ORGANENT SYMMETRY GROUP ON M. C ESHER’S SKETCH 'REPTILES' AND PLANE MOVEMENTS DESCRIBING THE FORMATION OF ITS FIGURED TILES

Abstract

Methods for constructing figured tiles stylizing images of animals and plants and completely filling the plane are not currently the subject of scientific research. This is due to the fact that the authors of many scientific papers consider M. C. Escher’s prints as a mosaic composed of polygons with a repeating pattern applied to them. Therefore, they look for fragments in them that fit into rhombuses, squares, regular triangles or regular hexagons, and with their help make a mosaic. But we went the other way − by opening the laws of symmetry, which allow us to build a flat figure stylizing the images of plants and animals and filling the plane without overlays and gaps. Thus, the purpose of the article is to establish a rule for constructing a figure stylizing images of animals and plants and filling the plane without overlays and gaps with translations and rotations of its repetitions. A rule for constructing figured tiles stylizing images of plants and animals and filling the plane without overlays and gaps with parallel transfers and rotations of its repetitions, in particular, figured tiles in the form of zoomorphic shapes on M. C. Escher’s sketches 'Reptiles' and 'Butterflies' is proposed. The proposed rule was applied to composition ornaments stylizing the M. C. Escher’s sketches 'Reptiles' and 'Butterflies'. It is shown that these ornaments have set of symmetry axes of the 3rd order, set of symmetry axes of the 6th order and six translation vectors. The connection between the movements of the plane leading to the formation of a figured tile and the symmetry group of the ornament obtained on its basis is revealed. It was established that the symmetry of the ornament and its repetitive figure are described by 6th-order rotation groups and groups of translations of the rotation axes. Therefore, if any group of transformations of the plane 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. The ornament 'Composition No. 1' which is not described in the literature on the history and theory of ornament was developed. 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 sketches.