Ce texte a deux objectifs :
– D’une part, donner un survol sans démonstration de la théorie classique de Bieberbach des pavages euclidiens périodiques ainsi que de ses analogues hyperboliques, affines et projectifs.
– D’autre part, exhiber quelques exemples concrets de pavages périodiques et apériodiques en dimension dans le contexte euclidien, mais aussi dans les contextes hyperboliques, affines et projectifs. En particulier, nous construisons des pavages affines du plan à l’aide d’heptagones affinement réguliers comme dans la fleur ci-dessous.
We survey without proofs Bieberbach’s theory for euclidean periodic tilings and its hyperbolic, affine and projective analogs.
We also describe explicit examples of periodic and aperiodic 2-dimensional tilings in the euclidean setting as well as in the hyperbolic, the affine and the projective setting.
For instance, we construct aperiodic affine tilings of the plane with affinely regular heptagons as in the flower below.
@incollection{XUPS_2001____1_0,
author = {Yves Benoist},
title = {Pavages du plan},
booktitle = {Pavages},
series = {Journ\'ees math\'ematiques X-UPS},
pages = {1--54},
publisher = {Les \'Editions de l{\textquoteright}\'Ecole polytechnique},
year = {2001},
doi = {10.5802/xups.2001-01},
language = {fr},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/xups.2001-01/}
}
Yves Benoist. Pavages du plan. Journées mathématiques X-UPS, Pavages (2001), pp. 1-54. doi : 10.5802/xups.2001-01. https://proceedings.centre-mersenne.org/articles/10.5802/xups.2001-01/
[1] Properly discontinuous groups of affine transformations with orthogonal linear part, C. R. Acad. Sci. Paris Sér. I Math., Volume 324 (1997) no. 3, pp. 253-258 | DOI | MR | Zbl
[2] Une nilvariété non affine, C. R. Acad. Sci. Paris Sér. I Math., Volume 315 (1992) no. 9, pp. 983-986 | MR | Zbl
[3] Tores affines, Crystallographic groups and their generalizations (Kortrijk, 1999) (Contemp. Math.), Volume 262, American Mathematical Society, Providence, RI, 2000, pp. 1-37 | DOI | MR | Zbl
[4] Convexes divisibles, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 5, pp. 387-390 | DOI | MR | Zbl
[5] Variétés localement affines, Séminaire de topologie et géométrie différentielle, Volume 2, Secrétariat mathématique, 1958-1960 (Exp. no. 7, 35 p.) | Zbl
[6] The undecidability of the domino problem, Mem. Amer. Math. Soc., Volume 66 (1966), p. 72 | MR | Zbl
[7] Affine structures on nilmanifolds, Internat. J. Math., Volume 7 (1996) no. 5, pp. 599-616 | DOI | MR | Zbl
[8] Un survol de la théorie des variétés affines, Séminaire de théorie spectrale et géométrie, Volume 6, Institut Fourier, Grenoble, 1987-1988, pp. 9-22 | DOI | Zbl
[9] Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc., Volume 118 (1993) no. 2, pp. 657-661 | DOI | MR | Zbl
[10] On three-dimensional space groups, Beitr. Algebra Geom., Volume 42 (2001) no. 2, pp. 475-507 | MR | Zbl
[11] Polycyclic-by-finite groups admit a bounded-degree polynomial structure, Invent. Math., Volume 129 (1997) no. 1, pp. 121-140 | DOI | MR | Zbl
[12] Le miroir magique de M.C. Escher, Le Chêne, Paris, 1976
[13] Tilings and patterns, W. H. Freeman and Company, New York, 1987
[14] International tables for crystallography. Vol. A (Theo Hahn, ed.), D. Reidel Publishing Co., Dordrecht, 1983 (Space-group symmetry) | Zbl
[15] Aperiodic tilings of the hyperbolic plane by convex polygons, Israel J. Math., Volume 107 (1998), pp. 319-325 | DOI | MR | Zbl
[16] The affine structures on the real two-torus. I, Osaka Math. J., Volume 11 (1974), pp. 181-210 http://projecteuclid.org/euclid.ojm/1200694718 | MR | Zbl
[17] Pentaplexity. A class of non-periodic tilings of the plane, Math. Intell., Volume 2 (1979), pp. 32-37 | DOI | Zbl
[18] Shadows of the mind. A search for the missing science of consciousness, Oxford University Press, Oxford, 1995
[19] Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995
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