Symmetries of equivelar 4-toroids
The quotient of a regular tiling by a subgroup of full rank of its translation group is an equivelar toroid. In this paper, the authors consider such toroids obtained from the tessellations \(\{4,4\}\) and \(\{3,6\}\) of the plane, and the cubic tiling \(\{4,3,4\}\) of ordinary space, particularly the latter. The symmetry group of a toroid depends on how symmetric the translation subgroup is. For example, for \(\{4,3,4\}\) the subgroup is generated by triples \(v_1,v_2,v_3\) of lattice (integer) vectors, and these fall into various classes, subdivided into families, according to how large a subgroup of the octahedral group continues to act on the toroid. Thus in class 1, with three families, one has the full octahedral group (these are the regular toroids). Class 2 would contain chiral toroids, but there are none, and so this class is empty. Otherwise, there is a class (for some, split into subclasses) for each divisor of \(24\), with only the central symmetry of the octahedra group surviving in class 24; the full results are too elaborate to reproduce in the review. In a similar way, the classification of the toroids of types \(\{4,4\}\) and \(\{3,6\}\) depend on the subgroup structure of the dihedral groups of orders \(8\) and \(12\), respectively.
scientific article; zbMATH DE number 6124644