Comptes Rendus
no. 8
Combinatorics/Geometry
The Heesch number for multiple prototiles is unbounded
[Le nombre de Heesch pour plusieurs proto-pavés est non borné]

Présenté par : the Editorial Board

Bojan Bašić1
1 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 665-669.

Dans cette note, on prouve que, pour tous entiers k⩾3 et n⩾1, il existe un proto-ensemble qui possède k proto-pavés et dont le nombre de Heesch est égal à n. Cela réfute une conjecture de Grünbaum et Shephard.

In this paper we show that, for all integers k⩾3 and n⩾1, there exists a protoset consisting of k prototiles, whose Heesch number is n. This disproves a conjecture by Grünbaum and Shephard.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.05.002
Bojan Bašić 1

1 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia 
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Bojan Bašić. The Heesch number for multiple prototiles is unbounded. Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 665-669. doi : 10.1016/j.crma.2015.05.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.05.002/

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[7] C.S. Kaplan Introductory Tiling Theory for Computer Graphics, Morgan & Claypool Publishers, 2009

[8] W. Lietzmann Lustiges und Merkwürdiges von Zahlen und Formen, Hirt, Breslau, 1928

[9] C. Mann Heesch's problem and other tiling problems, University of Arkansas, 2001 (PhD thesis)

[10] C. Mann Heesch's tiling problem, Am. Math. Mon., Volume 111 (2004), pp. 509-517

[11] M. Senechal Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995

[12] S.K. Stein; S. Szabó Algebra and Tiling: Homomorphisms in the Service of Geometry, Mathematical Association of America, Washington, DC, 1994

[13] A.S. Tarasov On the Heesch number for the Lobachevskiĭ plane, Mat. Zametki, Volume 88 (2010), pp. 97-104 (Russian); English translation in Math. Notes, 88, 2010, pp. 97-102

[14] W.P. Thurston Groups, tilings and finite state automata, Minneapolis, MN, USA (1989)

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