MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR B. N. DELONE

PROOF OF THE FUNDAMENTAL THEOREM OF THE THEORY OF STEREOHEDRA

Regular partitions of spaces have long interested mathematicians and find important applications in crystallography and in the physics of solids. The best known are the regular partitions of the sphere. They form the basis of the theory of the so-called simple forms of crystals considered in crystallography. On the theory of the regular Dirichlet partitions corresponding to the group of parallel translations of three-dimensional space is based the method, proposed by me in 1933, for reducing (regular setting) crystals, which is now accepted in international handbooks, etc.

However, up to now one of the most fundamental theorems of the theory of regular partitions had not been proved; namely, it was not known whether, as a function of the number of dimensions (n), the number of topologically distinct normal regular partitions of (n)-dimensional Euclidean space is bounded. This question had so far been solved only for the case (n = 2), i.e., for the plane, and there was no visible path to its solution even in the case of ordinary three-dimensional space. In the present note I give a complete solution of this question for arbitrary (n).

The solution proposed here is based on a certain modification of the famous argument of Minkowski, by means of which he proved that the number of facets of an (n)-dimensional parallelohedron, i.e., of a convex fundamental domain of a discrete group of parallel translations of (n)-dimensional Euclidean space, is not greater than (2 \cdot (2^n - 1)).

If the whole space is filled by certain bodies, which may also have common interior points, then we call this a filling; if, however, these bodies pairwise have no common interior points, i.e., fill the space uniquely, then we shall call their totality a partition of the space. A partition is called regular or isohedral if, for any two bodies of the partition, in the group of all those motions of the space (of the first kind, or of the first and second kind) that map the partition onto itself, there is at least one motion mapping the first of these bodies onto the second. The bodies themselves of a regular partition are called stereohedra. If the stereohedra of the partition are convex, then they are convex polyhedra, and the partition itself is called convex. If, moreover, they are adjacent along whole facets, the partition is called normal.

Theorem. The number of topologically distinct regular normal partitions of (n)-dimensional Euclidean space is bounded as a function only of (n).

Proof. If there is a normal regular partition, then the topology of this partition is completely determined by the following data:

(1^\circ). The topology of the edge-net of the stereohedron of the partition.

(2^\circ). The topology of the incidence of the stereohedra of the partition at the vertices of the stereohedron.

(3^\circ). The adjacency symbol(^*), i.e., an indication of which facets and how adjoin—

(^*) We do not know whether one can restrict oneself only to the data (1^\circ) and (2^\circ).

with the given stereohedron (S) of the partition those stereohedra (S_1, S_2, \ldots) which are adjacent to it along ((n-1))-dimensional faces. But since, when the first two topologies (1^\circ) and (2^\circ) are given, the number of different adjacency symbols is already bounded by purely combinatorial considerations, in order to prove the theorem it is enough to prove boundedness as a function only of (n) of the number of different topologies (1^\circ) and (2^\circ). To prove boundedness of the number of different topologies (1^\circ), it is plainly enough to prove boundedness, depending only on (n), of the number of faces of stereohedra; and to bound the number of different topologies (2^\circ), it is enough to prove boundedness, depending only on (n), of the number of stereohedra meeting at a vertex.

The full group (G) of motions of the space which carry the regular partition onto itself is, by assumption, transitive with respect to the stereohedra of the partition; i.e., whatever two stereohedra of the partition are taken, the group contains at least one motion under which the first of them passes into the second. Consider that subgroup (G_0) of the group (G) whose motions carry some fixed stereohedron (S) of the partition into itself; this is a certain finite group of rotations of the stereohedron (S) about its center of gravity (A). Let it be of order (l); then the number of motions from (G) which carry a certain given stereohedron of the partition into a certain given other stereohedron of it is equal to (l).

If the group (G) has a subgroup (G') such that its (G'_0) reduces to the identity transformation alone, then the stereohedra of the partition are fundamental regions for this group (G'). In this case we call the group (G') a fundamental group of the partition under consideration. However, we do not know whether the theorem is true that every normal regular partition of (n)-dimensional Euclidean space has at least one fundamental group. In the case of the Euclidean 2-dimensional plane this is so, and even in this case regular partitions have, generally speaking, several different fundamental groups. However, in the case of the 2-dimensional sphere this is not so: the regular partitions of the sphere corresponding to the icosahedron and the triacontahedron have no fundamental groups.

The group (G) is an (n)-dimensional Fedorov group (i.e., (n)-dimensional, discrete, and having a finite fundamental domain), and therefore, by the well-known theorem of Schoenflies—Bieberbach, it has an (n)-dimensional subgroup (T) of parallel translations. The index (h) of this subgroup relative to (G) is the order of the finite rotation group (\Psi), consisting of the rotational parts of all motions of the group (G). As the center of rotations of the group (\Psi) one may take the point (A). This group (\Psi) carries into itself the lattice ({A_t}) of points obtained from the point (A) by all translations (t) of the group (T). By Minkowski’s theorem, the number of faces of an (n)-dimensional parallelohedron is not greater than (2\cdot(2^n-1)); hence it follows that the order (h) of the group (\Psi) is not greater than some number depending only on (n), since the rotations of (\Psi) plainly carry into itself that parallelohedron which is the Dirichlet domain of the point (A) in the lattice ({A_t}).

The group (G_0) is the group of those motions of the group (G) which leave the point (A) fixed; therefore the group (G_0) is a subgroup of (\Psi). Let

[
\Psi = G_0 + \psi_2 G_0 + \psi_3 G_0 + \ldots + \psi_{h'} G_0 \quad (\text{where } h' = h/l)
]

be the decomposition of the group (\Psi) into cosets with respect to its subgroup (G_0), and let (1, g_2, g_3, \ldots, g_{h'}) be motions from (G) whose rotational parts are (1, \psi_2, \psi_3, \ldots, \psi_{h'}). In this case all stereohedra of the partition split into (h') nonintersecting “lattices of stereohedra,” namely into such aggregates of stereohedra whose centers of gravity form the lattices ({A_t}, {A_{g_2t}}, \ldots, {A_{g_{h'}t}}), and whose rotations relative to the initial stereohedron (S) under consideration are (1, \psi_2, \psi_3, \ldots, \psi_{h'}).

After these preliminary considerations we pass to the proof of the theorem.

We shall first show that the number of ((n-1))-dimensional faces of a stereohedron is not greater than

[
2\cdot(2^n-1) + (h'-1)\cdot 2^n .
]

Indeed, consider those ((n-1))-dimensional faces of the stereohedron (S) under consideration along which it is touched by stereohedra of any one and the same of the (h') “lattices of stereohedra” considered above. If the number of such stereohedra were greater than (2^n), then among them there would be two, (S_1) and (S_3), whose corresponding coordinates of the centers of gravity, relative to the fundamental parallelepiped of the lattice to which they belong, would be congruent modulo 2; then the vector going from the center of gravity of one of these stereohedra to the center of gravity of the other would have even coordinates, i.e., would be twice an integral vector of this lattice. But then there would exist a third stereohedron (S_2) such that (S_2) is obtained from (S_1) by the same translation (t) by which (S_3) is obtained from (S_2). Let the ((n-1))-dimensional faces of the stereohedron (S) along which it is touched by the stereohedra (S_1) and (S_3) be (b) and (a). Namely, the stereohedron (S_1) touches (S) along its face (b) with its own face (b_1), and the stereohedron (S_3) touches (S) along its face (a) with its own face (a_3). On the stereohedra (S_1, S_2, S_3) there are faces (a_1b_1, a_2b_2, a_3b_3), obtained from (b_1) and (a_3) by the corresponding parallel translations. Let (A) and (B) be some interior points of the faces (a) and (b), and let (A_1B_1, A_2B_2, A_3B_3) be the corresponding interior points of the faces (a_1b_1, a_2b_2, a_3b_3), with the points (B) and (B_1), and the points (A) and (A_3), coinciding. Join by straight-line segments the points (A_1B_1, A_2B_2, A_3B_3) and the points (B_1) and (A_3). The following figure is obtained:

on which (A_1B_1B_3A_3) is a parallelogram, (A_2B_2) is its midline, and (B_1A_3) is a diagonal. By virtue of the convexity of the stereohedra, all interior points of the segment (A_2B_2) are interior points of the stereohedron (S_2), and the interior points of the segment (B_1A_3) are interior points of the stereohedron (S). The point (C) of intersection of these segments, which is their common midpoint, is therefore a common interior point of the stereohedra (S_2) and (S), i.e., these stereohedra coincide.

Hence we conclude that if (S) does not belong to that “lattice of stereohedra” whose contacts along ((n-1))-dimensional faces with the stereohedron (S) we are studying, then the number of stereohedra of such a lattice touching (S) along ((n-1))-dimensional faces is not greater than (2^n). If, however, (S) belongs to this lattice, then there are not more than (2\cdot(2^n-1)) of them, since in this case to each stereohedron (S_1) of this lattice touching (S) along an ((n-1))-dimensional face there corresponds another stereohedron (S_3) of it, also touching (S) along an ((n-1))-dimensional face, such that the triple (S_1SS_3) is precisely the triple (S_1S_2S_3).

Let us now prove that the number of stereohedra meeting at one vertex is not greater than (h!2^n). Indeed, if more than (2^n) stereohedra of one and the same “lattice of stereohedra” came to the vertex, then in it there would be a triple (S_1S_2S_3) such that its stereohedra (S_1) and (S_3) would have a common point. But this is impossible, since if (P) is some ((n-1))-dimensional plane separating (S_1) and (S_2)—and such a plane always exists, since (S_1) and (S_2) are convex and have no common interior points—then the plane (P'), obtained from it by the translation (t) by which (S_2) is obtained from (S_1), or (S_3) from (S_2), separates (S_2) from (S_3), and therefore the stereohedra (S_1) and (S_3) are separated from one another by a layer (“slab”) (PP') of nonzero thickness. The latter is so because (S_2) lies between (P) and (P').

Received
24 III 1961