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.