PARQUETS OF RHOMBUSES FORMING A REGULAR FIVE-POINTED STAR AND RHOMBUSES SUPPLEMENTING IT TO A REGULAR DECAGON

Abstract

Despite the fact that it is impossible to fill a plane with regular pentagons without overlaps and gaps, the problem of tiling a plane with regular pentagons continues to occupy the minds of not only geometers, but also designers who create new types of ornaments. This is due to the fact that a regular pentagon is a figure that has the highest aesthetic qualities of all regular polygons. Therefore, the working out of ways to fill the plane with regular pentagons is an actual challenge for both geometers and designers.
 One of the solutions to this problem can be tiling the plane with regular pentagons, in which the inevitable overlaps and gaps do not destroy the composition of the ornament, but, on the contrary, become its intrinsic part. For example, if a regular pentagon is transformed into a regular five-pointed star made up of five rhombuses with angles 72° and 108° and supplemented with five rhombuses with angles 36° and 144° to form a regular decagon, then overlapping will be rhombuses with angles 36° and 144°, and by gaps - rhombuses with angles 72° and 108°.
 The purpose of the study is to work out new, previously unexplored ways of tiling a plane with rhombuses that make up a regular five-pointed star, and rhombuses that supplement it to a regular decagon. Moreover, the results obtained by us do not repeat the work carried out by other researchers, for example by Roger Penrose, but supplement them with new types of ornament, including elements of a regular pentagon.
 The classical variant of the plane tiling with rhombuses with angles 72° and 108° and rhombuses with angles 36° and 144° is considered and its connection with the mosaics designed by Roger Penrose is shown. Consequently, the parquet designed by Roger Penrose is not the only parquet that can be composed of rhombuses forming a regular five-pointed star and rhombuses supplementing it to a regular decagon. It is shown that there are only six ways to compose a regular decagon from rhombuses with angles 72° and 108° and rhombuses with angles 36° and 144°. Four previously unknown variants of the parquet that compose of rhombuses forming a regular five-pointed star and rhombuses that supplement it to a regular decagon are proposed. It is supposed that our further research will be directed towards the design of parquet composed of rhombuses that form five-pointed and ten-pointed stars.