Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 2, 205-238

Symmetry Breaking of the Cube Tiling and the Spatial Chess Board by D-Symbols

Emil Molnár

Technical University of Budapest, Department of Geometry Egry J. út 1. H.22, H-1521 Budapest XI molnar e@dora.inflab.bme.hu

Abstract. The purpose of this paper is to popularize the method of D-symbols (Delone–Delaney–Dress symbols) for classifying periodic tilings (T , Γ), where a group Γ acts on a d-dimensional combinatorial tiling T with prescribed transitivity properties. As a new result, we describe the equivariant classification of all cube tilings (T , Γ) in Euclidean space E^{3}, where the group Γ acts on the infinite face-to-face cube tiling T with 1 transitivity class (Γ-orbit) of faces. In Table 1–3 the 43 classes of these pairs (T , Γ) are characterized by giving for each the number of the space group Γ, the D-symbol, a fundamental domain F in Fig.2, face identifying generators and defining relations for F and the Γ-conjugacy classes for stabilizers of solids (Γ^{3}), faces (Γ^{2}), edges (Γ^{1}) and vertices (Γ^{0}), respectively. Such classification problems seem to have applications, e.g. in the theory and praxis of solid state matter (Dirichlet–Voronoi tilings and coordination polyhedra). Therefore, we survey the general theory and indicate some problems as well.

1. Introduction

The chess board can be extended to an infinite square tiling with or without black and white coloring. Each of both can be characterized by the group of its self-symmetries. For the black and white tiling T_{ch} this group is Γ_{ch} = p4m a plane crystallographic group, which acts on T_{ch} with 2 (black resp. white) classes (Γ_{ch}-orbits) of tiles, 1 class of edges and 1 class of vertices (Fig.1.3.1). For the usual square tiling T_{sq} we have the isomorphic symmetry group Γ_{sq} = p4m (Fig.1.1.1) but Γ_{sq} has only 1 tile orbit, 1 edge orbit and 1 vertex orbit. All the 14 classes of pairs (T , Γ) have been classified [6], [7], [11] where Γ acts on the infinite combinatorial square tiling T_{sq} = T with 1 edge orbit. From these ”isotoxal” square tilings we can choose particular symmetry breakings of T_{ch} or T_{sq} (Fig.1) and their realizations (C: convex, N: non-convex, M: marked).

Analogous classification problems arise from the metric cube tiling T_{cu} (with or without coloring) in Euclidean space E^{3}. In this paper all 43 classes of (T , Γ) pairs are described

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1994 Heldermann Verlag, Berlin

where a space group Γ acts on a combinatorial cube tiling T_{cu} = T with exactly 1 face class (face orbit under Γ). In Tables 1–3 we see also particular symmetry breakings (T , Γ). For instance, we obtain 17 classes with exactly 1 Γ-orbit of solids, faces, edges, vertices, respectively (in Tables 1–2, (1)–(17)).

Furthermore we find 4 pairs (T , Γ) where the space group Γ acts on the cube tiling T with 2 solid orbits, 1 face orbit, 1 edge orbit and 1 vertex orbit (Table 3 (1^{⋆})–(4^{⋆})). These are the 4 symmetry breakings of the infinite spatial chess board with black and white coloring, in a restricted sense.

In this paper we intend to popularize the method of D-symbols (Delaney symbols, generalized Schläfli symbols), developed for classifying d-dimensional tilings with prescribed transitivity properties. A.W.M. Dress initiated this method [3] completing some ideas of M.S. Delaney [1]. The origins go back to the adjacency symbols of B.N. Delone (Delaunay) [2]. G. Grünbaum and G.C. Shephard applied adjacency symbols in various classifications in the Euclidean plane [7, Ch.6]. A. Dress and his coworkers R. Scharlau, R. Franz, D. Huson have developed a nice theory which is connected to the theory of generalized Coxeter groups (see [4], [5], [8], [9] for other references). They applied this theory to classifying 2-dimensional tilings. O. Delgado Friedrichs and D. Huson elaborated important and effective computer programs for Delaney symbols. I suggest using the name D-symbol in honour of Delone–Delaney–Dress.

Recently in [5] we have combined this theory with my polyhedron algorithm which finds all the discontinuous transformation groups for combinatorially given d-dimensional fundamental polyhedron, together with their stabilizers for any m-face class 0 ≤ m ≤ d−1; d ≥ 2 [15]. In the joint work [5] we have classified the 88 face transitive tilings of Euclidean space E^{3} by computer. The algorithm promises to be useful in other classifications in Euclidean and non-Euclidean spaces, e.g. minimal presentations for space groups (see [13], [17] or polyhedron realizations for 3-manifolds [14], etc.

We intend to illustrate the method with the familiar square and cube tilings and to give general ideas how to find tilings with prescribed transitivity properties. The results, appearing in Tables 1–3 and Fig.2, have been obtained by hand. These experiences of the author, however, were helpful in finding the general algorithm in [5] which will be sketched in Section 4.

2. The Concept of D-Symbol Illustrated by Square Tilings

Figure 1 shows the 14 equivariant classes of edge transitive symmetry breakings of the usual square tiling T which are considered combinatorially (or topologically, with Schläfli symbol {4, 4}). That means T is an incidence structure of 2-dimensional faces (squares), 1dimensional sides (curved edges) and 0-dimensional vertices (points). The formal barycentric subdivision of T leads to the simplex (triangle) set C where the (formal) midpoints of i-dimensional pieces of T (faces, edges, vertices) are labelled by i ∈ I := {0, 1, 2} as are the opposite simplex sides as well. Thus, a flag of T , i.e. a triple of incident 0-, 1-, 2-pieces, can uniquely be characterized by a labelled simplex of C. Two simplices C_{1} and C_{2} of C are called i-adjacent iff they have a common i-side, i.e. only their i-vertices are different. We also say that the involutive i-operation σ_{i} carries C_{1} onto C_{2} =: σ_{i}C_{1}. That means σ_{i} E. Molnár: Symmetry Breaking of the Cube Tiling 207

Figure 1 is first a permutation of simplices of C which assigns to any simplex of C the i-adjacent one.

(Later σ_{i}C_{1} will be considered also as an image of C_{1} under the ”local topological reflection” σ_{i} in the i-facet of C_{1}. Then our (necessarily) strange notations

C^{γ_{1}γ_{2}} := (C^{γ_{1}})^{γ_{2}},  σ_{j}σ_{i}C := σ_{j}(σ_{i}C),  (σ_{j}C)^{γ} = σ_{j}(C^{γ})

show how to get the corresponding simplex. Here γ’s in the exponent are global mappings of C, preserving the j-adjacencies, j ∈ I = {0, 1, 2}; σ_{i} is a local mapping, then it always relates to a fixed simplex C of C, so that σ_{i} preserves the j-adjacencies of C and σ_{i}C. In this sense we say: if the transformations of C act from the right, then the σ_{i}-operations act on C from the left. Our convention is to be in accordance with our Tables 1–3 and with our earlier papers [13]–[18]. See Sect.4 for more details and draw pictures.)

At a k-vertex of any C ∈ C some simplices of C meet cyclically at their i- and j-sides (i, j, k ∈ I are different). Let 2 · m_{ij}(C) denote the number of these joining simplices (m_{ij}(C) ∈ N the set of natural numbers). Then we can write

(σ_{j}σ_{i})^{m_{ij}(C)}C = C (2.1)

and the symmetric matrix function M : C → N_{I×I}, C ↦ M(C) = (m_{ij}(C)) will be introduced to describe the combinatorial structure of T .

In case of the square tiling T = T_{sq}

(m_{ij}(C)) = ( \begin{matrix}1 & 4 & 2 \\ 4 & 1 & 4 \\ 2 & 4 & 1 \end{matrix} ) (2.2)

is a constant matrix on C (m_{ii} = 1 means σ_{i} is involutive, {m_{01}, m_{12}} = {4, 4} is the Schläfli symbol, m_{02} = 2 since |i − j| > 1).

Now we assume that an isometry group Γ acts on T (as in our 14 cases in Fig.1) which is compatible with the barycentric subdivision C and the σ_{i}-operations as well. Γ factors the simplices of C into Γ-orbits, and it factors the matrix function as well. Edgetransitivity of Γ easily implies [11] that any Γ-orbit will be an element of a finite set D := {D_{1}, D_{2},…, D_{n}} (in Fig.1 D_{k} ↔ k, n = 1, 2, 4). From n =: |D| representative simplices of D a fundamental domain F for Γ can be glued by the σ_{i}-operations. Any σ_{i}- operation, defined on C, induces a definition of σ_{i} on D in a natural way (i ∈ I := {0, 1, 2}). Furthermore, if D := C^{Γ} := {C^{γ} : γ ∈ Γ} is the Γ-orbit of C in C, then m_{ij}(D) := m_{ij}(C) by (2.2) defines the matrix function

M : D ⟶ N_{I×I}, D ⟼ M(D) = (m_{ij}(D)) (2.3)

on the D-diagram (D-graph). D and M together are called a D-symbol D(M).

In our example Fig.1.3.1 D(M) determines the group Γ = p4m which has a fundamental domain F consisting of 2 simplices of C. Thus the D-diagram D has 2 nodes (vertices). The σ_{i}-operations

σ_{0} : (1)(2) · · · · · · , σ_{1} : (1)(2) − −−, σ_{2} : (1, 2) −, (2.4) as involutive permutations, are indicated by the above i-lines as distinguished (colored) edges. These σ_{i}’s also show the line reflections

(2.5) m_{0} and m′_{0} (by σ_{0}), m_{1} and m′_{1} (by σ_{1})

in the sides of F as generators for Γ = p4m. Any partial D-symbol D^{k}(M) = D_{ij}(M) of D(M) (i, j, k ∈ I are different), obtained by cancelling the σ_{k}-operation, i.e. the k-lines, describes the simplicial neighbourhoods of the various k-vertices in C and the stabilizer subgroups Γ^{k} in Γ by a local fundamental domain F^{k}. We shall have precisely as many Γ-conjugacy classes for Γ^{k}, as we have Γ-orbits for k-vertices, as we have connected components of D^{k} = D_{ij} in D(M).

In our case 3.1 D(M) has 2 components of D^{2} = D_{01} each with 1 =: e_{01} node (Dsimplex). As m_{01}(D_{1}) = 4 = m_{01}(D_{2}), so we have the defining relations for our generators in (2.5):

(m_{0}m_{1})^{4} = 1 = (m′_{0}m′_{1})^{4}, since m_{01}/e_{01} = 4.

Similarly, we have 1 component of D^{1} = D_{02}, consisting of 2 =: e_{02} nodes, and m_{02}(D_{1}) = 2 = m_{02}(D_{2}) involves the relation

m_{0}m′_{0} = 1 or m_{0} = m′_{0}, since m_{02}/e_{02} = 1.

We have 1 component of D^{0} = D_{12} consisting of 2 =: e_{12} nodes, thus m_{12}(D_{1}) = 4 = m_{12}(D_{2}) implies the relation (m_{1}m′_{1})^{2} = 1 because m_{12}/e_{12} = 2 is just the exponent. So we see, the D-symbol D(M) determines the group Γ = p4m by the presentation

\begin{matrix}Γ = ( & m_{0}, m′_{0}, m_{1}, m′_{1} − 1 = m_{0}^{2} = m′_{0}^{2} = m_{1}^{2} = m′_{1}^{2} = \\ = (m_{0}m_{1})^{4} = (m′_{0}m′_{1})^{4} = m_{0}m′_{0} = (m_{1}m′_{1})^{2}). \end{matrix} (2.6)

We choose a common edge (connecting 0- and 1-vertices) of simplices D_{1} and D_{2} in the fundamental domain F, with great freedom, and apply the isometries of Γ. So we get the simplices of C and the square tiling (T , Γ) determined by D(M) up to an equivariant deformation.

In Sect.4.3 we shall give a general definition of the equivariant classification of ddimensional tilings (T , Γ), and the isomorphism classification of D-symbols D(M), and we will see that the two definitions correspond to each other (Th.4.1).

In the cases described in Fig.1 all the 14 square tilings (T_{s}, Γ) are edge-transitive, i.e. each D-symbol D(M) has a partial symbol D^{1} = D_{02} consisting of one component. Since any edge in T_{sq} belongs to 4 simplices in the barycentric subdivision C_{sq}, a fundamental domain F contains at most 4 simplices and so does the D-set D. Thus we can easily check by hand that our classification is complete.

We emphasize that combinatorially different fundamental domains F_{1} and F_{2} for the group Γ may provide the same (equivariant) tiling (see e.g. the cases 2.1–5 in Fig.1). This is in accordance with the fact that F_{1} may be rebuilt to F_{2} by cutting and gluing simplices at the labelled sides as the σ_{i}-operations dictate in the D-symbol D(M).

Summarizing, a D-symbol D(M) can geometrically be considered as an ”amoeba-like” fundamental domain F with a numbered simplex subdivision and identifications, where labelled sides are distinguished by the σ_{i}-operation (i ∈ I), for the possible cutting and gluing procedures.

In Sect.4.3 we shall also define the symmetry breakings of a given tiling T which will be related with the D-morphisms onto the D-symbol of (T , Γ = Aut T ).

Actually, we can say that all the 14 edge-transitive square tilings in Fig.1 can be obtained by symmetry breaking from the most symmetric one (T_{sq}, Γ_{sq} = Aut T_{sq} = p4m) under case 1.1.

3. Symmetry Breakings of the Cube Tilings in E^{3}

The concept of D-symbol for any 3-dimensional tiling (T , Γ) can be formed in full analogy to Sect.2. We consider a Γ-compatible barycentric subdivision of T into the set C of labelled simplices. The σ_{i}-operations are introduced for i ∈ I := {0, 1, 2, 3}. The new σ_{3}-operation or 3-adjacency is denoted by ∼∼∼ lines in any D-diagram. Here the D-set D := C/Γ is the set of Γ orbits of C, and the σ_{i}-operations (i ∈ I) are again induced from C.

D can be characterized by an ”amoeba-like” fundamental domain F consisting of labelled simplices, each of them representing Γ-orbit in C. Moreover, the σ_{i}-operations (i ∈ I) define how to glue these simplices together along their faces, and how to pair the faces of F, respectively. The face pairing isometries generate the group Γ. Again, the symmetric matrix function C ↦ M(C) = (m_{ij}(C)) i, j ∈ I := {0, 1, 2, 3} on C determines the combinatorial structure of T . By Γ-factorization we again obtain the matrix function in formula (2.3) and the D-symbol D(M) for (T , Γ).

If T_{cu} is the familiar cube tiling of the Euclidean space E^{3} then the matrix function is constant on C_{cu}:

(m_{ij}) = ( \begin{matrix}1 & 4 & 2 & 2 \\ 4 & 1 & 3 & 2 \\ 2 & 3 & 1 & 4 \\ 2 & 2 & 4 & 1 \end{matrix} ) . (3.1)

Here m_{01} = 4 describes the square faces, since 8 simplices meet at any 23-edge in the barycentric subdivision C_{cu} cyclically along their 0- and 1-faces. A 23-edge connects a (2dimensional) face centre with a (3-dimensional) solid centre. Analogously, m_{23} = 4 implies that 4 cubes of T_{cu}, i.e. 8 simplices of C_{cu} meet at any 01-edge cyclically along the 2- and 3-faces. To describe the 3 faces and edges at any vertex of the cube, we write m_{21} = 3 in (3.1). Indeed, 6 simplices of C_{cu} meet at any 03-edge cyclically along the 1- and 2-faces.

Any group Γ, acting on the cube tiling T_{cu}, induces a D-symbol D(M) with the constant matrix (3.1). The D-symbol also describes a fundamental domain F for Γ, consisting of |D| simplices as indicated above.

The automorphism group of the combinatorial cube tiling T can be considered as the crystallographic space group Γ = Pm¯3m of the metric cube tiling (T_{cu}, Γ = Pm¯3m). The fundamental domain F = AEKQ is the simplex illustrated in Fig.2.a. Thus the Ddiagram is a node, together with the four σ_{i}-loops (i ∈ I), as Table 1.No.1 shows it. All E. Molnár: Symmetry Breaking of the Cube Tiling 211

Figure 2 the identifications on F are plane reflections m_{0}, m_{1}, m_{2}, m_{3} in the corresponding simplex faces. The matrix (3.1) implies the defining relations for Γ in Table 1, as usual for so-called Coxeter groups generated by plane reflections.

In general, the partial D-symbol components obtained from D by cancelling any σ_{k}- operation, characterizes the lower-dimensional simplicial neighbourhoods of the k-midpoint orbit(s) in C and their stabilizer(s) Γ^{k} in Γ.

E.g. by cancelling σ_{3}-operation in Table 1.No.1, we get D^{3}(M) = D_{012}(M) with the corresponding 3×3 submatrix of (3.1). This describes a spherical tiling around any 3-vertex of C_{cu} with stabilizer Γ^{3}(Q) = m¯3m as a crystallographic point group, the symmetry group of a fixed cube. Analogously, we get the stabilizer Γ^{2}(K) of the 2-midpoint and every point in its Γ-orbit (and for the corresponding cube face orbit), resp. the stabilizer Γ^{0}(A) of the 0-vertex orbit in Fig.2.a.

We are searching for all the symmetry breakings (T , Γ) of the Euclidean cube tiling T_{cu} = T where, in addition, the isometry group Γ acts on T with one transitivity class of T -faces.

To do this, we classify all D-symbols D(M) (up to D-isomorphism) with constant matrix function M : D → N_{I×I}, D ↦ (m_{ij}(D)) by (3.1) such that the partial symbol D^{2}(M) = D_{013}(M), obtained from D(M) by cancelling the σ_{2}-operation, has a connected D-diagram. Moreover, all the stabilizers Γ^{l} < Γ, according to the components of the partial symbol D^{l}(M) = D_{ijk}(M) (i, j, k, l ∈ I are different), are finite (spherical) point groups.

In other words we will look for all the above D-symbols and the corresponding fundamental domains F. Such a domain F will contain a piece of a cube face as Fig.2 shows, i.e. F consists of |D| = 1, 2, 4, 8, 16 simplices supporting the cube face from one side or both sides. The face pairing on F will generate a crystallographic space group Γ in E^{3}.

As we will see, there are 43 cases, enumerated in Tables 1–3, referring to Fig.2.

3.1. Table 1

Consider the cases of Fig.2.a–h, only the upper domains with 3-midpoint (cube centre) Q. Then all the cube centres are in one Γ-orbit and the face stabilizer Γ^{2}(K) changes the sides of the face. Any such D-symbol D(M) has exactly one D^{23} = D_{01} component (if we cancel the σ_{2}- and σ_{3}-operations and keep σ_{0} and σ_{1} of D(M)).

For instance, let us consider the case No.11 where in Fig.2.e we have chosen the domain F = ABF GQ. The D-diagram dictates the face pairing generators r_{1}, r, r′, r_{3} as described there (see also Sect.4.2). The components of D_{01} = D^{23}, D_{02} = D^{13}, D_{03} = D^{12}, D_{12} = D^{03}, D_{13} = D^{02}, D_{23} = D^{01}, respectively, provide the defining relations:

\begin{aligned}1 & = r_{1}^{2} = r_{1}rr′ = r_{3}^{2} = (r_{1}r_{3})^{2} = r^{3} = r′^{3} = \\ & = trivial = (r_{3}rr_{3}r′)^{2} \end{aligned} (3.1.1)

which also describe the corresponding stabilizers

\begin{matrix}Γ^{23}(KQ) = 2, \\ Γ^{12}(EK) = 2, \\ Γ^{03}(AQ) = 3, \\ Γ^{02}(AK, BK) = 1, \end{matrix}  \begin{matrix}Γ^{13}(F Q, GQ, EQ) = 1, \\ Γ^{12}(F K, GK) = 2, \\ Γ^{03}(BQ) = 3, \\ Γ^{01}(BF, AG, AE, BE) = 2 \end{matrix}

of the simplex edge classes of C_{cu}, respectively.

How to get the last relation, say, from D_{23} = D^{01} whose diagram is

◯ 1 ; ∼∼∼∼∼∼∼∼∼ ; ◯ 2 ; |  | ; ◯ 4 ; ∼∼∼∼∼∼∼∼∼ ; ◯ 3 with m_{23} = 4.

This dictates the cyclic order of the Γ-images of the representing simplices from F around BF in C_{cu} as follows (see Fig.2.e):

D_{3} = BF KQ; D_{2}^{r′} = (BEKQ)^{r′}; D_{1}^{r_{3}r′} = (AEKQ)^{r_{3}r′};

D_{4}^{rr_{3}r′} = (AGKQ)^{rr_{3}r′}; D_{3}^{r_{3}rr_{3}r′} = (BF KQ)^{r_{3}rr_{3}r′}; (3.1.2)

and so on up to D_{3}^{(r_{3}rr_{3}r′)(r_{3}rr_{3}r′)} = D_{3},

since we have 8 simplices because of m_{23} = 4. So we have just obtained the relation (r_{3}rr_{3}r′)^{2} = 1. The general procedure for dimension d, where we go around a (d − 2)- dimensional simplex ”edge” in C to obtain a defining relation among the identification generators of Γ, are described in [15] and is called Poincaré algorithm, since it was first applied by him for d = 2, 3 (of course in other formulation [19], see also [12] and our Sect.4.2).

Table 1.No.11 contains also the corresponding stabilizers Γ^{k} (k ∈ I) for k-midpoint orbits in C_{cu}. For instance Γ^{0}(A, B) = 23 (as a point group) has the partial symbol D^{0} = D_{123} by cancelling σ_{0}-operation.

◯ 1 ∼∼∼∼∼∼∼∼∼◯ 2
| | ; |  | ; | |
◯ 4 ∼∼∼∼∼∼∼∼∼◯ 3

Thus we can produce a local fundamental domain F^{0} consisting of 4 spherical simplices glued, say, around vertex A. From D^{0} and from (3.1) we derive two 3-rotations and one 2-rotation as in case of point group 23. As we see, this tiling has exactly one Γ-orbit of 3-, 2-, 1-, 0-midpoints each. So we have obtained a symmetry breaking No.(8) of the cube tiling in a restricted sense among the 17 ones.

214 E. Molnár: Symmetry Breaking of the Cube Tiling

Table 1

{1 \atop (1)} \begin{matrix}221. \\ P m¯3m \end{matrix} \begin{matrix}Fig.2.a (C) \\ F = AEKQ \end{matrix} \frac{m_{0} : EKQ, m_{1} : AKQ, m_{2} : AEQ, m_{3} : AEK}{(m_{i}m_{j})^{m_{ij}} = 1 by (3.1); i, j ∈ I := \{0, 1, 2, 3\}}

{Γ^{3}(Q) = m¯3m, Γ^{2}(K) = 4/mmm, \atop Γ^{1}(E) = 4/mmm, Γ^{0}(A) = m¯3m}

{2 \atop (2)} \begin{matrix}200. \\ P m¯3 \end{matrix} \begin{matrix}Fig.2.c (M) \\ F = AEKF Q \end{matrix} \frac{m_{0} : EKQ, m′_{0} : F KQ, m_{3} : AEKF, r: AEQ ⟶ AF Q}{{m_{0}^{2}, m′_{0}^{2}, m_{3}^{2}, r^{3}, (m_{0}m′_{0})^{2}, (m_{0}m_{3})^{2}, (m′_{0}m_{3})^{2}, \atop m_{0}rm′_{0}r^{−1}, (m_{3}rm_{3}r^{−1})^{2}}}

{Γ^{3}(Q) = m¯3, Γ^{2}(K) = mmm, \atop Γ^{1}(E, F) = mmm, Γ^{0}(A) = m¯3}

{3 \atop (3)} \begin{matrix}226. \\ F m¯3c \end{matrix} \begin{matrix}Fig.2.c (N) \\ F = AEKF Q \end{matrix} \frac{{m_{0} : EKQ, m′_{0} : F KQ, r_{3} : AEKF ⟶ AF KE, \atop r: AEQ ⟶ AF Q}}{{m_{0}^{2}, m′_{0}^{2}, r_{3}^{2}, r^{3}; (m_{0}m′_{0})^{2}, m_{0}r_{3}m′_{0}r_{3}, m_{0}rm′_{0}r^{−1}, \atop (r_{3}r)^{4}}}

{Γ^{3}(Q) = m¯3, Γ^{2}(K) = ¯42m, \atop Γ^{1}(E, F) = 4/m, Γ^{0}(A) = 432}

E. Molnár: Symmetry Breaking of the Cube Tiling 215

Table 1 (continued)

4 \begin{matrix}225. \\ F m¯3m \end{matrix} \begin{matrix}Fig.2.b (M) \\ F = ABKQ \end{matrix} \frac{m_{1} : BKQ, m′_{1} : AKQ, m_{2} : ABQ, m_{3} : ABK}{{m_{1}^{2}, m′_{1}^{2}, m_{2}^{2}, m_{3}^{2}; (m_{1}m′_{1})^{2}, (m_{1}m_{2})^{3}, (m_{1}m_{3})^{2}, \atop (m′_{1}m_{2})^{3}, (m′_{1}m_{3})^{2}, (m_{2}m_{3})^{4}}}

{Γ^{3}(Q) = ¯43m, Γ^{2}(K) = mmm, Γ^{1}(E) = 4mm, \atop Γ^{0}(A) = m¯3m, Γ^{0}(B) = m¯3m}

{5 \atop (4)} \begin{matrix}215. \\ P ¯43m \end{matrix} \begin{matrix}Fig.2.b (N) \\ F = ABKQ \end{matrix} \frac{m_{1} : BKQ, m′_{1} : AKQ, m_{2} : ABQ, r: EAK ⟶ EBK}{{m_{1}^{2}, m′_{1}^{2}, m_{2}^{2}, r^{2}, (m_{1}m′_{1})^{2}, (m_{1}m_{2})^{3}, m_{1}rm′_{1}r, \atop (m′_{1}m_{2})^{3}, (m_{2}rm_{2}r)^{2}}}

{Γ^{3}(Q) = ¯43m, Γ^{2}(K) = ¯42m, \atop Γ^{1}(E) = ¯42m, Γ^{0}(A, B) = ¯43m}

{6 \atop (5)} \begin{matrix}226. \\ F m¯3c \end{matrix} \begin{matrix}Fig.2.b (M) \\ F = ABKQ \end{matrix} \frac{r_{1} : KAQ ⟶ KBQ, r_{2} : EAQ ⟶ EBQ, m: ABK}{r_{1}^{4}, r_{2}^{2}, m^{2}; (r_{1}r_{2})^{3}, mr_{1}mr_{1}^{−1}, (mr_{2}mr_{2})^{2}}

{Γ^{3}(Q) = 432, Γ^{2}(K) = 4/m, \atop Γ^{1}(E) = ¯42m, Γ^{0}(A, B) = m¯3}

\begin{matrix}Fig.2.c (M) \\ F = AEKF Q \end{matrix} \frac{r_{1} : KF Q ⟶ KEQ, r: AEQ ⟶ AF Q, m: AEKF}{r_{1}^{4}, r^{3}, m^{2}; (r_{1}r)^{2}, mr_{1}mr_{1}^{−1}, (mrmr^{−1})^{2}}

{Γ^{3}(Q) = 432, Γ^{2}(K) = 4/m, \atop Γ^{1}(E, F) = ¯42m, Γ^{0}(A) = m¯3}

216 E. Molnár: Symmetry Breaking of the Cube Tiling

Table 1 (continued)

{7 \atop (6)} \begin{matrix}207. \\ P 432 \end{matrix} \begin{matrix}Fig.2.b (N) \\ F = ABKQ \end{matrix} \frac{{r_{1} : KAQ ⟶ KBQ, r_{2} : EAQ ⟶ EBQ, \atop r_{3} : EAK ⟶ EBK}}{r_{1}^{4}, r_{2}^{2}, r_{3}^{2}; (r_{1}r_{2})^{3}, (r_{1}r_{3})^{2}, (r_{2}r_{3})^{4}}

{Γ^{3}(Q) = 432, Γ^{2}(K) = 422, \atop Γ^{1}(E) = 422, Γ^{0}(A, B) = 432}

\begin{matrix}Fig.2.c (N) \\ F = AEKF \end{matrix} \frac{{r_{1} : KF Q ⟶ KEQ, r: AEQ ⟶ AF Q, \atop r′_{3} : AF K ⟶ AEK}}{r_{1}^{4}, r^{3}, r′_{3}^{2}; (r_{1}r)^{2}, (r_{1}r′_{3})^{2}, (rr′_{3})^{4}}

{Γ^{3}(Q) = 432, Γ^{2}(K) = 422, \atop Γ^{1}(E, F) = 422, Γ^{0}(A) = 432}

8 \begin{matrix}229. \\ Im¯3m \end{matrix} \begin{matrix}Fig.2.d (M) \\ F = ABCQ \end{matrix} \frac{m_{1} : ACQ, m_{2} : ABQ , m_{3} : ABC, r: F BQ ⟶ F CQ}{{m_{1}^{2}, m_{2}^{2}, m_{3}^{2}, r^{2}; (m_{1}m_{2})^{3}, (m_{1}m_{3})^{2}, \atop m_{1}rm_{2}r, (m_{2}m_{3})^{4}, (m_{3}rm_{3}r)^{2}}}

{Γ^{3}(Q) = ¯3m, Γ^{2}(K) = mm2, Γ^{1}(E) = 4/mmm, \atop Γ^{1}(F) = ¯42m, Γ^{0}(A) = m¯3m, Γ^{0}(B, C) = 4/mmm}

{9 \atop (7)} \begin{matrix}166. \\ R¯3m \end{matrix} \begin{matrix}Fig.2.g(d) (C) \\ F = ABCQ (N) \end{matrix} \frac{{m_{1} : ACQ, m_{2} : ABQ, r: F BQ ⟶ F CQ, \atop r_{3} : KAB ⟶ KCB}}{{m_{1}^{2}, m_{2}^{2}, r^{2}, r_{3}^{2}; (m_{1}m_{2})^{3}, m_{1}rm_{2}r, \atop m_{1}r_{3}m_{1}r_{3}, m_{2}r_{3}r_{2}r_{3}m_{2}r_{3}r_{2}r_{3}}}

{Γ^{3}(Q) = ¯3m, Γ^{2}(K) = 2/m, Γ^{1}(E, F) = 2/m, \atop Γ^{0}(A, B, C) = ¯3m}

E. Molnár: Symmetry Breaking of the Cube Tiling 217

Table 1 (continued)

10 \begin{matrix}202. \\ F m¯3 \end{matrix} \begin{matrix}Fig.2.d(e) (M) \\ F = ABCQ \end{matrix} \frac{r_{1} : KAQ ⟶ KCQ, r: BCQ ⟶ BAQ, m: ABC}{r_{1}^{2}, r^{3}, m^{2}; (r_{1}r)^{3}, mr_{1}mr_{1}, (mrmr^{−1})^{2}}

{Γ^{3}(Q) = 23, Γ^{2}(K) = 2/m, Γ^{1}(E, F) = mm2, \atop Γ^{0}(A, C) = m¯3, Γ^{0}(B) = m¯3}

{11 \atop (8)} \begin{matrix}195. \\ P 23 \end{matrix} \begin{matrix}Fig.2.e(d) (N) \\ F = ABF GQ \end{matrix} \frac{{r_{1} : KF Q ⟶ KGQ, r: AGQ ⟶ AEQ, \atop r′: BEQ ⟶ BF Q, r_{3} : KF BE ⟶ KGAE}}{r_{1}^{2}, r^{3}, r′^{3}, r_{3}^{2} ; r_{1}rr′, (r_{1}r_{3})^{2}, (r_{3}rr_{3}r′)^{2}}

{Γ^{3}(Q) = 23, Γ^{2}(K) = 222, \atop Γ^{1}(E, F, G) = 222, Γ^{0}(A, B) = 23}

{12 \atop (9)} \begin{matrix}219. \\ F ¯43c \end{matrix} \begin{matrix}Fig.2.d(e) (N) \\ F = ABCQ \end{matrix} \frac{{z: KCB ⟶ KBA, r_{1} : KAQ ⟶ KCQ, \atop r: BCQ ⟶ BAQ}}{r_{1}^{2}, r^{3}; zzr_{1}, (zr^{−1}zr^{−1})^{2}, (r_{1}r)^{3}}

{Γ^{3}(Q) = 23, Γ^{2}(K) = ¯4, Γ^{1}(E, F) = ¯4, \atop Γ^{0}(A, B, C) = 23}

Table 1 (continued)

13 \begin{matrix}209. \\ F 432 \end{matrix} \begin{matrix}Fig.2.d(e) (N) \\ F = ABCQ \end{matrix} \frac{{r_{1} : KAQ ⟶ KCQ, r: BCQ ⟶ BAQ, \atop r_{3} : BAK ⟶ BCK}}{r_{1}^{2} , r^{3}, r_{3}^{2} ; (r_{1}r)^{3}, (r_{1}r_{3})^{2}, (rr_{3})^{4}}

{Γ^{3}(Q) = 23, Γ^{2}(K) = 222, Γ^{1}(E, F) = 4, \atop Γ^{0}(A, C) = 432, Γ^{0}(B) = 432}

14 \begin{matrix}223. \\ P m¯3n \end{matrix} \begin{matrix}Fig.2.f (M) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, \atop r_{2} : GCQ ⟶ GDQ, m: ABCD ⟶ ABCD}}{{r^{3}, r_{1}^{2} , r_{2}^{2} , m^{2} ; rr_{1}r_{2} , (mrmr^{−1})^{2}, \atop (mr_{1}mr_{1})^{2}, (mr_{2}mr_{2})^{2}}}

Γ^{3}(Q) = 32, Γ^{2}(K) = m, Γ^{1}(E, H) = mm2, Γ^{1}(F) = ¯42m, Γ^{1}(G) = ¯42m, Γ^{0}(A) = m¯3, Γ^{0}(B, C, D) = mmm

15 \begin{matrix}211. \\ I432 \end{matrix} \begin{matrix}Fig.2.f (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, \atop r_{2} : GCQ ⟶ GDQ, r_{3} : ABC ⟶ ADC}}{r^{3}, r_{1}^{2} , r_{2}^{2} , r_{3}^{2} ; rr_{1}r_{2} , (rr_{3})^{4}, (r_{3}r_{1}r_{3}r_{2})^{2}}

σ_{0} : (1, 2)(3, 4)(5, 6)(7, 8) ; σ_{1} : (1, 8)(2, 3)(4, 5)(6, 7) ; σ_{2} : (1, 8)(2, 7)(3, 4)(5, 6) ; σ_{3} : (1, 8)(2, 7)(3, 6)(4, 5)  {Γ^{3}(Q) = 32, Γ^{2}(K) = 2, Γ^{1}(E, H) = 4, \atop Γ^{1}(F, G) = 222, Γ^{0}(A) = 432, Γ^{0}(B, C, D) = 422}

{16 \atop (10)} \begin{matrix}167. \\ R¯3c \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, \atop r_{2} : GCQ ⟶ GDQ, z: ABCD(K) ⟶ CDBA(K)}}{r^{3}, r_{1}^{2} , r_{2}^{2} , z^{2}; rr_{1}r_{2} , zr_{2}zr^{−1}zr_{1}zr}

σ_{0} , σ_{1} , σ_{2} as before σ_{3} : (1, 5)(2, 6)(3, 7)(4, 8)

{Γ^{3}(Q) = 32, Γ^{2}(K) = ¯1, Γ^{1}(E, F, G, H) = 2, \atop Γ^{0}(A, B, C, D) = ¯3}

E. Molnár: Symmetry Breaking of the Cube Tiling 219

Table 1 (continued)

{17 \atop (11)} \begin{matrix}155. \\ R32 \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, \atop r_{2} : GCQ ⟶ GDQ, r_{3} : BCD ⟶ BAD}}{r^{3}, r_{1}^{2} , r_{2}^{2} , r_{3}^{2} ; rr_{1}r_{2}, rr_{3}r_{1}r_{3}r^{−1}r_{3}r_{2}r_{3}}

σ_{0} , σ_{1} , σ_{2} as before σ_{3} : (1, 4)(2, 3)(5, 8)(6, 7)

{Γ^{3}(Q) = 32, Γ^{2}(K) = 2, Γ^{1}(E, F, G, H) = 2, \atop Γ^{0}(A, B, C, D) = 32}

18 \begin{matrix}204. \\ Im¯3 \end{matrix} \begin{matrix}Fig.2.f (M) \\ F = ABCDQ \end{matrix} \frac{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, m: ABCD}{r^{3}, m^{2}; rzz, (mrmr^{−1})^{2}, (mzmz^{−1})^{2}}

σ_{0} , σ_{1} as before σ_{2} : (1, 8)(2, 7)(3, 5)(4, 6) σ_{3} : (1)(2)(3)(4)(5)(6)(7)(8)

{Γ^{3}(Q) = ¯3, Γ^{2}(K) = m, Γ^{1}(E, H) = mm2, \atop Γ^{1}(F, G) = mm2, Γ^{0}(A) = m¯3, Γ^{0}(B, C, D) = mmm}

{19 \atop (12)} \begin{matrix}206. \\ Ia¯3 \end{matrix} \begin{matrix}Fig.2.f (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop r_{3} : EADG ⟶ EBCG}}{r^{3}, r_{3}^{2} ; rzz, r_{3}r^{−1}r_{3}zr_{3}z^{−1}r_{3}r}

σ_{0} , σ_{1} , σ_{2} as before σ_{3} : (1, 2)(3, 8)(4, 7)(5, 6)

{Γ^{3}(Q) = ¯3, Γ^{2}(K) = 2, Γ^{1}(E, F, G, H) = 2, \atop Γ^{0}(A, B, C, D) = ¯3}

20 \begin{matrix}222. \\ P n¯3n \end{matrix} \begin{matrix}Fig.2.f (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop r_{3} : ABC ⟶ ADC}}{r^{3}, r_{3}^{2} ; rzz, (rr_{3})^{4}, (zr_{3}zr_{3})^{2}}

σ_{0} , σ_{1} , σ_{2} as before σ_{3} : (1, 8)(2, 7)(3, 6)(4, 5)

{Γ^{3}(Q) = ¯3, Γ^{2}(K) = 2, Γ^{1}(E, H) = 4, \atop Γ^{1}(F, G) = ¯4, Γ^{0}(A) = 432, Γ^{0}(B, C, D) = 422}

{21 \atop (13)} \begin{matrix}148. \\ R¯3 \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop z_{3} : ABCD(K) ⟶ CDAB(K)}}{r^{3}, z_{3}^{2} ; rzz, z_{3}z^{−1}z_{3}rz_{3}z^{−1}z_{3}r}
σ_{3} : (1, 5)(2, 6)(3, 7)(4, 8)  {Γ^{3}(Q) = ¯3, Γ^{2}(K) = ¯1, Γ^{1}(E, F, G, H) = ¯1, \atop Γ^{0}(A, B, C, D) = ¯3}
{22 \atop (14)} \begin{matrix}167. \\ R¯3c \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ F = ABCDQ \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop r_{3} : BAD ⟶ BCD}}{r^{3}, r_{3}^{2} ; rzz, r_{3}zr_{3}rr_{3}zr_{3}r}
σ_{3} : (1, 4)(2, 3)(5, 8)(6, 7)  {Γ^{3}(Q) = ¯3, Γ^{2}(K) = 2, Γ^{1}(E, F, G, H) = ¯1, \atop Γ^{0}(A, B, C, D) = 32}

If we cut off the simplex D_{4} = AGKQ along AKQ, and glue it GKQ → F KQ by the 2-rotation r_{1}, then we get a new fundamental domain ABCQ in Fig.2.d with the same D-symbol but another presentation for Γ = P23 as follows:

\begin{matrix}r_{1} : KCQ ⟶ KAQ, & r′ : BAQ ⟶ BCQ, \\ r_{3} : KAE ⟶ KBE, & r′_{3} : KBF ⟶ KCF \end{matrix}

are the generators, and

\begin{aligned}1 & = r_{1}^{2} = trivial = r_{3}^{2} = r′_{3}^{2} = (r_{1}r′) = \\ & = r′^{3} = r_{3}r′_{3}r_{1} = r_{3}r′r′_{3}r′^{−1} \end{aligned} (3.1.3)

are the defining relations. Furthermore, we have the stabilizers

Γ^{3}(Q) = 23, Γ^{2}(K) = 222, Γ^{1}(E, F) = 222, Γ^{0}(A, B, C) = 23

for the 3-, 2-, 1-, 0-midpoint orbits under Γ = P23.

As we see, different fundamental domains may represent the same D-symbol and tiling as we indicated in Sect.2. These phenomena appear in Table 1.No.6–7, and they are indicated at No.10–13 as well.

3.2. Table 2

Consider the cases of Fig.2.i–j. Again, all the cube centres will be in one Γ-orbit, although the face stabilizer Γ^{2}(K) does not change the sides of the face. Thus, any D-symbol D(M) has two D^{23} = D_{01} components, but only one D^{3} = D_{012} component, describing a symmetry group Γ^{3}(Q) of the cube by a local spherical fundamental domain F^{3}. There are 2 cube face orbits (K, K′; by D_{01}) in Γ^{3}(Q) (by D_{012}), as Fig.2.i or 2.j illustrate. They have to be connected by σ_{3}-operation of D, i.e. by identifying the 2 cube faces under Γ.

For example, consider No.25 with Fig.2.j where the D-symbol D(M) contains 16 elements as simplices

\begin{matrix}D_{1} = ALKQ, D_{2} = XLKQ, …, & D_{8} = AEKQ, \\ D_{9} = AEK′Q, D_{10} = BEK′Q, …, & D_{16} = AHK′Q. \end{matrix}

We prescribe the following operations in D (D_{x} ↔ x):

\begin{matrix}σ_{0} : & (1, 2)(3, 4)(5, 6)(7, 8)(9, 10)(11, 12)(13, 14)(15, 16) · · · · · · \\ σ_{1} : & (1, 8)(2, 3)(4, 5)(6, 7)(9, 16)(10, 11)(12, 13)(14, 15) − −− \\ σ_{2} : & (8, 9)(7, 10)(1, 4)(2, 3)(5, 16)(6, 15)(11, 14)(12, 13) − \\ σ_{3} : & (1, 15)(2, 16)(3, 9)(4, 10)(5, 11)(6, 12)(7, 13)(8, 14) ∼∼∼ \end{matrix} (3.2.1)

E. Molnár: Symmetry Breaking of the Cube Tiling 221

Table 2

{23 \atop (15)} \begin{matrix}160. \\ R3m \end{matrix} \begin{matrix}Fig.2.i(g) (N) \\ F = ABCX \end{matrix} \frac{m_{1} : BCX, m_{2} : AXC, s: XAB ⟶ ABC}{m_{1}^{2} , m_{2}^{2} ; (m_{1}m_{2})^{3}, m_{1}sm_{2}s^{−1}, m_{1}s^{−2}m_{2}s^{2}}
σ_{0} : (1, 2)(3, 4)(5, 6)(7, 8) ; σ_{1} : (1)(2, 3)(4)(5)(6, 7)(8) ; σ_{2} : (1)(2)(3, 5)(4, 6)(7)(8) ; σ_{3} : (1, 5)(2, 6)(3, 7)(4, 8)  {Γ^{3}(Q) = 3m, Γ^{2}(K, K′) = m, Γ^{1}(E, F, G) = m, \atop Γ^{0}(A, B, C, X) = 3m}
24 \begin{matrix}217. \\ I¯43m \end{matrix} \begin{matrix}Fig.2.i (N) \\ F = ABCX \end{matrix} \frac{m_{1} : BCX, m_{2} : AXC, z: EAXB ⟶ EBCA}{m_{1}^{2} , m_{2}^{2} , (zz)^{2}; (m_{1}m_{2})^{3}, m_{1}zm_{2}z^{−1}, (m_{1}z^{−1}m_{2}z)^{2}}

σ_{0} , σ_{1} , σ_{2} as before σ_{3} : (1, 8)(2, 7)(3, 6)(4, 5)

{Γ^{3}(Q) = 3m, Γ^{2}(K, K′) = m, Γ^{1}(E) = ¯4, \atop Γ^{1}(F, G) = mm2, Γ^{0}(A, B) = ¯42m, Γ^{0}(C, X) = ¯43m}

{25 \atop (16)} \begin{matrix}161. \\ R3c \end{matrix} \begin{matrix}Fig.2.j(h) (N) \\ AXYBCDQ=F \end{matrix} \frac{r: XADC(Q) ⟶ XY BC(Q), c: XY BA ⟶ ABCD}{r^{3}, ccrc^{−1}r^{−1}c^{−1}r}
Formula (3.2.1)  {Γ^{3}(Q) = 3, Γ^{2}(K, K′) = 1, Γ^{1}(E, F, G, H, L, M, N) = 1, \atop Γ^{0}(A, X, Y, B, C, D) = 3}
26 \begin{matrix}205. \\ P a¯3 \end{matrix} \begin{matrix}Fig.2.j (N) \\ AXYBCDQ=F \end{matrix} \frac{r: XADC(Q) ⟶ XY BC(Q), z: AXY B ⟶ ABCD}{r^{3}, zzrzrz^{−1}r^{−1}}
{σ_{3} : (1, 9)(2, 10)(3, 11)(4, 12) \atop (5, 13)(6, 14)(7, 15)(8, 16)} {Γ^{3}(Q) = 3, Γ^{2}(K, K′) = 1, Γ^{1}(E, F, G, H, L, M, N) = 1, \atop Γ^{0}(A, Y, C) = ¯3, Γ^{0}(X, B, D) = ¯3}
27 \begin{matrix}218. \\ P ¯43n \end{matrix} \begin{matrix}Fig.2.j (N) \\ AXYBCDQ=F \end{matrix} \frac{{r: XADC(Q) ⟶ XY BC(Q), \atop z: (E)AXY B ⟶ (E)BCDA}}{r^{3}, (zz)^{2}; (rzrz)^{2}, (rzrz^{−1})^{2}}
{σ_{3} : (1, 11)(2, 12)(3, 13)(4, 14) \atop (5, 15)(6, 16)(7, 9)(8, 10)} Γ^{3}(Q) = 3, Γ^{2}(K, K′) = 1, Γ^{1}(E) = ¯4, ; Γ^{1}(H, N) = ¯4, Γ^{1}(L, M, G, F) = 2, ; Γ^{0}(X, C) = 23, Γ^{0}(A, B, D, Y) = 222
28 \begin{matrix}197. \\ I23 \end{matrix} \begin{matrix}Fig.2.j (N) \\ AXYBCDQ=F \end{matrix} \frac{r: XADC(Q) ⟶ XY BC(Q), s: XY BA ⟶ CBAD}{r^{3}, (rsr^{−1}s^{−1})^{2}, (ssr)^{2}}
{σ_{3} : (1, 14)(2, 13)(3, 12)(4, 11) \atop (5, 10)(6, 9)(7, 16)(8, 15)} Γ^{3}(Q) = 3, Γ^{2}(K, K′) = 1, Γ^{1}(L, M, F, G) = 2, ; Γ^{1}(N, E, H) = 2, Γ^{0}(X, C) = 23, ; Γ^{0}(A, Y, B, D) = 222

Table 2 (continued)

{29 \atop (17)} \begin{matrix}146. \\ R3 \end{matrix} \begin{matrix}Fig.2.j(h) (N) \\ AXYBCDQ=F \end{matrix} \frac{r: XADC(Q) ⟶ XY BC(Q), s :XABY ⟶ ABCD}{r^{3}, rsrsrs^{−1}s^{−1}}

{σ_{3} : (1, 10)(2, 9)(3, 16)(4, 15) \atop (5, 14)(6, 13)(7, 12)(8, 11)} {Γ^{3}(Q) = 3, Γ^{2}(K, K′) = 1, Γ^{1}(L, M, N, E, F, G, H) = 1, \atop Γ^{0}(A, X, Y, B, C, D) = 3}

Drawing the D-diagram is left to the reader. These induce the identifications on the fundamental domain F. The curved face XADC(Q) is paired with XY BC(Q) by rotation r about the diagonal XQC. The second generator c : XY BA → ABCD is a glide reflection with glide plane LHF N and glide vector \overrightarrow{LH}. We have the partial symbol D^{03} = D_{12} with 4 components but only one non-trivial relation 1 = r^{3}, and the partial symbol D^{01} = D_{23} with 2 components for the relation 1 = c^{2}rc^{−1}r^{−1}c^{−1}r, as Table 2.No.25 shows, again by Poincaré algorithm as in (3.1.1–2). In the same manner we get every k-midpoint stabilizer Γ^{k} of D^{k}, obtained from D(M) by cancelling the σ_{k}-operation, respectively k ∈ I. No.(16) refers to the list of symmetry breakings in the restricted sense.

Now the space group 161.R¯3c = Γ indicates how to deform the cube tiling to a rhombohedron tiling (in analogy to Fig.2.h) by expanding it in direction XC. Such a phenomenon also occurs in other cases. The sign (N) means that non-convex realization of (T , Γ) is possible without marking, but with deformation of faces XY BA and ABCD = (XY BA)^{c}.

We remark that the fundamental domain F in Fig.2.j provides the minimal presentation for the space group R¯3c with 2 generators and 2 defining relations in the sense of [13]. Cutting F at ABQ and gluing ABCDQ to XY BAQ by glide reflection c^{−1} : ABCD ↦ XY BA, we get another fundamental domain F′ with other (redundant) presentation but the same D-symbol. This F′ (as in Fig.2.f or h) would be a Dirichlet– Voronoi cell corresponding to the R¯3c-orbit of K.

3.3. Table 3

The upper and lower domains together show our fundamental domain F in Fig.2.a–h. The face stabilizer Γ^{2}(K) does not change the sides of the face and we have two orbits for cube centres Q and Q_{2}. Thus, any such D-symbol D(M) has two D^{3}-components describing the solid stabilizers Γ^{3}(Q) and Γ^{3}(Q_{2}), respectively. These facts mean that we are describing cube tilings in 2 colours, i.e. the spatial chess board, with isometry group Γ acting transitively on the faces.

The automorphism group Aut T of this chess board T is just the space group Γ = Fm¯3m under No.225, as described in Table 3.No.30, (1^{⋆}). Γ is a Coxeter group, generated by plane reflections in the side faces of F = AEQQ_{2}. In F we can choose a piece AEK of the cube face with great freedom, and we also get a non-convex realization of (T , Γ = Fm¯3m).

The other tilings (T , Γ) in Table 3 are all the face transitive symmetry breakings of chess board (T , Γ = Fm¯3m) as indicated in the Introduction (see also our Sect.4.3).

E. Molnár: Symmetry Breaking of the Cube Tiling 223

Table 3

{30 \atop (1*)} \begin{matrix}225. \\ F m¯3m \end{matrix} \begin{matrix}Fig.2.a (N) \\ F = AEQQ_{2} \end{matrix}  \frac{m_{0} : EQQ_{2}, m_{1} : AQQ_{2}, m_{2} : AEQ, m′_{2} : AEQ_{2}}{{m_{0}^{2} , m_{1}^{2} , m_{2}^{2} , m′_{2}^{2}; (m_{0}m_{1})^{4}, (m_{0}m_{2})^{2}, \atop (m_{0}m′_{2})^{2}, (m_{1}m_{2})^{3}, (m_{2}m′_{2})^{2}, (m_{1}m′_{2})^{3}}}

{Γ^{3}(Q) = m¯3m, Γ^{3}(Q_{2}) = m¯3m, Γ^{2}(K) = 4mm, \atop Γ^{1}(E) = mmm, Γ^{0}(A) = ¯43m}

{31 \atop (2*)} \begin{matrix}202. \\ F m¯3 \end{matrix} \begin{matrix}Fig.2.c (N) \\ AEKF QQ_{2}=F \end{matrix} \frac{{m_{0} : EQQ_{2} , m′_{0} : F QQ_{2} , r: AEQ ⟶ AF Q, \atop r′: AF Q_{2} ⟶ AEQ_{2}}}{{m_{0}^{2} , m′_{0}^{2}, r^{3}, r′^{3}; (m_{0}m′_{0})^{2}, m_{0}rm′_{0}r^{−1}, \atop m′_{0}r′m_{0}r′^{−1}, (rr′)^{2}}}

{Γ^{3}(Q) = m¯3, Γ^{3}(Q_{2}) = m¯3, Γ^{2}(K) = mm2, \atop Γ^{1}(E, F) = 2/m, Γ^{0}(A) = 23}

32 \begin{matrix}216. \\ F ¯43m \end{matrix} \begin{matrix}Fig.2.b (N) \\ ABQQ_{2} = F \end{matrix} \frac{m_{1} : BQQ_{2}, m′_{1} : AQQ_{2}, m_{2} : ABQ, m′_{2} : ABQ_{2}}{{m_{1}^{2} , m′_{1}^{2} , m_{2}^{2} , m′_{2}^{2}; (m_{1}m′_{1})^{2}, (m_{1}m_{2})^{3}, \atop (m_{1}m′_{2})^{3}, (m′_{1}m_{2})^{3}, (m′_{1}m′_{2})^{3}, (m_{2}m′_{2})^{2}}}

{Γ^{3}(Q) = ¯43m, Γ^{3}(Q_{2}) = ¯43m, Γ^{2}(K) = mm2, \atop Γ^{1}(E) = mm2, Γ^{0}(A) = ¯43m, Γ^{0}(B) = ¯43m}

224 E. Molnár: Symmetry Breaking of the Cube Tiling

Table 3 (continued)

{33 \atop (3*)} \begin{matrix}209. \\ F 432 \end{matrix} \begin{matrix}Fig.2.b (N) \\ ABQQ_{2} = F \end{matrix} \frac{{r_{1} : QAQ_{2} ⟶ QBQ_{2} , r_{2} : EAQ ⟶ EBQ, \atop r′_{2} : EAQ_{2} ⟶ EBQ_{2}}}{r_{1}^{4} , r_{2}^{2} , r′_{2}^{2}; (r_{1}r_{2})^{3}, (r_{1}r′_{2})^{3}, (r_{2}r′_{2})^{2}}
σ_{0} : (1, 2)(3, 4) ; σ_{1} = σ_{2} = σ_{0} ; σ_{3} : (1, 4)(2, 3)  {Γ^{3}(Q) = 432, Γ^{3}(Q_{2}) = 432, Γ^{2}(K) = 4, \atop Γ^{1}(E) = 222, Γ^{0}(A, B) = 23}
34 \begin{matrix}224. \\ P n¯3m \end{matrix} \begin{matrix}Fig.2.d (N) \\ ABCQQ_{2} = F \end{matrix} \frac{{m_{1} : AQ_{2}CQ, m_{2} : ABQ, m′_{2} : ABQ_{2}, \atop r_{2} : F BQ ⟶ F CQ, r′_{2} : F CQ_{2} ⟶ F BQ_{2}}}{{m_{1}^{2} , m_{2}^{2} , m′_{2}^{2}, r_{2}^{2} , r′_{2}^{2}; (m_{1}m_{2})^{3}, (m_{1}m′_{2})^{3}, \atop (m_{2}m′_{2})^{2}, m_{1}r_{2}m_{2}r_{2}, m_{1}r′_{2}m′_{2}r′_{2} , (r_{2}r′_{2})^{2}}}

σ_{0} : (1, 2)(3, 4)(5, 6)(7, 8) ; σ_{1} : (1)(2, 3)(4)(5)(6, 7)(8) ; σ_{2} : (1)(2)(3, 4)(5)(6)(7, 8) ; σ_{3} : (1, 5)(2, 6)(3, 7)(4, 8)  {Γ^{3}(Q) = ¯3m, Γ^{3}(Q_{2}) = ¯3m, Γ^{2}(K) = m, Γ^{1}(E) = mm2, \atop Γ^{1}(F) = 222, Γ^{0}(A) = ¯43m, Γ^{0}(B, C) = ¯42m}

35 \begin{matrix}166. \\ R¯3m \end{matrix} \begin{matrix}Fig.2.g(d) (N) \\ ABCQQ_{2} = F \end{matrix} \frac{{m_{1} : AQ_{2}CQ, m_{2} : QAB, r_{2} : F BQ ⟶ F CQ, \atop m′_{2} : BCQ, r′_{2} : EAQ_{2} ⟶ EBQ}}{{m_{1}^{2} , m_{2}^{2} , r_{2}^{2} , m′_{2}^{2}, r′_{2}^{2}; (m_{1}m_{2})^{3}, m_{1}r_{2}m_{2}r_{2}, \atop (m_{1}m′_{2})^{3}, m_{1}r′_{2}m′_{2}r′_{2}, m_{2}r′_{2}m_{2}r′_{2}, r_{2}m′_{2}r_{2}m′_{2}}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1)(2)(3, 4)(5, 6)(7)(8)

{Γ^{3}(Q) = ¯3m, Γ^{3}(Q_{2}) = ¯3m, Γ^{2}(K) = m, \atop Γ^{1}(E) = 2/m, Γ^{1}(F) = 2/m, Γ^{0}(A, B, C) = 3m}

36 \begin{matrix}196. \\ F 23 \end{matrix} \begin{matrix}Fig.2.d(e) (N) \\ ABCQQ_{2} = F \end{matrix} \frac{{r_{1} : QAQ_{2} ⟶ QCQ_{2}, r: BCQ ⟶ BAQ, \atop r′: BAQ_{2} ⟶ BCQ_{2}}}{r_{1}^{2} , r^{3}, r′^{3}, (r_{1}r)^{3}, (r_{1}r′)^{3}, (rr′)^{2}}
σ_{0} : (1, 2)(3, 4)(5, 6)(7, 8) ; σ_{1} : (1, 4)(2, 3)(5, 8)(6, 7) ; σ_{2} : (1, 4)(2, 3)(5, 8)(6, 7) ; σ_{3} : (1, 5)(2, 6)(3, 7)(4, 8)  {Γ^{3}(Q) = 23, Γ^{3}(Q_{2}) = 23, Γ^{2}(K) = 2, \atop Γ^{1}(E, F) = 2, Γ^{0}(A, C) = 23 Γ^{0}(B) = 23}

E. Molnár: Symmetry Breaking of the Cube Tiling 225

Table 3 (continued)

37 \begin{matrix}208. \\ P 4_{2}32 \end{matrix} \begin{matrix}Fig.2.f (N) \\ ABCDQQ_{2}=F \end{matrix} \frac{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, ; r_{2} : GCQ ⟶ GDQ, r′: ADQ_{2} ⟶ ABQ_{2}, ; r′_{1} : F BQ_{2} ⟶ F CQ_{2}, r′_{2} : GCQ_{2} ⟶ GDQ_{2},}{{r^{3}, r_{1}^{2} , r_{2}^{2} , r′^{3}, r′_{1}^{2}, r′_{2}^{2}; rr_{1}r_{2} , r′r′_{1}r′_{2} , \atop (r^{−1}r′)^{2}, (r_{1}r′_{1})^{2}, (r_{2}r′_{2})^{2}}}

Formula (3.3.1)

Γ^{3}(Q) = 32, Γ^{3}(Q_{2}) = 32, Γ^{2}(K) = 1,
Γ^{1}(E, H) = 2, Γ^{1}(F) = 222, Γ^{1}(G) = 222,
Γ^{0}(A) = 23, Γ^{0}(B, C, D) = 222
38 \begin{matrix}155. \\ R32 \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ ABCDQQ_{2}=F \end{matrix} \frac{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, ; r_{2} : GCQ ⟶ GDQ, r′: CDQ_{2} ⟶ CBQ_{2}, ; r′_{1} : EBQ_{2} ⟶ EAQ_{2}, r′_{2} : HAQ_{2} ⟶ HDQ_{2}}{{r^{3}, r_{1}^{2} , r_{2}^{2} , r′^{3}, r′_{1}^{2}, r′_{2}^{2}; rr_{1}r_{2} , r′r′_{1}r′_{2}, \atop rr′_{1}r^{−1}r′_{2} , r′r_{1}r′^{−1}r_{2}}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1, 8)(2, 7)(3, 4)(5, 6) (12, 13)(11, 14)(9, 10)(15, 16)

{Γ^{3}(Q) = 32, Γ^{3}(Q_{2}) = 32, Γ^{2}(K) = 1, \atop Γ^{1}(E, H) = 2, Γ^{1}(F, G) = 2, Γ^{0}(A, B, C, D) = 3}

39 \begin{matrix}228. \\ F d¯3c \end{matrix} \begin{matrix}Fig.2.f (N) \\ ABCDQQ_{2}=F \end{matrix} \frac{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, ; r_{2} : GCQ ⟶ GDQ, r′: ADQ_{2} ⟶ ABQ_{2}, ; z: Q_{2}BC ⟶ Q_{2}CD}{r^{3}, r_{1}^{2} , r_{2}^{2} , r′^{3}; rr_{1}r_{2} , r′zz, (r^{−1}r′)^{2}, r_{1}zr_{2}z^{−1}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1, 8)(2, 7)(3, 4)(5, 6) (9, 16)(10, 15)(11, 13)(12, 14)

Γ^{3}(Q) = 32, Γ^{3}(Q_{2}) = ¯3, Γ^{2}(K) = 1, ; Γ^{1}(E, H) = 2, Γ^{1}(F, G) = 2, ; Γ^{0}(A) = 23, Γ^{0}(B, C, D) = ¯4

40 \begin{matrix}167. \\ R¯3c \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ ABCDQQ_{2}=F \end{matrix} \frac{r: ADQ ⟶ ABQ, r_{1} : F BQ ⟶ F CQ, ; r_{2} : GCQ ⟶ GDQ, r′: CDQ_{2} ⟶ CBQ_{2}, ; z: Q_{2}BA ⟶ Q_{2}AD}{r^{3}, r_{1}^{2} , r_{2}^{2} , r′^{3}; rr_{1}r_{2} , r′zz, rzrz, r′r_{1}r′^{−1}r_{2}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1, 8)(2, 7)(3, 4)(5, 6) (12, 13)(11, 14)(9, 15)(10, 16)

Γ^{3}(Q) = 32, Γ^{3}(Q_{2}) = ¯3, Γ^{2}(K) = 1, ; Γ^{1}(E, H) = ¯1, Γ^{1}(F, G) = 2, ; Γ^{0}(A, B, C, D) = 3

Table 3 (continued)

41 \begin{matrix}201. \\ P n¯3 \end{matrix} \begin{matrix}Fig.2.f (N) \\ ABCDQQ_{2}=F \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop r′: ADQ_{2} ⟶ ABQ_{2}, z′: Q_{2}BC ⟶ Q_{2}CD}}{r^{3}, r′^{3}; (r^{−1}r′)^{2}, rzz, r′z′z′, (z^{−1}z′)^{2}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1, 8)(2, 7)(3, 5)(4, 6) (9, 16)(10, 15)(11, 13)(12, 14)

Γ^{3}(Q) = ¯3, Γ^{3}(Q_{2}) = ¯3, Γ^{2}(K) = 1, ; Γ^{1}(E, H) = 2, Γ^{1}(F, G) = 2, ; Γ^{0}(A) = 23, Γ^{0}(B, C, D) = 222

{42 \atop (4*)} \begin{matrix}205. \\ P a¯3 \end{matrix} \begin{matrix}Fig.2.f (C,N) \\ ABCDQQ_{2}=F \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop r′: DAQ_{2} ⟶ DCQ_{2}, z′: Q_{2}CB ⟶ Q_{2}BA}}{r^{3}, r′^{3}; rzz, r′z′z′, rz′^{−1}zr′^{−1}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1, 8)(2, 7)(3, 5)(4, 6) (14, 15)(13, 16)(9, 11)(10, 12)

{Γ^{3}(Q) = ¯3, Γ^{3}(Q_{2}) = ¯3, Γ^{2}(K) = 1, \atop Γ^{1}(E, F, G, H) = 1, Γ^{0}(A, B, C, D) = 3}

43 \begin{matrix}148. \\ R¯3 \end{matrix} \begin{matrix}Fig.2.h(f) (N) \\ ABCDQQ_{2}=F \end{matrix} \frac{{r: ADQ ⟶ ABQ, z: QBC ⟶ QCD, \atop r′: CBQ_{2} ⟶ CDQ_{2}, z′: Q_{2}DA ⟶ Q_{2}AB}}{r^{3}, r′^{3}; rzz, r′z′z′, rz′^{−1}rz′^{−1}, r′z^{−1}r′z^{−1}}

σ_{0} , σ_{1} , σ_{3} as before σ_{2} : (1, 8)(2, 7)(3, 5)(4, 6) (12, 13)(11, 14)(9, 15)(10, 16)

Γ^{3}(Q) = ¯3, Γ^{3}(Q_{2}) = ¯3, Γ^{2}(K) = 1, ; Γ^{1}(E, H) = ¯1, Γ^{1}(F, G) = ¯1, ; Γ^{0}(A, B, C, D) = 3

For example, consider Table 3.No.37 with Fig.2.f. The D-symbol D(M) contains 16 elements as simplices

\begin{matrix}D_{1} = AEKQ, D_{2} = BEKQ, …, & D_{8} = AHKQ, \\ D_{9} = AEKQ_{2}, D_{10} = BEKQ_{2}, …, & D_{16} = AHKQ_{2}, \end{matrix}

with the matrix function M constant (3.1), where the following operations are introduced in D (D_{x} ↔ x):

\begin{matrix}σ_{0} : & (1, 2)(3, 4)(5, 6)(7, 8)(9, 10)(11, 12)(13, 14)(15, 16) · · · · · · \\ σ_{1} : & (1, 8)(2, 3)(4, 5)(6, 7)(9, 16)(10, 11)(12, 13)(14, 15) − −− \\ σ_{2} : & (1, 8)(2, 7)(3, 4)(5, 6)(9, 16)(10, 15)(11, 12)(13, 14) − \\ σ_{3} : & (1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16) ∼∼∼ \end{matrix} (3.3.1)

We do not draw the D-diagram here. The σ_{i}-operations, i ∈ I, induce the face pairing generators of F given in Table 3.No.37. The face and its midpoint K have trivial stabilizer. The partial symbols D_{02}, D_{12}, D_{23} provide the defining relations for the group Γ = P4_{2}32 as space group No.208. The stabilizers Γ^{k}, given by the partial symbols D^{k}, obtained from D by cancelling σ_{k}-operations (k ∈ I), helped us in choosing this space group from International Tables [10].

We have obtained 4 symmetry breakings of the spatial chess board in the restricted sense, as enumerated (1^{⋆})–(4^{⋆}) in Table 3.

We remark that I. Prok [20] classified all the fundamental cube tilings in E^{3}, i.e. with exactly one trivial solid stabilizer Γ^{3}(Q) = 1. By computer he found 298 equivariant classes (T , Γ) distributed into 130 non-isomorphic space groups. Our first 29 tilings in Table 1–2 do not belong to these classes, because our solid stabilizers are not trivial.

4. The General Strategy for Classifying d-Dimensional Tilings with Given Transitivity Properties by D-Dymbols

We consider combinatorially equivariant classes (T , Γ) of d-tilings in certain simply connected d-spaces S^{d} to be described by the following requirements stated in Assumptions 1– 3. One can think of their realizations in Euclidean plane E^{2} or space E^{3} (see again Sect.2 and 3).

4.1. The Description of a Tiling (T , Γ) by a D-Symbol

Assumption 1 A tiling T of a simply connected d-space S^{d} is characterized by its barycentric subdivision C of d-simplices with I-labelled i-vertices and opposite i-facets ((d − 1)-hyperfaces) for any i ∈ I := {0, 1, …, d − 1, d}. A starting simplex C_{1} ∈ C has a coordinate mapping x : C_{1} → Δ_{I} onto the standard real coordinate simplex

(4.1.1) Δ_{I} := {x = (x^{0},…, x^{d}) ∈ R^{I} : x^{i} ≥ 0 for every i ∈ I, ∑_{i=0}^{d} x^{i} = 1}

The i-facet of C_{1} is defined by the i-facet of Δ_{I}

Δ_{I}^{i} := {x ∈ Δ_{I} : x^{i} = 0}, i ∈ I. (4.1.2)

Then the topological space S^{d} is defined as the union Δ_{I} × C of as many copies of Δ_{I} as there are simplices C ∈ C with i-facet identifications

(Δ_{I}^{i}, C) = (Δ_{I}^{i}, σ_{i}C) i ∈ I. (4.1.3)

So the i-adjacent simplex σ_{i}C of any C ∈ C, i.e. the involutive σ_{i}-operation on C is also introduced for any i ∈ I. This σ_{i} is first a permutation of C, which orders to any simplex its own i-adjacent one. Furthermore, σ_{i} can be considered a ”local hyperface reflection” in the i-facet of an actually fixed simplex C as well that fixes all the points of the common facet (4.1.3), and

σ_{i}(x, C) := (x, σ_{i}C) for any x ∈ Δ_{I} (4.1.4)

228 E. Molnár: Symmetry Breaking of the Cube Tiling holds for the points of simplices C and σ_{i}C.

Then the (free) Coxeter group Σ_{I} := (σ_{0}, σ_{1},…, σ_{d} − 1 = σ_{i}^{2} for any i ∈ I) (4.1.5) acts on C from the left by formula

(σ_{i}σ)C := σ_{i}(σC)

for any C ∈ C, σ ∈ Σ_{I} and i ∈ I. That means, if

σ = σ_{i_{r}} ⋯ σ_{i_{2}}σ_{i_{1}} ∈ Σ_{I}

is a product of generators {σ_{j} : j ∈ I}, then the path through simplices C, σ_{i_{1}}C, σ_{i_{2}}σ_{i_{1}}C, …, σ_{i_{r}} ⋯ σ_{i_{2}}σ_{i_{1}}C, σ_{i}σ_{i_{r}} ⋯ σ_{i_{2}}σ_{i_{1}}C (4.1.6) arrives at σ_{i}σC entering the i-facet of σC. These also show how to define the action of Σ_{I} as local mappings of a fixed simplex C ∈ C onto the corresponding simplices of C. Assumption 2 To describe the tiling T and so its space S^{d}, the matrix function M : C ⟶ N_{I×I}, C ⟼ M(C) := (m_{ij}(C)) (4.1.7) is introduced (with entries from the set N of natural numbers) by the following properties (4.1.8–10): Any entry m_{ij}(C) is minimal for the equality

m_{ij}(C)−times

C = (σ_{j}σ_{i})^{m_{ij}(C)}C with product \overbrace{(σ_{j}σ_{i}) ⋯ (σ_{j}σ_{i})(σ_{j}σ_{i})};

(4.1.8)

m_{ii}(C) = 1, m_{ij}(C) = m_{ji}(C) = m_{ji}(σ_{i}C); (4.1.9) m_{ij}(C) = 2 if |i − j| > 1; m_{ij}(C) > 2 if |i − j| = 1 (4.1.10) hold for any i, j ∈ I and for any C ∈ C.

We require C and S^{d} to be simply connected. That means Σ_{I} acts transitively on C, and that moreover, any relation σC = C, σ ∈ Σ_{I} (4.1.11) is a consequence of the ”loop relations” τ^{−1}(σ_{j}σ_{i})^{m_{ij}(τC)}τC = C (4.1.12) for any C ∈ C, τ ∈ Σ_{I}, i, j ∈ I (in the sense of (4.1.6)).

Thus, the generalized Coxeter group Σ_{I}(C, M) =: Σ_{I}(T ) acting freely on C, is defined at (4.1.5), the matrix function (4.1.7–10) and by a fixed simplex C_{1}. Different C_{1}’s lead to conjugated Coxeter groups (not distinguished in the following [4]).

Assumption 3 In addition, we require a group Γ of simplicial (piecewise linear) homeomorphisms which acts on T = (T , Γ) and freely on simplices of C with right action as

C^{γ_{1}γ_{2}} := (C^{γ_{1}})^{γ_{2}} (4.1.13)

denotes for any C ∈ C and γ_{1}, γ_{2} ∈ Γ. The transformations in Γ shall preserve all incidences of 0-faces=vertices, 1-faces=edges, …, (d − 1)-faces=facets, d-faces=solids of (T , Γ). That means, we require the Γ-invariance of Σ_{I}, as

(σC)^{γ} = σ(C^{γ}) (4.1.14)

shows for any σ ∈ Σ_{I} and γ ∈ Γ. In particular, the γ-image of the i-adjacent of C is the same as the i-adjacent of C^{γ}. The group Γ factorizes C into Γ-orbits (Γ-images)

C^{Γ} := {C^{γ} ∈ C : γ ∈ Γ} =: D, (4.1.15)

required to constitute a finite set D called D-set of (T , Γ). Moreover, there exists an induced action of Σ_{I} on D as left σ_{i}-operations for i ∈ I defined by

σ_{i}D := σ_{i}C^{Γ} := {σ_{i}(C^{γ}) = (σ_{i}C)^{γ} : γ ∈ Γ} for C^{Γ} = D (4.1.16)

and the matrix function M = (m_{ij}) by

(4.1.17) M : D ⟶ N_{I×I}, D ⟼ M(D) = M(C) with any C ∈ D.

This function has the properties (4.1.8–10) with C’s replaced by D’s, but without requiring the minimality in (4.1.8).

The system D(M), as a finite set with the I-adjacencies i.e. Σ_{I}-operations and the matrix function M, is called a D-symbol (Delone–Delaney–Dress symbol).

As we have seen, all the 14 Euclidean square tilings in E^{2} =: S^{2} and all the 43 cube tilings in E^{3} =: S^{3} and their D-symbols satisfy these requirements Ass.1–3 with appropriate crystallographic isometry groups Γ of E^{2} resp. E^{3}. The properties and algorithmic procedures, described in Sect.2–3 by examples, can be formulated in the following Propositions 1–3 of principial importance.

4.2. Fundamental Domain and Presentation of the Group Γ for a Tiling (T , Γ)

Proposition 1 Any representative set, containing one simplex C_{x} from each orbit D_{x} ∈ D constitutes a fundamental set

F := ⋃_{x=1}^{|D|} C_{x}

which is compact in S^{d} since |D| is finite. Any such F, by gluing its simplices and pairing its free (not glued) facets as the σ_{i}-operations dictate, represents the D-symbol D(M). Any fixed F (may be not connected) presents a finite system of generators for the group Γ. Each of them pairs some σ_{i}-corresponded free simplex facets of F in C (i ∈ I).

If τ_{1}C_{1} ∈ D_{1} and τ_{2}C_{1} ∈ D_{2} and σ_{i}D_{1} = D_{2}, with D_{1}, D_{2} ∈ D, τ_{1}, τ_{2} ∈ Σ_{I}, i ∈ I, then we have two cases:

◯ D⊃i (C.1)

i.e. D ∋ τ_{1}C_{1}, D ∋ τ_{2}C_{1} = σ_{i}τ_{1}C_{1} then τ_{1}C_{1} ↦ τ_{2}C_{1} = σ_{i}τ_{1}C_{1} =: (τ_{1}C_{1})^{m} defines a simplicial reflection m for C_{1}: τ_{1}^{−1}σ_{i}τ_{1}C_{1} =: C_{1}^{m}. If C = τC_{1} with τ ∈ Σ_{I}(M, C_{1}) then the m-image of C is defined by

C^{m} := (τC_{1})^{m} := (ττ_{1}^{−1}σ_{i}τ_{1}τ^{−1})C = τ(C_{1}^{m}). (4.2.1)

In this sense, after having fixed C_{1} ∈ C, we have defined the facet reflection m ∈ Γ with relation m^{2} = 1 expressed in Σ_{I}(M, C_{1}) by Ass.1–3.

◯ D_{1}−^{i} ◯ D_{2} (C.2)

Then τ_{1}C_{1} ↦ τ_{2}C_{1} = σ_{i}τ_{1}C_{1} =: (τ_{1}C_{1})^{γ} defines γ ∈ Γ with the required extension to C: For any C = τC_{1} with τ ∈ Σ_{I}(M, C_{1}) let its γ-image

C^{γ} := (τC_{1})^{γ} := τ(C_{1}^{γ}) = (ττ_{2}^{−1}σ_{i}τ_{1}τ^{−1})C (4.2.2)

be defined. Thus the mapping γ ∈ Γ and its inverse γ^{−1} by

C^{γ^{−1}} := (τC_{1})^{γ^{−1}} := τ(C_{0}^{γ^{−1}}) = (ττ_{1}^{−1}σ_{1}τ_{2}τ^{−1})C

are expressed in Σ_{I}(M, C_{1}). In both cases we obtain generators for Γ.

Proposition 2 Any free (d − 2)-face class of F under Γ-equivalence provides a defining relation for the previous generators of Γ by the Poincaré algorithm.

Here ”free” means that any element in the (d−2)-face Γ-class lies on the (d−1)-surface of F, i.e. it is not surrounded by simplices of C, belonging to F. If i- and j-facets of an F-simplex C ∈ D ∈ D join this (d − 2)-face class, and m_{ij}(D) := m_{ij}(C) is the matrix entry, then we have two cases again:

j⊂\overbrace{◯ D−^{i} ◯−^{j} ⋯ −^{j} ◯−^{i} ◯}⊃j_{or ⋯ −^{i} ◯−^{j} ◯⊃i}^{e_{ij}(D)} (C.A)

That is, we have a ⟨σ_{i}, σ_{j}⟩-chain D_{ij} in D with chain length e_{ij}(D). This describes a local fundamental domain F_{ij}(D) for the stabilizer Γ_{ij}(D) of this (d − 2)-face Γ-class, generated by two facet reflections from Γ. Since 2 · m_{ij}(D) simplices from C meet around this (d − 2)-face of C ∈ D, the natural number

v_{ij}(D) := \frac{2 · m_{ij}(D)}{2 · e_{ij}(D)} (4.2.3)

is just the order of the ”rotation subgroup” of the stabilizer Γ_{ij}(D).

\begin{matrix}◯ D & −^{i} & ◯ & −^{j} & ◯ \\ ^{j}| & |^{i} \\ ◯ & −^{i} & ⋯ & −^{j} & ◯ \end{matrix} (C.B)

That means, we have in D a ⟨σ_{i}, σ_{j}⟩-cycle D_{ij} of length e_{ij}(D) which is even. This describes F_{ij}(D) for the stabilizer Γ_{ij}(D) of the (d − 2)-face Γ-class, which is a ”rotation group” of order

v_{ij}(D) := \frac{2 · m_{ij}(D)}{e_{ij}(D)}. (4.2.4)

(A non trivial ”rotation” about a (d − 2)-face means that exactly this (d − 2)-face is pointwise fixed.)

Similarly to Pr.1, the generators of Γ_{ij}(D) can be expressed by the generators of Γ in both cases, and we get the relation

R_{ij}^{v_{ij}} = 1. (4.2.5)

Considering all the free (d − 2)-face Γ-classes of F, we obtain a complete set of defining relations for the generators of Γ in accordance with Ass.2 (4.1.11–12). That means, any relation of Γ is a consequence of these (d−2)-face relations (4.2.5) in the form γ^{−1}R_{ij}^{v_{ij}}γ = 1, for certain γ ∈ Γ.

Proposition 3 Any k-face of (T , Γ) is characterized by a common k-vertex of the joint simplices in the barycentric subdivision C, k ∈ I. The action of Γ on T , and so on C, induces these k-face orbits, or k-midpoint orbits, and their stabilizers Γ^{k} by the components of the partial D-symbol D^{k}. This D^{k} is derived from D(M) by cancelling the σ_{k}-operation and keeping the others.

Any component D_{c}^{k} describes the local tiling round the corresponding k-vertex of C, its barycentric subdivision C_{c}^{k} and the stabilizer Γ_{c}^{k}, furthermore the (d − 1)-dimensional local tiling (T_{c}^{k}, Γ_{c}^{k}). We can choose a local fundamental domain F_{c}^{k}, glued from those representing C_{c}^{k}-simplices of D_{c}^{k}, which lie also in the fundamental domain F of Γ. Then we get the generators of Γ_{c}^{k} expressed by generators of Γ. We can derive defining relations for Γ_{c}^{k} as consequences of the presentation of Γ by F, analogously as in Pr.1–2.

Proofs are not difficult and omitted here, see [3], [4], [5], [15]. In Sect.2–3 we have used these propositions in particular cases.

4.3. Equivariant Classifications and Symmetry Breakings of Tilings

Definition 1 Two tilings (T_{1}, Γ_{1}) and (T_{2}, Γ_{2}) are combinatorially (topologically) equivariant and belong to the same equivariance class, iff there is a combinatorial bijection φ : T_{1} → T_{2}, T ↦ T^{φ} (a topological mapping of the corresponding topological realizations), preserving all the incidences in T_{1} resp. T_{2} (i.e. the flag structures and barycentric subdivisions of T_{1} resp. T_{2}) such that Γ_{2} = φ^{−1}Γ_{1}φ, i.e, φ is equivariant isomorphism (homeomorphism).

The latter requirement means: if γ_{1} : T_{1} → T_{1}, T_{1} ↦ T_{1}^{γ_{1}} is any element of Γ_{1} then

γ_{2} := φ^{−1}γ_{1}φ : T_{2} ⟶ T_{2}, T^{φ} ⟼ (T^{γ_{1}})^{φ}

also belongs to Γ_{2}. Moreover, we obtain all the elements of Γ_{2} in this manner.

By Def.1 we can classify various tilings, and we can ask for their realizations, i.e. their equivariantly isomorphic (homeomeorphic) images, e.g. in a space of constant curvature with a group Γ of isometries. The concept of D-symbol D(M), defined to a tiling (T , Γ), and their D-isomorphisms are of great importance, since we have the characteristic

Theorem 1 [4, §3] The tilings (T_{1}, Γ_{1}) and (T_{2}, Γ_{2}) are combinatorially equivariant, iff their D-symbols D_{1}(M_{1}) and D_{2}(M_{2}) are D-isomorphic.

Definition 2 A D-isomorphism is a bijection φ : D_{1} → D_{2}, D ↦ D^{φ} with (σ_{i}D)^{φ} = σ_{i}(D^{φ}) and M_{1}(D) = M_{2}(D^{φ}) for any D ∈ D_{1} and i ∈ I.

Definition 3 A tiling (T , Γ) can be obtained by symmetry breaking from the tiling (T^{⋆}, Γ^{⋆}), iff there is an isomorphism

φ : T ⟶ T^{⋆}, such that φ^{−1}Γφ ⊆ Γ^{⋆}.

In terms of D-symbols D(M) and D^{⋆}(M^{⋆}), this means that there exists a surjective mapping ψ : D → D^{⋆}, D ↦ D^{⋆} with |D| ≥ |D^{⋆}|, (σ_{i}D)^{ψ} = σ_{i}(D^{ψ}) and M(D) = M^{⋆}(D^{ψ}) for any D ∈ D and i ∈ I. Any such ψ is called a D-morphism, it will be called proper iff |D| > |D^{⋆}|, otherwise it will be called D-isomorphism.

We remark that the above φ : T → T^{⋆} is determined by any simplex C ∈ C(T ) and its φ-image C^{φ} ∈ C(T^{⋆}) uniquely if such φ exists. Analogously ψ : D → D^{⋆} above is uniquely determined by any D and D^{ψ} if such a D-morphism ψ exists.

These facts are very useful in searching either for symmetry breakings of a given tilings (T , Γ) or for the so-called maximal tiling (T , Γ = Aut T ). Here Aut T denotes the set of all self-isomorphisms of T in a combinatorial sense. But the existence of a topological, etc., metric realization of such a maximal tiling is not clear yet, in general.

4.4. Tilings for a Given D-Symbol and Their Realizations

We shall consider the most important inverse problem.

Problem 1 Let a finite D-symbol D(M) be given with its Σ_{I}-operations and matrix function M described in (4.1.17) and (4.1.8–10) as in Ass.2 with C’s replaced by D’s but without requiring minimality in (4.1.8). Does there exist a tiling (T , Γ) in certain simply connected space S^{d} so that the D-symbol of (T , Γ) is D-isomorphic to the prescribed D-symbol D(M)?

If such a tiling exists, it would be unique up to combinatorial (topological) equivariance.

In general, the answer is no, if D(M) represents a ”bad orbifold” in the sense of Thurston (see [8], [9] and [21]). As a simple example, we mention the following. Let D = {1, 2, 3, 4} be a D-set of 4 elements (now we write D_{x} ↔ x for brevity) and we define

σ_{0} : (1, 2)(3)(4); σ_{1} : (1)(2, 3)(4); σ_{2} : (1)(2)(3, 4)

with the matrix function

m_{ij}(D_{1}) = m_{ij}(D_{2}) = m_{ij}(D_{3}) = ( \begin{matrix}1 & 3 & 2 \\ 3 & 1 & 3 \\ 2 & 3 & 1 \end{matrix} ); m_{ij}(D_{4}) = ( \begin{matrix}1 & 6 & 2 \\ 6 & 1 & 3 \\ 2 & 3 & 1 \end{matrix} ) . (4.4.1)

This is a 2-dimensional D-symbol, I = {0, 1, 2}. (Draw a picture!) The procedure in Subsection 4.2 provides a (hypothetic) fundamental domain F by gluing the simplices: D_{1} and D_{2} along their 0-facets as σ_{0} dictates, D_{2} and D_{3} along their 1-facets by σ_{1}, D_{3} and D_{4} along their 2-facets by σ_{2}. The free facets of F

f_{0}(D_{3}), f_{0}(D_{4}), f_{1}(D_{1}), f_{1}(D_{4}), f_{2}(D_{1}), f_{2}(D_{2})

would serve 6 facet-reflections denoted by

μ_{03}, μ_{04}, μ_{11}, μ_{14}, μ_{21}, μ_{22},

respectively. Fixing the simplex D_{3}, say, we have the generators for a (hypothetic) transformation group Γ as follows

\begin{matrix}μ_{03} : D_{3} ⟼ D_{3}^{μ_{03}} := σ_{0}D_{3}; \\ μ_{11} : D_{3} ⟼ D_{3}^{μ_{11}} := (σ_{1}σ_{0})σ_{1}(σ_{0}σ_{1})D_{3}; \\ μ_{21} : D_{3} ⟼ D_{3}^{μ_{21}} := (σ_{1}σ_{0})σ_{2}(σ_{0}σ_{1})D_{3}. \end{matrix} \begin{matrix}μ_{04} : D_{3} ⟼ D_{3}^{μ_{04}} := σ_{2}σ_{0}σ_{2}D_{3}; \\ μ_{14} : D_{3} ⟼ D_{3}^{μ_{14}} := σ_{2}σ_{1}σ_{2}D_{3}; \\ μ_{22} : D_{3} ⟼ D_{3}^{μ_{22}} := σ_{1}σ_{2}σ_{1}D_{3}. \end{matrix}

Now μ_{03} = μ_{04} follows from σ_{0}D_{3} = σ_{2}σ_{0}σ_{2}D_{3} i.e. D_{3} = σ_{0}σ_{2}σ_{0}σ_{2}D_{3}, since m_{02}(D_{3}) = 2. Similarly μ_{03} = μ_{11} holds by

σ_{0}D_{3} = σ_{1}σ_{0}σ_{1}σ_{0}σ_{1}D_{3} because of m_{01}(D_{3}) = m_{01}(D_{2}) = m_{01}(D_{1}) = 3.

Thus we have a reflection

m_{1} := μ_{03} = μ_{04} = μ_{11}

in a facet of F. In the same manner we have another reflection

m_{2} := μ_{14} = μ_{22} = μ_{21}

in the other facet of F. This is the consequence of

m_{12}(D_{3}) = m_{12}(D_{4}) = m_{12}(D_{2}) = 3 and m_{02}(D_{2}) = m_{02}(D_{1}) = 2.

So, our group Γ should be generated by two reflections m_{1} and m_{2}. The action of Γ, however, is not compatible now, because by (4.2.2) the product

\begin{aligned}m_{1}m_{2} & = μ_{11}μ_{21} : D_{3} ⟼ (D_{3}^{μ_{11}})^{μ_{21}} := [(σ_{1}σ_{0})σ_{1}(σ_{0}σ_{1})D_{3}]^{μ_{21}} = \\ & = (σ_{1}σ_{0})σ_{1}(σ_{0}σ_{1})(σ_{1}σ_{0})σ_{2}(σ_{0}σ_{1})D_{3} = (σ_{1}σ_{0})σ_{1}σ_{2}(σ_{0}σ_{1})D_{3} \end{aligned}

would have the order m_{12}(D_{1}) = 3, from one side, and the order m_{01}(D_{4}) = 6 from the other side by

\begin{aligned}m_{1}m_{2} & = μ_{04}μ_{14} : D_{3} ⟼ (D_{3}^{μ_{04}})^{μ_{14}} := (σ_{2}σ_{0}σ_{2}D_{3})^{μ_{14}} \\ & = σ_{2}σ_{0}σ_{2}σ_{2}σ_{1}σ_{2}D_{3} = σ_{2}(σ_{0}σ_{1})σ_{2}D_{3}. \end{aligned}

Hence, this example does not give rise to a realizable tiling.

If we choose m_{01}(D_{4}) = 3 in (4.4.1), then the fundamental domain F and its images under the dihedral group

Γ = (m_{1}, m_{2} − 1 = m_{1}^{2} = m_{2}^{2} = (m_{1}m_{2})^{3})

provide a tetrahedron tiling on the sphere S^{2}. The sphere would be the universal covering manifold of our modified orbifold. The original D-symbol describes a tetrahedron tiling on the ”tear-drop” (+, 0; [2]; { }) which is a bad orbifold with the sphere as underlying manifold. This tear-drop is the universal cover of our bad orbifold (+, 0; [ ]; {(3, 6)}) represented by the original D-symbol, i.e. it is orientable with genus 0, no rotation centre and with dihedral corners of order 3 and 6 on its one boundary component [21]. In the paper [15] the algorithmic extension of Poincaré’s polyhedron theorem (Th.8.1) has been formulated for the existence of a tiling with prescribed fundamental polyhedron and for the realization in a space S^{d} of constant curvature.

A hopeful strategy of the proof, which is due to A.D. Aleksandrov, can be applied to attack our Problem 1 as well. We can follow the opposite way as in Sect.4.1–2.

First we glue a connected fundamental domain F from |D| representing simplices. Choose a starting simplex C_{1} corresponding to a vertex D_{1} in the D-graph D, and glue to C_{1} the other simplices C_{x} as for D_{x} the Σ_{I}-operations dictate (I = {0, 1,…, d}). Though F is not unique, Σ_{I} also pairs the free simplex facets of F, starting to generate a group Γ, moreover a combinatorial tiling (F, Γ) and the barycentric simplex tiling C(Σ_{I}, M, C_{1}). The matrix function M describes the branching numbers of C and also the defining relations for the generators of Γ in the same time, as we indicated at Pr.1–2 and at Ass.1–3 in the other direction.

Our conjecture is, the topological construction is possible (then unique up to equivariance) iff it is compatible round (d − 2)-faces of F. This seems to be satisfactory to exclude bad orbifolds [21]. As our example (4.4.1) illustrated, this condition is necessary. Of course, we could define the (d−1)-facets of F and also the other dimensional faces in the combinatorial terminology of C(Σ_{I}, M, C_{1}). The partial D-symbol D^{k}(M) obtained from D(M) by cancelling the σ_{k}-operation (k ∈ I), consists of as many components, as many Γ-equivalence classes of the k-labelled vertices of C we have. Any component of D^{k}(M) describes the stabilizer Γ^{k} of the corresponding k-vertex class and the local tiling (T^{k}, Γ^{k}) by means of a (d − 1)-dimensional fundamental domain F^{k}. So we may have an inductive procedure to define the tiling (F, Γ) so C(Σ_{I}, M, C_{1}) and (T , Γ) further and further from the starting simplex C_{1}. If the fundamental domain F, with its face structure, has a nice topology, then we can guarantee the space S^{d} = Δ_{I} × C at (4.1.2–4) to be simply connected by Ass.2 (4.1.11–12). We can go by induction on the dimension and on the simplicial distance from C_{1}.

Of course, this was only a sketch of a hypothetic procedure for constructing the topological realization of our tiling. Together with the colleagues in Bielefeld we are working on this problem [5].

In our case, where the Euclidean space E^{3} and its combinatorial cube tiling T is given, the procedure is straightforward. E.g. the D-symbol under formulas (3.3.1) and (3.1) in Sect.3 implied the fundamental domain F := ABCDQQ_{2} (Fig.2.f), the facet pairing generators, given in Table 3.No.37, the defining relations and the vertex stabilizers. The only problem was to read off the group Γ = P4_{2}32 from the International Tables [10]. In [16] and [18] we have described some tilings in non-Euclidean metric spaces of constant curvature, moreover tilings in other homogenous Riemannian spaces as in S^{2}×R, H^{2}×R, S̃L_{2}R, Nil [19]. Our intention is to give criteria for realizations of D-symbols by polyhedral tilings in E^{3} where Γ is a crystallographic space group. This seems to have applications also in non-mathematical fields as in the theory of solid state matter.

4.5. Algorithm for Classifying Euclidean Space Tilings (T , Γ) in E^{3} with Given Transitivity Properties

We describe principial steps which will be performed in various computer programs. We follow the above strategy in Sect.4.4. Our joint paper [5] realized this algorithm first for facet-transitive 3-dimensional tilings in E^{3}.

Step 1 The required transitivity properties determine the substructures and so the structures of the possible D-diagrams D with Σ_{I}-operations, I = {0, 1, 2, 3}. The number |D| has to be finite as a consequence of the following steps.

Step 2 For any D-diagram D we consider the ⟨σ_{i}, σ_{j}⟩-orbits i.e. D_{ij}-components (in Pr.1), their types (C.A) or (C.B) and their corresponding parameters e_{ij}. For each D_{ij} component we choose a rotation order v_{ij}(D) = 1, 2, 3,… and so we choose by (4.2.3–4) a corresponding matrix element m_{ij}(D) for each D ∈ D_{ij} according to (4.1.17) and (4.1.8– 10). The possible v_{ij}’s will be bounded by the Step 3 or finally by Step 4. So we have completed the D-diagram to a D-symbol D(M).

Step 3 The partial symbols D^{k}, obtained by cancelling σ_{k}-operations from D, k ∈ I, shall have components determining finite stabilizers Γ^{k} for any k-midpoint orbit. So each component D_{c}^{k} describes a spherical tiling (T_{c}^{k}, Γ_{c}^{k}). Then the Euler theorem yields positive curvature for any connected D^{k}-component D_{c}^{k}, i.e.

∑_{D}^{D_{c}^{k}}[( ∑_{i<j}^{I\\{k\}} \frac{1}{m_{ij}(D)}) − 1] > 0. (4.5.1)

This 2-dimensional ”curvature formula” is deciding for 2-dimensional tilings (see [4] and [9] for the existence criteria of 2-tilings). E.g. if k = 0, then any component D_{c}^{0}(M_{c}^{0}) of D(M), with the corresponding submatrix function M_{c}^{0} satisfying (4.5.1), shall describe a spherical tiling T_{c}^{0} around a vertex (0-face) and a symmetry group Γ_{c}^{0} which will be restricted further in Step 4.

Step 4 The group Γ determined by D(M) has to be a crystallographic space group from among the 219 isomorphism classes. Thus the rotation orders in Step 2 correspond to the Barlow condition v_{ij}(D) = 1, 2, 3, 4, 6. Moreover, the substructures of stabilizers Γ^{k}, k ∈ I, help choosing the candidate Γ on the base of Wyckoff positions from [10].

Step 5 The space group candidate Γ corresponds to D(M) indeed, if a combinatorially given fundamental domain F by D(M) can be realized also metrically (up to equivariant bijection) as a metric fundamental domain in E^{3}. The generalized Poincaré theorem (in the sense of Sect.4.4 and [15]) says that all the partial D-symbol components D_{ij} and the corresponding stabilizers Γ_{ij} have to be realized metrically in E^{3} by the space group Γ (see e.g. [15] for some details). These criteria guarantee the existence of the equivariant bijection in question. As all the space groups are well-known, such a copy algorithm can be constructed by the coordinate presentation of the stabilizers in Γ.

The critical points of our algorithm are Step 1 and Step 5. Indeed, in searching all the solid transitive tilings in E^{3} we cannot guarantee the finite set of possible D-diagrams. This fact is known since Heesch’s and Zamorzayev’s examples. But assuming e.g. face transitivity, we can prove that |D| ≤ 20 (as in [5]).

In non-Euclidean spaces, as in the spherical space S^{3} and the Bolyai–Lobachevskian hyperbolic space H^{3}, the rotational orders maybe not bounded in Step 2.

A given stabilizer collection for D(M) in Step 3 may lead to different Γ’s of E^{3} (as the 10 fixed point free space groups show), moreover, stabilizers in different spaces may coincide.

We are working on the algorithmic problems mentioned in Sect.4.4 together with A. Dress, D. Huson, O. Delgado Friedrichs, I. Prok, J. Szirmai, Z. Lučić, M. Stojanović and other colleagues.

We hope that analogous algorithms have applications in various fields of investigation. I am indebted to Andreas Dress and Daniel Huson for their valuable remarks and

suggestions, to I. Prok for typesetting the text in TEXand to E. Buchsteiner-Kiessling for setting the tables in TEX.

References

[1] M.S. Delaney: Quasi symmetries of space group orbits. Proceedings of the ZiF-Conference on Crystallographic Groups, Match 9 (1980), 73–80.

[2] B.N. Delone: Teoriya planigonov. Izv. Akad. Nauk SSSR Ser. Matem. 23 (1959), 365–386. (Russian)

[3] A.W.M. Dress: Regular polytopes and equivariant tessellations from a combinatorial point of view. In Algebraic Topology, pages 56–72. SLN 1172, Göttingen, 1984.

[4] A.W.M. Dress: Presentation of discrete groups, acting on simply connected manifolds in term of parametrized systems of Coxeter matrices. Advances in Math. 63 (1987), 196–212.

[5] A.W.M. Dress, D.H. Huson, and E. Molnár: The classification of face-transitive 3Dtilings. Acta Crystallographica A49: 806–817, 1993.

[6] B. Grünbaum and G.C. Shepard: Isotoxal tilings. Pacific J. of Math. 76 (1978), 407–430.

[7] B. Grünbaum and G.C. Shepard: Tilings and Patterns. W.H. Freeman and Company, New York, 1987.

[8] D.H. Huson: Tile-transitive partial tilings of the plane. Beiträge zur Algebra und Geometrie 34 (1993), No.1. 87–118.

[9] D.H. Huson: The generation and classification of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane. Geometriae Dedicata 47 (1993), 269–296.

[10] International Tables for X-Ray Crystallography, Vol.1, N.F.M. Henry – K. Lonsdale, Symmetry groups. Kynoch Press, Birmingham 1969. New edition by Theo Hahn, Vol.A, Reidel Co. Dordrecht 1983.

[11] Z. Lučić, E. Molnár, and M. Stojanović: The 14 infinite series of isotoxal tilings in the planes of constant curvature. Manuscript, 1990. Periodica Math. Hung.

[12] B. Maskit: On Poincaré’s theorem for fundamental polygons. Advances in Math. 7 (1971), 219–230.

[13] E. Molnár: Minimal presentation of crystallographic groups by fundamental polyhedra. Crystall Symmetries 2, edited by I. Hargittai and B.K. Vainshtein, Special Issue in Comput. Math. Applic. 16, No.5–8 (1988), 507–720.

[14] E. Molnár: Two hyperbolic football manifolds. In Proceedings of Int. Conf. on Diff. Geom and Applic. 1988, Dubrovnik (Yugoslavia), 217–241.

[15] E. Molnár: Polyhedron complexes with simply transitive group actions and their realizations. Acta Math. Hung. 59 (1–2) (1992), 175–216.

[16] E. Molnár: Klassifikation der hyperbolischen Dodekaederpflasterungen von maximalen flächentransitiven Bewegungsgruppen. Math. Pannonica 4/1 (1993), 113–136.

[17] E. Molnár and I. Prok: A polyhedron algorithm for finding space groups. Proceedings of Third Int. Conf. on Engineering Graphics and Descriptive Geometry. Vienna 1988, Vol 2, 37–44.

[18] E. Molnár and I. Prok: Classification of solid transitive simplex tilings in simply connected 3-spaces, Part 1. Colloquia Math. Soc. J. Bolyai 63, Intuitive Geometry, Szeged (Hungary) 1991, North–Holland, 311–362.

[19] H. Poincaré: Theorie des groupes Fuchsiennes. Acta Math. 1 (1882), 1–62.

[20] I. Prok: The Euclidean space has 298 fundamental tilings with marked cubes by 130 space groups. Colloquia Math. Soc. J. Bolyai 63, Intuitive Geometry, Szeged (Hungary) 1991, North–Holland, 363–390.

[21] P. Scott: The geometries of 3-manifols. Bull. London Math. Soc. 15 (1983), 401–487.

References [8], [9], [16], [18], [21] have been inserted at the revision.

Work supported in part by

• the Center for Interdisciplinary Research of the University of Bielefeld (ZiF) (research group on COMBINATORICS AND ITS APPLICATIONS (1990/91),

• the Sonderforschungsbereich 343 ”DISKRETE STRUKTUREN IN DER MATHE-MATIK”, Bielefeld University

• and the HUNGARIAN FOUNDATION FOR SCIENTIFIC RESEARCH (OTKA) No.T 7351 (1993).

Received: Oktober 15, 1992, revised version May 25, 1994.