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In Search of Demiregular Tilings Helmer Aslaksen

Summary: In Search of Demiregular Tilings
Helmer Aslaksen
Department of Mathematics
National University of Singapore
Singapore 117543
Many books on mathematics and art discuss a topic called demiregular tilings and claim that there are 14 such tilings.
However, many of them give different lists of 14 tilings! In this paper we will compare the lists from some standard
references that give a total of 18 such tilings. We will also show that unless we add further restrictions, there will
in fact be infinitely many such tilings. The "fact" that there are 14 demiregular tilings has been repeated by many
authors. The goal of this paper is to put an end to the concept of demiregular tilings.
1 Introduction
Several authors, including Ghyka (1946, [5]), Critchlow (1969, [2]), Williams (1979, [13]) and Lundy (2001,
[9]), introduce a concept called demiregular tilings They define an edge-to-edge tiling by regular polygons to
be demiregular if it has more than one type of vertex, and claim that there are only 14 such tilings. However,
they all give different lists of demiregular tilings! In addition, some of them cite Steinhaus (1937, [11]) who
says: "[demiregular tilings] are perhaps even more beautiful ... their number is unlimited. (Why?)". Even


Source: Aslaksen, Helmer - Department of Mathematics, National University of Singapore


Collections: Mathematics