Beitr\"age 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 Moln{\'a}r
Abstract.
The purpose of this paper is to popularize the method of
D-symbols (Delone--Delaney--Dress symbols) for classifying
periodic tilings $(\cT,\Ga)$, where a group $\Ga$ acts on a
$d$-dimensional combinatorial tiling $\cT$ with prescribed
transitivity properties. As a new result, we describe the equivariant
classification of all cube tilings $(\cT,\Ga)$ in Euclidean space
$\bE^3$, where the group $\Ga$ acts on the infinite face-to-face
cube tiling $\cT$ with 1 transitivity class ($\Ga$-orbit) of
faces. In Table~1--3 the 43 classes of these pairs $(\cT,\Ga)$
are characterized by giving for each the number of the space group
$\Ga$, the D-symbol, a fundamental domain $\cF$ in Fig.2, face
identifying generators and defining relations for $\cF$ and the
$\Ga$-conjugacy classes for stabilizers of solids $(\Ga^3)$,
faces $(\Ga^2)$, edges $(\Ga^1)$ and vertices $(\Ga^0)$,
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.