Pavages du plan

Benoist, Yves

Journées mathématiques X-UPS, Pavages (2001), pp. 1-54.

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 $2$ 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.

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