Quelles tuiles! (Pavages apériodiques du plan et automates bidimensionnels). (Aperiodic tilings of the plane and two-dimensional automata). (French) Zbl 0726.11020

Salon, Olivier

Sémin. Théor. Nombres Bordx., Sér. II 1, No. 1, 1-26 (1989).

It is proved that the regular Robinson tiling of the plane [see B. Gruenbaum and G. C. Shephard, Tilings and patterns (New York 1987; Zbl 0601.05001)], restricted to a quadrant, can be viewed as a two- dimensional fixed point \((t(m,n))_{(m,n)\in {\mathbb{N}}^ 2}\) of a (2,2)- substitution. Actually, there are 48 subsequences of the form \((t(p^ km+r,p^ kn+s))_{m,n},k\in {\mathbb{N}},\quad 0\leq r,s<p^ k\) with \(p=2\) and the result follows from a theorem due to the author [Sémin. Théor. Nombres, Univ. Bordeaux I 1986/1987, Exp. No.4 (1987; Zbl 0653.10049)] which characterizes fixed points t of (p,p)-substitutions by the finiteness of the set of subsequences of the above type.

Reviewer: P.Liardet (Marseille)

MSC:

11B85	Automata sequences
05B45	Combinatorial aspects of tessellation and tiling problems
68Q80	Cellular automata (computational aspects)

Keywords:

aperiodic tiling; 2-dimensional finite automata; regular Robinson tiling; fixed point; substitution

Citations:

Zbl 0601.05001; Zbl 0653.10049

References:

[1]	Allouche, J.-P., Salon, O., Quasiperiodic tilings and finite automata. preprint
[2]	Berger, M.,, Géométrie, Cédic/FernandNathan, Tome I (1977), 33-43. · Zbl 0382.51011
[3]	Blanchard, A. et Mendes France, M., Symétrie et transcendance, Bull. Sci. Math.106 (1982), 325-335. · Zbl 0492.10027
[4]	Christol, G., Kamae, T., Mendes France, M. et Rauzy, G., Suites algébriques, automates et substitutions, Bull. Soc. Math. France108 (1980), 401-419. · Zbl 0472.10035
[5]	Cobham, A., Uniform tag sequences, Mathem. Syst. Theory6 (1972), 164-192. · Zbl 0253.02029
[6]	Grunbaum, B. et Shephard, G.C., Tilings and patterns, W.H. Freeman and CompanyNew-York (1987). · Zbl 0601.05001
[7]	Penrose, R., The role of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl.10 (1974), 266-271.
[8]	Schechtman, D., Quasiperiodic crystals - Experimental evidence, Journal de Physique, Colloque C3 Supplément au n° 7, Tome 47. juillet 1986
[9]	Schechtman, D., Blech, I.A., Gratias, D. et Cahn, J.W., Physical Review Letters, n° 2053 (1984), 1951-1953.
[10]	Salon, O., Suites automatiques à multi-indices” Séminaire de Théorie des Nombres de Bordeaux (1986- 1987). exposé n° 4. · Zbl 0653.10049
[11]	Voderberg, H., Zur Zerlegung eines ebenen Bereiches in kongruente Bereiche in Form einer Spirale, Jber. dtsch. Math. Ver., 46 (1936), 229-231. et aussi 46, (1937), 159-160. · JFM 63.1183.05