PLANE’S MOTIONS DESCRIBING THE CONSTRUCTION OF A FIGURED TILE OF AN ORNAMENT ON M. C. ESCHER’S LITHOGRAPH ‘REPTILES’, AND ITS SYMMETRY GROUP

Abstract

In the foreign press you can find numerous articles whose authors attempt to answer the question: ‘How did M. C. Escher create such famous prints as ‘Horsemen’, ‘Day and Night’, ‘Sky and Water’ or ‘Reptiles’? It is noteworthy that all their attempts boiled down to the fact that they placed rhombuses, squares, regular triangles or regular hexagons on the print, cut out a repetitive fragment from it with their help and completely filled the plane with it.

In our opinion, the method of regularly dividing a plane, consisting in the fact that a repetitive pattern fits into some regular polygon, not only does not lead to an explanation of how M. C. Escher worked, but also leads away from it in the opposite direction. Therefore, before looking in the prints of M. C. Escher for fragments that fit into rhombuses, squares, regular triangles, and so on, it is necessary to understand how M. C. Escher created figures that, with the help of its translations, rotations or mirror reflections cover the plane without overlaps and gaps.

Thus, our purpose is to classify ornaments according to crystallographic symmetry groups on the plane, discovered by the Russian scientist E. S. Fedorov, and to connect the symmetry groups of ornaments with the groups of plane’s motions that describe the construction of their repetitive figures.

A rule has been proposed for constructing figured tiles that stylize images of plants and animals and fills the plane without overlaps and gaps during translations and rotations of its repetitions, in particular a figured tile that generalizes the zoomorphic form in M. C. Escher’s lithograph ‘Reptiles’. The construction of a figured tile that generalizes the zoomorphic form in M. C. Escher’s lithograph ‘Reptiles’ is considered. The proposed rule was applied to compose an ornament stylizing M. C. Escher’s lithograph ‘Reptiles’. It is shown that this ornament has set of axes of 3rd order symmetry and six translation axes. A connection has been revealed between the symmetry group of the ornament and the motions of the plane leading to the formation of its figured tiles. It is assumed that our next work will be devoted to the application of one of the E. S. Fedorov’s crystallographic symmetry groups to the construction of a figured tile that stylizes a zoomorphic form in one of the graphic works of M. C. Escher.

Keywords: tessellation of a plane, figured tiles in the form of animals and plants, stylization of M. C. Escher’s prints.