A four-color theorem for periodic tilings
type: Item
instance of: scholarly article
title: A four-color theorem for periodic tilings
zbMATH Open document ID: 0823.52015
DOI: 10.1007/BF01264100
published in: Geometriae Dedicata
publication date: 1994-10-18T00:00:00Z
review text: 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.).
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 52-04
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 68U05
zbMATH DE Number: 663733
zbMATH Keywords: periodic tilings
zbMATH Keywords: plane \(k\)-homeohedral tilings
zbMATH Keywords: classification
zbMATH Keywords: coloring
zbMATH Keywords: enumeration
author: D. Huson
MaRDI profile type: Publication
cites work: Q3911038
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full work available at URL: https://doi.org/10.1007/bf01264100
OpenAlex ID: W1963691306
description: scientific article; zbMATH DE number 663733