In Search of Demiregular Tilings

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Helmer Aslaksen: In Search of Demiregular Tilings. In: Bridges 2006. Pages 533–536



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.

Extended Abstract


Used References

[1] D. P. Chavey, Periodic tilings and tilings by regular polygons, Ph.D. thesis, Univ. of Wisconsin, Madi- son, 1984.

[2] Keith Critchlow, Order in Space, A Design Source Book, Thames and Hudson, 1969.

[3] Brian L. Galebach, Number of n-uniform tilings, The On-Line Encyclopedia of Integer Sequences,

[4] Brian L. Galebach, N-uniform Tilings,

[5] Matila Ghyka, The Geometry of Art and Life, Sheed and Ward, 1946 (second edition, Dover Publica- tions, 1977).

[6] Branko Grünbaum and G.C. Shephard, Tilings by Regular Polygons, Mathematics Magazine 50 (1977), 227–247.

[7] Branko Grünbaum and G.C. Shephard, Tilings and Patterns, W. H. Freeman and Company, 1987.

[8] O. Krötenheerdt, Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene. I, Wiss. Z. Martin- Luther-Univ. Halle-Wittenberg, Math.-Natur. Reihe 18 (1969), 273–290.

[9] Miranda Lundy, Sacred Geometry, Walker and Company, 2001.

[10] NG Lay Ling, Tilings and Patterns, Honours Project, Department of Mathematics, National Univ. of Singapore, 2004,

[11] Hugo Steinhaus, Mathematical Snapshots, Dover Publications, 1999 (originally published in Polish, 1937; first English edition, 1938; Oxford University Press, 1950).

[12] Eric W. Weisstein, Demiregular Tessellation, from MathWorld — A Wolfram Web Resource, mathworld.

[13] Robert Williams, The Geometrical Foundation of Natural Structure: A Source Book of Design, Dover Publications, 1979.


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