4-Regular Vertex-Transitive Tilings of E 3
Discrete Comput Geom
4-Regular Vertex-Transitive Tilings of E3
O. Delgado Friedrichs 1
D. H. Huson 0
0 Applied and Computational Mathematics, Princeton University , Princeton, NJ 08544 , USA
1 Fakulta ̈t fu ̈r Mathematik, Universita ̈t Bielefeld , 33501 Bielefeld , Germany
There exist precisely 149 topological types of semipolytopal tile-transitive tilings of E3 by “extetrahedra” (obtained from tetrahedra by introducing certain new vertices of degree 2). Dualization gives rise to 149 types of 4-regular vertex-transitive tilings. The 4-coordinated networks carried by these tilings are closely related to crystal structures such as zeolites or diamond. These results are obtained using “combinatorial tiling theory.”
In Tilings and Patterns [GS2], Gr u¨nbaum and Shephard present in detail the full range
of problems and methods associated with (mainly two-dimensional) tilings and patterns
and discuss in depth their relevance for art and science. They address the problem of
tiling three-dimensional space in a number of papers including [GS1], [Gr2], [GMLS],
It seems obvious that a classification of periodic tilings of three-dimensional Euclidean
space E3 will have applications in crystal chemistry, ideally by supplying an enumeration
of all mathematically feasible crystal-structures of a given type, up to a certain degree
However, the problem of classifying periodic tilings of E3 is considerably more
difficult than the problem of classifying two-dimensional periodic tilings, and indeed
touches on one of mathematics great open problems: the Poincare´ Conjecture.
¤ Current address: Celera Genomics, 45 West Gude Drive, Rockville, MD 20850, USA.
More than 10 years ago Dress introduced the method of Delaney symbols [Dr1], [Dr2]
and developed the foundation of what we propose to call combinatorial tiling theory.
This has given rise to a number of papers that investigate different questions and aspects
of this theory.
Based on combinatorial tiling theory, we have developed a computer-aided approach
[DHM], [DH3] to the problem of classifying periodic tilings of E3, involving
combinatorial topology [DH1], computational geometry [EM], [DH2], computational algebra
[ScC], and other tools [Mc], [LMPC], [NM]. Its viability was recently demonstrated in
[DH3] by classifying all tilings of Euclidean space by combinatorial cubes, tetrahedra, or
octahedra, establishing, in particular, the existence of precisely 11, 9, and 3 (respectively)
topological types of tile-transitive tilings by such tiles.
One aim of the current paper is to show that our approach is not restricted to
combinatorially regular tiles, but also applies in the case of combinatorially less regular ones.
In a future paper we shall demonstrate that it can also be used to classify tilings with two
or more types of tiles.
Crystal structures are often interpreted as atom-bond networks, or, topologically,
simply as graphs embedded in E3. Given such a network, it is a highly nontrivial task to
decide whether a periodic tiling exists that carries it in the sense that the edge-skeleton
of the tiling is (topologically) the given network.
A zeolite is an aluminosilicate in which the Al and Si atoms occupy 4-coordinated
(i.e., 4-valent) vertices of a three-dimensional network, and the oxygen atoms occupy
2coordinated positions between the 4-coordinated vertices [Sm]. Neglecting the 2-valent
oxygen atoms, zeolites are 4-valent networks, as is the diamond network, too. They have
many important applications in chemistry [Sm].
Currently, the online version of the Atlas of Zeolite Structure Types [OMB] (see
also [MO]) lists 121 approved zeolite structures. Precisely 18 of these are uninodal, i.e.,
have symmetry groups that act transitively on the set of 4-valent atoms. In turns out that
precisely six of these are carried by duals of tile-transitive tilings by combinatorial
tetrahedra [DH3] This inspires us to consider the following question: Do there exist periodic
tilings that carry the remaining 12 uninodal zeolite structures?
In an attempt to answer this, we introduce the concept of an extetrahedron of level h,
which is obtained by “extending” a tetrahedron by inserting new vertices of degree 2
into some of the original edges, up to h in each. We classify all tile-transitive tilings of
Euclidean space by extetrahedra of level 1 and will see that by dualization we obtain
carriers for all 18 uninodal zeolites.
Some further definitions are introduced in Section 2. We then give a short summary
of our approach in Section 3. Finally, in Section 4, we describe our classification results
in tabulated form and depict a number of interesting examples.
Combinatorial tiling theory and the methods and results described in this paper
represent a major step forward toward the goal of systematically enumerating mathematically
feasible crystal structures [DDHC], [O’K].
Although there is a common general understanding of what a tiling should be, definitions
differ in their details. Within the framework of “combinatorial tiling theory,” tilings
are naturally and very generally defined in terms of their chamber systems as those
subdivisions of space that possess a “Delaney-symbol” [Dr1], [Dr2]. For the purposes
of this paper, we define a tiling of some d-dimensional manifold X without boundary
as the collection of cells of a regular CW-complex with total space X . The cells are also
called faces of the tiling. This definition is narrower in that every tiling that satisfies it
possesses a Delaney symbol, but not vice versa.
We define the terms vertex, edge, facet, and tile in the usual way. Obviously, the set
of tiles covers X . Two faces are said to be incident if one is included in (the boundary
of) the other. Two nonincident faces are adjacent if their intersection is nonempty. If f
is any fixed face, then the set of all faces contained in the boundary of f form a tiling of
We define the graph carried by a tiling to be the graph naturally induced by the
vertices and edges of the 1-skeleton of the tiling. As usual, a graph is called polytopal if
it is isomorphic to the graph of a convex 3-polytope, i.e., if it is planar and 3-connected
[Gr1]. We call a tiling of a topological 2-sphere a combinatorial polytope if its graph
is polytopal, and we call it semipolytopal if its graph can be derived from a polytopal
graph by subdivision of edges. We call a tiling of E3 semipolytopal if all its tiles and the
tiles of its dual are semipolytopal.
The degree of a vertex is the number of edges incident to it and we call a tiling
n-regular if all its vertices have degree n. The smallest degree that can appear in a
semipolytopal tiling is 4.
We call two tilings topologically equivalent if there exists a homeomorphism between
their total spaces that takes faces onto faces. The symmetry group of a tiling of some
metric space such as E3 consists of all isometries of that space that map faces onto faces.
A tiling is called vertex-transitive if for each pair of vertices there exists a symmetry that
maps one onto the other. In general, transitivity classes of tiles, faces, edges, or vertices
are to be understood with respect to the symmetry group of the tiling.
Ultimately we are interested in vertex-transitive 4-regular tilings. These arise by
dualization from tile-transitive tilings by 4-faced tiles and we now focus on the latter.
There exist precisely 11 different topological types of extetrahedra of level 1, see
Fig. 1. Each gives rise to one or more different equivariant types, which are distinguished
by taking the possible symmetry groups into account, as in [DH3]. The 11 depicted
topological types t; t1; t2a; t2b; : : : give rise to 11, 5, 2, 8, 2, 4, 4, 2, 8, 5, and 11 equivariant
For each equivariant type T , we apply the combinatorial enumeration approach
described in [DH3] to classify all periodic tilings of E3 by tiles of type T . In total we obtain
1720 different topological types, including the nine types of tilings by tetrahedra. To be
precise, in terms of combinatorial tiling theory, this produces a list of “maximal Delaney
symbols” that describe the tilings uniquely up to topological equivalence.
For each such combinatorial description we are then faced with the task of
constructing a geometric realization of the encoded tiling. In [De] and [DH3] we
indicate how a straight-edge realization can often be obtained by first determining and
parameterizing the linear hull of the space of admissible vertex positions and then
using standard optimization techniques to find “preferable” parameter values.
We prefer parameters values that give rise to realizations with high volume
(measured as the ratio of the volume of a fundamental domain and the cubed average edge
length) and small variation of edge lengths. For a given positioning of vertices, each
higher-dimensional face is constructed inductively as the linear cone on its boundary
with apex at the center of gravity of its vertices. Some simple steps were taken to give
the two-dimensional faces a smoother appearance in Figs. 2–4.
This simple form of optimization is not guaranteed always to produce correctly
embedded tilings and indeed for 11 of the 149 cases listed in Table 2 (numbers 53, 58,
67, 68, 77, 78, 79, 100, 101, 102, and 145) it fails to do so. However, we
emphasize that the existence of Euclidean Delaney symbols for these exceptional cases
implies that they all possess geometric realizations, although not necessarily with straight
We remark that all tilings are realized with full symmetry, i.e., in such a way that all
combinatorial symmetries are isometries. It follows from a nontrivial result in geometric
topology [MS] that this is always possible for periodic tilings of E3 with “maximal
Delaney symbols” [De].
Using the approach indicated in the preceding section, we obtain the following result:
Theorem 4.1. There exist precisely 149 topological types of semipolytopal
tiletransitive tilings of three-dimensional Euclidean space by extetrahedra of level 1, of
which exactly nine are by combinatorial regular tetrahedra. Their duals are summarized
in Table 2. If tilings are not required to be semipolytopal, then there exist 1571 further
By dualization, Theorem 4 gives rise to 149 types of semipolytopal, vertex-transitive
4-regular tilings of E3. Of the 18 uninodal networks listed in the current online version
of the Atlas of Zeolite Structure Types [OMB], 16 are carried by at least one of these
(a) Tiling #047
(c) Tiling #050
(e) Tiling #075
tilings, as indicated in Table 2. Tilings that carry the remaining two uninodal zeolites
ANA (Analcime) and DFT can be found among (the duals of) the 1571 additional
Our results clearly do not give a complete classification of all semipolytopal,
vertextransitive 4-regular tilings of E3, as we only considered extetrahedra of level 1. By
results in combinatorial tiling theory [DHM], for any fixed level h, there exist only a
finite number of types of tile-transitive tilings by extetrahedra of level h. We state the
following open problem: Is there an upper bound for the possible number of additional
vertices? In other words, do there exist only finitely many types of semipolytopal,
tiletransitive tilings by extetrahedra of arbitrary level?
We depict a number of examples in Figs. 2–4. For each tiling a finite patch of tiles is
shown. Tiles are shrunk slightly toward their centers to make their faces visible. Note that
tiling number 149 (and all tilings that contain tiles of type 3k) represents the diamond
network. A complete description of our results is available on the World Wide Web at
We summarize the classification in Table 2. Each row describes one of the 149
topological types of tilings. The data in each column is:
) The number of the tiling.
) The vertex type, i.e., the topological type of the tiles of the dual tiling, as defined
in Fig. 1. Tilings with the same vertex type are listed consecutively.
) The “orbifold name” [Co], [CH] of the vertex stabilizer, the group of all
symmetries of the tiling that leave a given vertex fixed, see Table 1.
(4) A four-digit code listing the number of transitivity classes of vertices, edges,
facets and tiles.
(5) The topological types of the tiles, as defined in Fig. 5.
(6) The international number and Hermann–Mauguin name for the symmetry group
[Ha]. This refers to a representative of the topological class with maximal
(7) The “fibrifold name” (in the case of reducible groups) or “Conway name” (in the
case of the 35 irreducible groups) for the symmetry group [CDHT].
(8) For those tilings that carry a known zeolite, the “structure code” for the zeolite
[OMB]. The first occurrence of each zeolite is underlined.
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