RELATIONSHIP OF THE SYMMETRY GROUP OF THE ORNAMENT ON THE SKETCH OF M. C. ESHER’S SKETCH ‘SEAHORSES’ WITH THE MOTIONS OF THE PLANE DESCRIBING THE CONSTRUCTION OF ITS FIGURED TILE

Abstract

The first thing that catches your eye when you look at the of M. C. Escher’s sketch ‘Seahorses’ is its special feature, which consists in the fact that if you take any zoomorphic form for the original, then in order to get its copies, you need to complete the central symmetries of the original and its translations, with translations being carried out in six directions. We assume that the same central symmetries and translations are related to individual parts of the zoomorphic contour, which completely filling the plane. Our assumption is based on the fact that the connection between the group of symmetry of the ornament and the group of movements of the plane, describing the construction of figured tiles that fill the plane without overlaps and gaps, was discovered by us both in the M. C. Escher’s print ‘Horsemen’, and in his lithograph “Reptiles”. Thus, our purpose is to classify the ornaments according to the crystallographic symmetry groups on the plane, discovered by the Russian scientist E. S. Fyodorov, and to connect the symmetry groups of the ornaments with the groups of plane movements describing the construction of their repeating figures. A rule is proposed for constructing figured tiles that stylize images of plants and animals and fill the plane without overlaps and gaps by translations and rotations of its repetitions, in particular, figured tile that generalize the image of a zoomorphic form on the M. C. Escher’s sketch “Seahorses”. The proposed rule was applied to compose an ornament stylizing the M. C. Escher’s sketch “Seahorses”. It is shown that this ornament has set of centers of symmetry and six translation vectors. The connection between the symmetry group of the ornament and the movements of the plane, leading to the formation of its figured tiles was revealed. It is shown that if any group of plane transformations corresponds to any figure, then the ornament obtained by translations and rotations of its repetitions will correspond to the same group of transformations of the plane. It is assumed that the subject of further research will be the application of one of the crystallographic symmetry groups of E. S. Fyodorov to the construction of a figured tile stylizing a zoomorphic shape on one of M. C. Escher’s prints.