Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n. This expands on our earlier works (Fukuda et al 2006, 2008)

Escher: mathématicien malgré lui

Une imagevaut rnille mots. .. c’est ainsi que I’oeuvre graphique de I’artiste hollandais M.C. Escher (1898-1972) a gag ne I’intérêt des mathématiciens par son expression visuelle éclairée de concepts abstraits. Toutefois, seulement une fraction de I’habileté mathématique d’Escher transparaît dans son oeuvre achevee. Escher niait violemment toute compréhension des mathématiques, malgré le fait qu’il reconnaissait avoir plus d’affinités avec les mathématiciens qu’avec les autres artistes. Sans doute que son idée des mathématiques en était une de manipulation de symboles et de creation de formu les, ou de production ou de déchiffrement de textes abracadabrants. Toutefois, Escher était un mathématicien, un véritable chercheur, qui explora plusieurs questions mathématiques liées aux pavages coloriés du plan. Il se créa lui-même un ensemble de catégories, il inventa une notation de classification et, tout en faisant fi du systéme et des restrictions “reconnus”. et imposés par les mathématiciens et les cristallographes, il explora plus en profondeur que tous ces “professionnels” les questions soulevées par les pavag es périodiques du plan. Ses cahiers de notes de 1941 -1942 révèlent I’ampleur des recherches de ce mathématicien d’arrière-boutique.A picture is worth a thousand words.. , and so the graphic work of the Dutch artist, M.C. Escher (1898-1972), has been captured by mathematicians for its clever visual expression of abstract concepts. However, Escher’sfinished graphics show onlyafraction of his mathematical ability. Escher hotly denied understanding any mathematics, although he acknowledged more affinity with mathematicians than other artists. No doubt his idea of mathematics was manipulating symbols and creating formulas, or producing or deciphering “hocus pocus” texts. However, Escher was a mathematician, a true researcher, exploring many mathematical questions arising from colored tilings of the plane. He created for himself a set of categories, invented a notation of classification, and ignoring the “accepted” system and restrictions imposed by mathematicians and crystallographers, explored more deeply than any of these ‘professionals’, questions of colored periodic tilings of the plane. His notebooks of 1941-1942 reveal the extent of the explorations of this closet mathematician.Peer Reviewe

Applying Burnside’s lemma to a one-dimensional

Our point of departure is the paper [7] in which a problem of M. C. Escher is solved using methods of contemporary combinatorics, in particular, Burnside’s lemma. Escher originally determined (by laborously examining multitudes of sketches) how many different patterns would result by repeatedly translating a 2 × 2 square having its four unit squares filled with copies of an asymmetric motif in any of four aspects. In this note we simplify the problem from two dimensions to one dimension but at the same time we generalize it from the case in which a 2 × 2 block stamps out a repeating planar pattern to the case in which a 1 × n block stamps out a repeating strip pattern. The 1 × 2 case Suppose we are tiling a strip by a single rectangle containing an asymmetric motif, say, a motif taken from South African beadwork which is a rectangle divided by a diagonal into two triangles, one solid red, and the other yellow with a green stripe. The original motif has three additional aspects, namely the motif rotated by 180 ◦ , reflected in a vertical line and in a horizontal line. We note the motif by b and its other aspects as follows: b = p = q = d = since the letters p, q and d are the corresponding aspects of the letter b under these transformations. This notation was first introduced in [9] to encode the symmetry groups of strip patterns. Assume that we may select any two aspects from {b, q} (with repetition allowed) to form a signature for a 1 × 2 block of two rectangle