On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups

A stereohedron is any bounded convex polyhedron which tiles the space by the action of some crystallographic group. A Dirichlet stereohedron for a certain crystallographic group G is the Voronoi region Vor Gp (p) of a point p ∈ R 3 in the Voronoi diagram of an orbit Gp.

The study of the maximum number of facets for stereohedra is related to Hilbert’s 18th problem, “Building up the space with congruent polyhedra” (see [12,16]). Bieberbach (1910) and Reinhardt (1932) answered completly the first two of Hilbert’s specific questions, but other problems related to monohedral tessellations (i. e., tessellations whose tiles are congruent) remain open. An exhaustive account of this topic appeared in a survey article by Grünbaum and Shephard [11], where our problem, to determine the maximum number of facets-or, at least, a “good” upper bound-for Dirichlet stereohedra in R 3 , is mentioned as an important one. Previous results on this problem are:

• The fundamental theorem of stereohedra (Delaunay, 1961 [6]) asserts that a stereohedron of dimension d for a crystallographic group G with a aspects cannot have more than 2 d (a + 1) -2 facets. The number of aspects of a crystallographic group G is the index of its translational subgroup. Delone’s bound for three-dimensional groups, which have up to 48 aspects, is 390 facets. • The three-dimensional stereohedron with the maximum number of facets known so far was found in 1980 by P. Engel (see [7] and [11, p. 964 (1) Bounds after processing triad rotations.

(2) Bounds after diad rotations with axes parallel to the coordinate axes.

(3) Bounds after diagonal diad rotations.

(4) Bounds after intersecting with planar projections.

There is agreement among the experts (see [7, page 214], [11, page 960], [18, page 50]) that Engel’s sterohedron is much closer than Delone’s upper bound to having the maximum possible number of facets. Our results confirm this.

In 2000, the second author and D. Bochiş gave upper bounds for the number of facets of Dirichlet stereohedra. They did this by dividing the 219 affine conjugacy classes of three-dimensional crystallographic groups into three blocks, and using different tools for each. Their main results are:

• Within the 100 crystallographic groups which contain reflection planes, the exact maximum number of facets is 18 [1]. • Within the 97 non-cubic crystallographic groups without reflection planes, they found Dirichlet stereohedra with 32 facets and proved that no one can have more than 80. Moreover, they got upper bounds of 50 and 38 for all but, respectively, 9 and 21 of the groups [2]. • They also considered cubic groups, but they were only able to prove an upper bound of 162 facets for them [3]. In [17] we improved the bound for 14 of the 22 cubic groups without reflections planes, the 14 “full groups”: Theorem 1.1. Dirichlet stereohedra for full cubic groups cannot have more than 25 facets.

In this paper we give an upper bound for the remaining cubic groups: the 8 “quarter groups”. It has to be noted that to get these bounds, contrary to the ones in the previous papers of this series, computers are used. The upper bound we obtain for each quarter group is shown in Table 1. Columns (1) to (4) are the bounds obtained in different phases or our method, the column labeled “Final” is our final bound. Globally, we get the following. Theorem 1.2. Dirichlet stereohedra for quarter cubic groups cannot have more than 92 facets.

For the sake of completeness, we include here the full list of other crystallographic groups for which the bounds proved in this series of papers is bigger than 38 (Table 2). This list is the same as

2.1. “Full” and “quarter” cubic groups. Our division of cubic groups into “full” and “quarter” ones comes from the recent classification of three-dimensional crystallographic groups developed in [5] by Conway et al. They divide crystallographic groups into “reducible” and “irreducible”, were irreducible groups are those that do not have any invariant direction. It turns out that they coincide with the cubic groups of the classical classification. Conway et al. define odd subgroup of an irreducible group G as the one generated by the rotations of order three, and show that:

Theorem 2.1 (Conway et al. [5]).

(1) There are only two possible odd subgroups of cubic groups, that we denote F and Q.

(2) Both F and Q are normal in Isom(R 3 ). Hence, every cubic group lies between its odd subgroup and the normalizer N (F ) and N (Q) of it.

The second property reduces the enumeration of cubic space groups to the enumeration, up to conjugacy, of subgroups of the two finite groups N (F )/F and N (Q)/Q. N (Q)/Q is dihedral of order 8 and N (F )/F has order 16 and contains a dihedral subgroup of index 2.

The main difference between F and Q is that Q only contains triad

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