφ Γ = Aut When the rotational order parameters u, v, are such that the simplex i used enumeration for the simplices is the same as in [6, 8, 9] is hyperbolic and vertices in an equivalence class t are out of the absolute, then it is possible to truncate the simplex by polar planes of these vertices We get the trunc-simplex of finite volume possibly with 8 faces, as octahedron denoted by O t Dihedral angles around the new edges i are π/ It means there are four congruent trunc-simplices around such an edge in the fundamental space filling We can equip O t i with additional pairings of the new triangular faces trunc-faces hen Ot will be i a fundamental domain for the new group G j O t i which will be a supergroup of Γ i, for each value of edge parameters he trivial group extension is always possible with plane reflections in polar planes of the outer vertices Let us denote this group by G O t i If there is a further possibility to equip the new triangular faces to outer vertices in class t, represented by A k, with face pairing isometries, then the new additional face pairings of O t have to satisfy the following i criteria he polar plane of A k and so the stabilizer ΓA k will be invariant under these new transformations and exchange the half-spaces obtained by the polar plane hus, the fundamental domain P Ak is divided into two parts, and the new stabilizer of the polar plane will be a supergroup for ΓA k of index two Inner symmetries of the P Ak -tiling give us the idea how to introduce the new generators Note that truncations of vertices in different equivalence classes can independently be combined, even it is possible to truncate vertices in a class and leave others without truncating 3 Supergroups of the simplex and trunc-simplex tilings Here will be discussed the supergroups of simplicial fundamental domains from the families F6, F3, F4, and the corresponding trunc-simplicial domains In all cases Coxeter groups will be the maximal ones Coxeter groups will be the maximal supergroups of the simplicial and related trunc-simplicial domains in the family F, and also in all other families for special cases of parameters [, 9, ]

3 M Stojanović / Filomat 8:3 4, Family F6 his family is represented by its maximal group m Γ ū, v, w, x, ū, v w, 3 x with minimal fundamental domain a simplex F6 : A A A A 3 Fig his domain is also described in [9], by D-symbols and Schlegel diagram he group is special case of the reflection group Γ a, b, c, d, e, f, where for the parameters hold a = x, b = w, c =, d = ū e =, f = v [9] When parameters are such that ū + v + w < vertices A, A are outer Also if + v + x <, or + w + x <, vertices A and A 3 are outer and we can truncate them by polar planes of these vertices, respectively Vertex domains F A, F A and F A 3 shown in Fig are such that they haven t more symmetries keeping the adjacencies, ie the types of the side lines are invariant, so the additional trunc-side pairings Fig are trivial by reflections m i, i =,, 3 It means, the obtained trunc-simplex is fundamental domain of the new Coxeter group with Coxeter diagram given in Fig Vertices of this diagram indicate the reflection side planes he edges marked eg with ū means an angle π ū between the corresponding reflection planes Not connected vertices means orthogonal planes Vertices connected with dashed lines indicate not intersecting and not parallel, with a common perpendicular line to both planes Similar conventions will be applied later on as well Figure : F6: Simplex F6 ; F A i, i =,, 3 In family F6 there are four face paired simplices, shown in Fig3,4 Possible trunc-simplices for them have been described in [4] It would be obvious that groups for these four fundamental simplices are subgroups of m Γ ū, v, w, x relationship between parameters are given in table if we halve those simplices with plane α through edge A A 3 and midpoint M = A of A A Fig 5 In all cases we equip the new simplex A A A 3 M with face pairing by reflections M A A m α : 3 A A, m M A A : A 3 A M A, m 3 A A A : 3 A M A, m 3 A M A 3 : 3 A M A Since it is possible to express generators of each starting group by generators of group m Γ ū, v, w, x, the last one is a supergroup for appropriate parameters Similarly, after halving the corresponding trunc-simplex with plane α the supergroup of G j O t is the i Coxeter group ΓO, ū, v, w, x = m α, m, m, m 3, m, m, m 3 m α = m = m = m 3 = m =

4 M Stojanović / Filomat 8:3 4, Figure : F6: trunc-simplex and its Coxeter diagram Figure 3: Simplices 6 and obtained by gluing two copies of F6

5 M Stojanović / Filomat 8:3 4, Figure 4: Simplices 4 and 35 obtained by gluing two copies of F6 Figure 5: Halving a simplex from F6 with its symmetry plane α

6 M Stojanović / Filomat 8:3 4, = m = m 3 = m α m = m α m 3 = m m 3 ū = m 3 m v = m m w = = m α m x = m m = m m = m m 3 = m m α = m m = = m m 3 = m 3 m α = m 3 m = m 3 m =, ū + v + w <, + v + x <, + + < w x Here it is assumed that all vertices A, A, A 3 are truncated, although it is possible that eg A is not an outer vertex, since ū + v + w Note that the vertex M is always proper In able for each simplex from this family is given: Generators for the simplex group Γ i i = 6,, 4, 35, with fundamental simplex i Relationship between parameters for isometries of Γ i and their Coxeter s supergroup m Γ ū, v, w, x 3 Additional generators pairing new triangular faces for groups of truncated simplex O i i = 6,, 4, 35 given separately for each of the vertex classes l : {A, A }, m : {A }, n : {A 3 } If there are more possibilities, the trivial one is denoted by I Vertices of trunc-face of vertex A i are denoted by j i, where j i A i A j able : Family 6 6 :Γ 6 u, 4v, 4w, x A A m : A 3 A A ; m A A A : A 3 A A 3 A A A : A 3 A A 3 A A A 3 : A 3 A A A u = ū, v = v, w = w, x = x = m α m, r 3 = m α m 3 3 l I: m, m ; II: s: 3 3 m I: m ; II: h : 3 3 n I: m 3 ; II: h 3 : :Γ u, 4v, 4w, x A A r : A 3 A A ; m A A A : A 3 A A ; m 3 A A A 3 : A 3 A A A u = ū, v = v, w = w, x = x; r = m α m 3 l I: m, m ; II: s: 3 3 m I: m ; II: h : 3 3 n I: m 3 ; II: h 3 : :Γ 4 4u, v, 4w, x A A r : A 3 A A ; m A A A : A 3 A A 3 A A A 3 : A 3 A A A u = ū, v = v, w = w, x = x; r 3 = m α m 3, r = m α m 3 l I: m, m ; II: s: 3 3 m I: m ; II: h : 3 3 n I: m 3 ; II: h 3 :

7 M Stojanović / Filomat 8:3 4, :Γ 35 u, v, w, x A A r : A 3 A A A A A : A 3 A A 3 A A A 3 : A 3 A A A u = ū, v = v, w = w, x = x = m α m, r 3 = m α m 3, r = m α m 3 l I: m, m ; II: s: 3 3 m I: m ; II: h : 3 3 n I: m 3 ; II: h 3 : Family F3 Family is represented by its maximal group 3m Γū, v, ū, 3 v, which is a Coxeter 6 group [9] If the parameters are such that + ū + v < and v <, then vertices A and A respectively, are outer and we can truncate them by polar planes of these vertices Since vertex domains F A and F A Fig6 haven t more symmetries, the additional trunc-side pairings are trivial by reflections m and m Fig6 So, such trunc-simplex is the fundamental domain of the new Coxeter group with Coxeter diagram given in Fig6 Figure 6: F3: F A i, i =, ; trunc-simplex; Coxeter diagram he simplex group Γ 33 u, 6v is subgroup of the former maximal group iff 6u = ū and 6v = v hat would be obvious after splitting the fundamental simplex 33 which has the generators A A m : A 3 A A, r A A A : A 3 A A, z : A 3 3 A A A 3 A A A

8 M Stojanović / Filomat 8:3 4, into three congruent parts, we can obtain the supergroup of the starting one he new fundamental simplex A A A 3, where is the formal barycenter of face A A A 3 with face pairing isometries A A m : A 3 A A, r A A A : 3 A3 A, r : 3 A 3 A A A is simplex denoted by 4 It means that the new group and also the starting one are subgroups of the Coxeter group 3m Γ he old generators, expressed by the new ones, are 6 r = r r r, z = rm r Similar splitting of the trunc-simplex O 33 for each variant of paired faces provides the fundamental domain of the supergroup G O 4 So, again the Coxeter groups 3m 6 Γ and GO are supergroups of Γ 33 and G j O 33 j =, Figure 7: Symmetries of simplices 33, 4 from F3, according to face pairings Although another simplex 4 from family 3 has the same symmetries as 33, the faces of the same splitting is not possible to equip with pairing isometries agreeing with those of 4, since its isometry group Γ 4 6u, 3v has generators A A r : A 3 A A, r A 3 A A : A 3 A A, z : A 3 A A A 3 A A A A single one supergroup of isometry group with fundamental domain 4, is Coxeter group 3m Γū, v, 6 3u = ū, 3v = v with fundamental simplex eg A A M where M is midpoint of edge A A Also, the only supergroup of G j O 4 j =, is GO Family F4 Family F4 is represented by mm 4 Γ ū, v, w 3 ū w, v, the Coxeter group Vertices A, A are outer if + ū + v < and vertices A, A 3 are such that if + ū + w < hen it is possible to truncate these vertices Since there are no symmetries of F A and F A Fig8, the additional trunc-side pairings are trivial and new trunc-simplex is fundamental domain of the new Coxeter group with Coxeter diagram given in Fig8 Each of the six simplices Fig 9,,, from family F4, is possible to halve either with plane α as before, or with plane β through edge A A and mid point N of A A 3 Fig It is also possible to halve with both of them at the same time he new faces of simplices A A A 3 M and A A A N are paired with either plane reflection m α and m β, or with half-turn h around the axis MN If h is used as a face-pairing of face MA A 3 or NA A, let us denote it by h α or h β, respectively In the case of halving with α and β at the same time, a

9 M Stojanović / Filomat 8:3 4, Figure 8: F4: F A i, i =, ; trunc-simplex; Coxeter diagram new, quarter simplex A A MN, with plane reflections m, m 3, m α, m β pairing the faces, provides the Coxeter group mm 4 Γ ū, v, w as a supergroup of the starting ones When simplices and 8 are divided by β it appears simplex, with the group Γ 4a, 4b, c, d, e by [9], a = u, b = v, c = w, e = d =, from family 4 Face pairings of this simplex are: A A m : A 3 A A, r A A A : A 3 A A, m 3 A A 3 A : A 3 A A, m A A A 3 : A 3 A A A he group is maximal in general case, but here it is d = e and so, there are more symmetries by plane α Such new group Γ, has group mm 4 Γ ū, v, w as a supergroup In the case of simplices 7 and 38, with groups Γ 7 u, 4v, w and Γ 38 u, 4v, w, it is possible to obtain by notations in [6], [8], [9], [4] Γ 4û, ˆv, ŵ, with ū = û, v = ˆv, w = ŵ and Γ 3û, 4 ˆv, ŵ, with ū = û, v = ˆv, w = ŵ respectively, as a supergroups of index In these cases half-turn h around axis MN is identifying couples of points inside of simplices in such a way that we do not have a new face It means that new groups are solid transitive although there are no simplices as their fundamental domain Let us call their fundamental domains half-simplices and denote them by h 4 and h 3 Fig3 he generators of the group Γ 4u, v, w are A A r : A 3 A A A A 3 A 3 : A A A ; h : A A 3 A A A A A A 3 A and the generators of Γ 7 u, 4v, w expressed by those of Γ 4û, ˆv, ŵ are r = hr h, r = hr 3 h

10 M Stojanović / Filomat 8:3 4, Figure 9: Simplices and 7 from F4 Figure : Simplices 8 and 38 from F4

11 M Stojanović / Filomat 8:3 4, Figure : Simplices 54 and 57 from F4 Figure : Halving a simplex from F4 with planes α and β

12 M Stojanović / Filomat 8:3 4, Figure 3: Half-simplices h 4 and h 3 For the group Γ 3û, 4 ˆv, ŵ the generators are A A r : A 3 A A ; m A A 3 A 3 : A A A ; h : A A 3, A A A A A A 3 A while the generators of Γ 38 u, 4v, w expressed by them are r = hr h, z = m 3 h he only supergroup of both Γ 4û, ˆv, ŵ and Γ 3û, 4 ˆv, ŵ is the Coxeter group mm 4 Γ ū, v, w with corresponding parameters We can also truncate half-simplices, when rotatory parameters are such that vertices are out of the absolute If we pair then the trunc-faces with plane reflections m, m trivial group extension the new relations are by the Poincaré algorithm: for G ho 4 and identified vertices A, A : m r = m r 3 = ; for identified vertices A, A 3 : m r 3 = m r = ; for G ho 3 and A, A : m r = m m 3 = ; for A, A 3 : m r = m m 3 = Other possibility to pair the trunc-faces of half-simplices, is with half-turns r, r hen we have for G ho 4 and A, A : r 3 r = r r = ; for A, A 3 : r r = r 3 r = ; for G ho 3 and A, A : r m 3 = r r = ; for A, A 3 : r r = r m 3 = Results for Family 4 are collected in able, where 3 are organized as before Here, the vertex classes are denoted by l : {A, A } and m : {A, A 3 } In 4 6 there are given data about possible supergroups of index : splitting isometry; notation of simplex supergroup; notations of trunc-simplex supergroups If splitting of O i gives O j or ho j, then in all variants of pairings trunc-faces of O i which are given, the trivial pairing of O j ie ho j provides a supergroup Beside that, for some cases of non-trivial pairings of trunc-faces of O i, there is non-trivial pairing of trunc-faces of O j ie ho j which gives supergroup hat cases are also indicated in the table

13 M Stojanović / Filomat 8:3 4, able : Family 4 :Γ u, 8v, 4w A A m : A 3 A A A A A : A 3 A A 3 A A 3 A : A 3 A A A A A 3 : A 3 A A A u = ū, 4v = v, 4w = w; m = m α m m α, r = m m β, r = m β m 3 m α m β, r 3 = m 3 m α 3 l I: m, m m I: m, m 3 ; II: s: m β ; Γ ; G O 7 :Γ 7 u, 4v, w A A r : A 3 A A A A 3 A : A 3 A A A A 3 A : A 3 A A A A A 3 : A 3 A A A u = ū, v = v, w = w; m = m α m β m m α, r = m β m, r = m β m 3 m α m β, r 3 = m α m l I: m, m ; II: s : m I: m, m 3 ; II: s : ; III: : ; III: : h α ; Γ 35 ; G O 35, G O m 7 G O l 35 5 h β, Γ 35 ; G O 35, G O l 7 G O l 35 6 h, Γ 4; G ho 4, G j O t 7 G ho t, j =, 3, t {l, m} 4 8 :Γ 8 u, 8v, 4w A A m : A 3 A A A A A : A 3 A A ; z : A 3 3 A A 3 A A A A u = ū, 4v = v, 4w = w; m = m α m m α, r = m m β, z = m 3 m α m β 3 l I: m, m m I: m, m 3 ; II: s: 3 l I: m, m ; II: z: m β ; Γ ; G O 38 :Γ 38 u, 4v, w A A r : A 3 A A A A 3 A : A 3 A A ; z : A 3 A A 3 A A A A u = ū, v = v, w = w; r = m α m β m m α, r = m β m, z = m β m α m m I: m, m 3 ; II: z : h 3 : ; IV: s: ; III: s : ; III: h : h α ; Γ 6 ; G O 6, G 4 O m 38 G O l 6 6 h, Γ 3; G ho 3, G 3 O l 38 G ho l 3, G jo m 38 G ho m, j =, 3, 4 3,

14 M Stojanović / Filomat 8:3 4, :Γ 54 u, 4v, w A A r : A 3 A A A A A : A 3 3 A A A u = ū, v = v, w = w; r = m α m, r = m β m 3 3 l I: m, m ; II: s : 3 ; III: h : 3 h : 3 ; IV: z : m I: m, m 3 ; II: s : h 3 : ; IV: z : l I: m, m ; II: z : h : 3 ; IV: s : 3 ; III: h : , 3 3, 3 4 m α ; Γ ; G O, G j O m 54 G O l, j =, 4 5 m β ; Γ ; G O, G j O l 54 G O l, j =, 4 57 :Γ 57 u, 4v, w A A z : A 3 A A : A 3 A A 3 A A A A u = ū, v = v, w = w; z = m β m α m, r = m β m m I: m, m 3 ; II: s : h 3 : ; IV: z : ; III: h : 3 3 ; III: h : h α ; Γ 4 ; G O 4, G O m 57 G O l 35 3, 3 3, 3 Note that symmetries of a simplex induce symmetries of the fundamental domain of its vertex figure So, it is obvious that if the group of a simplex is the maximal one, then there are no symmetries of the fundamental domain of the vertex figure according to the face pairing Consequently, only trivial pairing of trunc-faces is possible Also symmetries of 7 and 38 leading to h 4 and h 3 show new, unexpected symmetry for each of the vertex figures of 7 and 38, respectively Namely, in all cases this is rotation in the polar plane of the vertex, as it is indicated in Figures 4 and 5 It means that there is one more possibility for pairing trunc-faces of O 7 and O 38 with face pairing identifications, not mentioned yet in [4] In the case of trunc-simplex O 7 Fig 6 the identifying isometries are glide reflections,, and the new relations for A variant III in able are while for A variant III these are r 3 r = r r =, r r = r r 3 = Here, are expressed by generators of ho I 4 = h m, = h m For trunc-simplex O 38 screw-motion s identify the trunc-faces of vertices A and A variant III, so the relations are s zs z = s r s r =

15 M Stojanović / Filomat 8:3 4, Figure 4: Vertex figures of simplex 7 Figure 5: Vertex figures of simplex 38 Figure 6: runc-simplices O 7 and O 38

16 M Stojanović / Filomat 8:3 4, and s is expressed by generators of ho II 3 s = h r Identification of vertices A and A 3 in this way is equivalent to one of the considered cases in [4] here denoted by II 4 Realization of family F6 on the base of Coxeter groups 4 If the 4-dimensional real vector space is denoted by V 4 and its dual space of its linear forms by V 4, then the projective 3-space P 3 V 4, V 4 can be introduced in the usual way he -dimensional subspaces of V 4 or the 3-subspaces of V 4 represent the points of P 3, and the -subspaces of V 4 or the 3-subspaces of V 4 represent the planes of P 3 he point Xx and the plane αa are incident iff xa =, ie the value of the linear form a on the vector x is equal to zero x V 4 \ {}, a V 4 \ {} he straight lines of P 3 are characterized by -subspaces of V 4 or of V 4, respectively If {e i } is a basis on V 4 and { e j} is its dual basis on V 4, ie e i e j = δ j i the Kronecker symbol, then the form a = e j a j takes the value xa = x i a i on the vector x = x i e i We use the summation convention for the same upper and lower indices We can introduce projective metric in P 3 by giving a bilinear form ; : V 4 V 4 R, b i u i ; b j v j = ui b ij v j where b i ; b j = b ij is the Schläfli matrix, and the basis { b i} in V 4 represents planes containing simplex faces opposite to the vertices A i, respectively Vectors a j of the dual basis { a j } in V 4, defined by a j b i = δ i j, represent the vertices A j of the simplex he induced bilinear form ; : V 4 V 4 R, x i a i ; y j a j = x i a ij y j is defined by the matrix a i ; a j = aij the inverse of b ij We assume, that the bilinear form ; is of signature +, +, +, which characterizes the hyperbolic metric, or of +, +, +, + for elliptic spherical metric, or of +, +, +, for Euclidean geometry see [9] for the other cases It is well-known that the bilinear form induces the distance and the angle measure of the 3-space Let Xx and Yy be two points in the projective space P 3 hen their distance dx, y is determined by x; y x; y cos dx, y = and ch dx, y = x; x y; y x; x y; y for the elliptic and hyperbolic case, respectively For the four simplices in family F6 the Schläfli matrix is b ij = b i ; b j = cos π x cos cos cos π x cos cos cos cos cos π ū cos cos cos π ū But, because of symmetries of these simplices, the Schläfli matrix for the half of them, the simplex F6 Fig can serve for investigating the space of realization hat matrix is b ij = cos π x cos π x cos cos cos cos π ū cos cos π ū

17 M Stojanović / Filomat 8:3 4, cos π x cos π ū cos cos π ū cos, cos π x cos cos the latter is by reordering the basis It is obvious that it has positive definite 3 3 minor cos π ū cos π ū so, the signature depends of the sign of the determinant Its value is = = det b ij = { cos π ū cos cos cos π } cos cos + ū cos π x [ cos π x cos π ] = ū { } cos cos cos π x π cos π x ū cos cos π cos cos = ū { } cos cos cos π x π cos π x ū cos cos π cos cos, ū where the terms in brackets { } are equal to the minor determinants B, B and B 3 of vertex figures for vertices Ā, Ā and Ā 3, respectively he above expression show that, if a vertex figure is of hyperbolic signature +, +,, ie the vertex is outer, then the determinant det b ij is negative It means, then the signature of b ij is +, +, +,, so the geometry is already hyperbolic In considering sign of B, B and B 3 it will be used the following identity = D = cos α cos β cos γ cos α cos β cos γ = cos α + cos β + cos γ cos α cos β + cos γ = = cosα + β cosα β cos γ [ cosα + β + cosα β + cos γ ] = = [ cosα + β + cos γ ] [ cosα β + cos γ ] = = 4 cos α + β + γ cos α + β γ cos α β + γ cos α + β + γ ], so cos α+β γ >, cos α β+γ >, cos α+β+γ > and We are working with angles in the interval [, π cos α+β+γ It follows that D > for α + β + γ > π, D = for α + β + γ = π, D < for α + β + γ < π If D is determinant of vertex figure it describes a vertex which is proper, ideal, outer, respectively Condition for Ā is outer vertex, if above we take α = π ū, β =, γ = with angel sum smaller then π Similar condition stands for Ā and Ā 3 when α = π x, β =, γ =, and α = π π x, β =, γ =, respectively π 4 Similarly as in [] we can express the generators of the Coxeter group m Γ ū, v, w, v = m, m, m, m 3 m = m = m = m 3 = = m m x = m m = m m 3 = m m w = = m m 3 v = m m 3 ū =, ū, w v, x 3

18 M Stojanović / Filomat 8:3 4, by matrices in basis Ā i a i i =,,, 3 Matrices for plane reflections m, m, m, m 3 are denoted by M, M, N, P 3, respectively: M = N = n n n 3, M =, P 3 = m m m 3 p 3 p 3 p 3 Here, m = cos π x, m = cos, m3 = cos, n = n = cos, n3 = cos π ū, p 3 = p 3 = cos, p 3 = cos π ū Namely, we have applied the above relations to the corresponding matrices, eg M N = n n n n n n n 3 = = If the polar plane of the vertex A 3 a 3 exists ie A 3 is hyperbolic, its form will be denoted by ā 3 It holds for the scalar products of forms ā 3, b i = for i =,,, ā 3, b 3 ; ā 3 = b a 3 + b a 3 + b a 3 + b 3 a 33 =: b i a i3 in the basis { b i} of forms in V 4 such that a i b j = δ j i, and bi = b ij a j express the plane - point polarity by b ij = b i, b j Its inverse b ij = aij, if exists, expresses the above equation āi = b j a ij for i = 3, as example a he reflection m 3 in the plane ā 3 can be expressed in the form basis { b i}, first by a matrix M 3 e i j as follows M 3 : b i b j e i j M 3 : b b b b 3 b b b b 3, e 3 e 3 e 3 and by its inverse matrix, ej i in the vector basis {aj } It is the same, since m 3 is an involution as a reflection in a plane a e 3 a a M 3 : e 3 a a e 3 a a 3 a 3 Since m 3 commutes with m, m and m, eg from m 3m = m m 3, ie M 3 M = M M 3 yields e 3 m m m 3 + e3 e 3 e 3 m m m e3 e3 + m e3 m3 e 3,

19 M Stojanović / Filomat 8:3 4, ie m 3 + e3 = m e3 + m e3 Similarly, we get from M 3 N = N M 3, n 3 + e3 = n e3 + n e3 From M 3 M = M M 3 we obtain e 3 e 3 e 3 e e 3 3 e 3 ie e 3 = e3 Solving the system of the unknown parameters e3, e3 and e3, the determinant of the system is D 3 = n m 4 + n m + π m = cos 4 cos + cos π x w = = 8 cos + 8 π x cos cos 8 = 8 + π x cos π x > hus, we can express e 3 = e3 = D 3 e 3 = D 3 n 3 n m 3 m n m n n 3 m 3 = cos π ū cos + cos cos π w + π x cos, π x = cos cos + cos π ū cos π w + π x cos π x b he reflection m in the plane ā can be described in the form basis { b i}, by a matrix M dj i, M : b i b j d i j, M : b b b b 3 b b b b d 3 d, d 3 and in the vector basis { a j } by M : a a a a 3 d d d 3 Again, m commutes with m, m and m 3, these determine d, d, d 3 Similarly as before d = d, m + d = m d + m3 d 3, d p = p 3 d + p 3 d Here, determinant of system is D = 4 m 3 p 3 m m3 p 3 cos π x = 8 + cos π x a a a a 3 and we get d = d = D d 3 = D m m p 3 p 3 p 3 = cos + cos cos π ū cos π x + cos π x, = cos π ū + cos cos cos π x cos π ū cos π x + cos π x m m 3 p p 3 3

20 M Stojanović / Filomat 8:3 4, c Finally, the reflection m in the plane ā can be expressed by M : b i b j c i j, M : b b b b 3 b b b b 3 c c c 3, and by M : a a a a 3 c c c 3 a a a a 3 Reflection m commutes with m, m and m 3 So, c + m = m c + m3 c 3, c + n = n c + n3 c 3, c p = p 3 c + p 3 c Determinant D is equal to Hence, D = 8 + m n3 p 3 + m3 n p 3 + n3 p 3 + m n + m3 p 3 = 3 cos π + + ū cos π + cos ū π + cos ū π + + ū c = m m m 3 n n 3 D p p cos π ū cos cos cos π cos cos ū c = D c 3 = D m m 3 n n n 3 p 3 p 3 m m n n p 3 p 3 p 3 = [ 3 cos π x cos π D ū 3 cos π x = 3 cos π x = ], cos π ū cos + cos, D cos π v + cos π ū cos D Summary he matrices above, expressed by the D-matrix ū, v, w, x with ū, v < w, 3 x, ie by angles π ū,,, π x, describes the extended reflection groups m Γ ū, v, w, x, which exist if conditions ū + v + w <, x + v <, x + w <, for vertices to be hyperbolic, are fulfilled Acknowledgment I thank Prof Emil Molnár for helpful comments to this paper References [] B Maskit, On Poincarés theorem for fundamental polygons, Adv in Math [] E Molnár, Eine Klasse von hyperbolischen Raumgruppen, Beiträge zur Algebra und Geometrie [3] E Molnár, Polyhedron complexes with simply transitive group actions and their realizations, Acta Math Hung [4] E Molnár, he projective interpretations of the eight 3-dimensional homogeneous geometries, Beiträge zur Algebra und Geometrie Contributions Alg Geom [5] E Molnár, IProk, A polyhedron algorithm for finding space groups, Proceedings of hird Int Conf On Engineering Graphics and Descriptive Geometry, Vienna 988, Vol, [6] E Molnár, I Prok, Classification of solid transitive simplex tilings in simply connected 3-spaces, Part I, Colloquia Math Soc János Bolyai 63 Intuitive Geometry, Szeged Hungary, 99 North-Holland

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