A four-color theorem for periodic tilings
Continuing previous investigations, the author has enumerated (by computer) those topological types of plane tilings (for \(k = 2, 3, 4\)) with \(k\) transitivity classes of tiles (\(k\)-homeohedral tilings) for which no tile shares an edge with an equivalent one and which, therefore, form a map coloured (in the sense of the Four Colour Theorem) with \(k\) colours (corresponding to the \(k\) classes of tiles). In this paper he presents the results of this classification, and describes the theoretical background (Delaney-Dress symbols of equivariant tilings) and the tools (vertex-truncation and edge contraction for these symbols) on which the algorithm is based. The results are summarized in two tables: The first one shows how these tilings (1 for \(k = 2\), 143 for \(k = 3\), and 4044 for \(k = 4\)) are distributed among the plane crystallographic groups. The second one lists (for \(k = 4\)) the number of types grouped according to the degrees (i.e., the number of edges) of the 4 classes of tiles. (Computer graphics of these tilings are available as a part of RepTiles, a shareware program for MacIntosh computers.).
scientific article; zbMATH DE number 663733