PARQUETS FROM REGULAR FIVE-POINTED AND TEN-POINTED STARS

Obviously, regular pentagons cannot fill a plane without overlaps and gaps. However, the plane can be filled without overlaps and gaps with polygons whose angles are multiples of 36°. For example, if a regular pentagon is changed into a regular five-pointed star, composed of five rhombuses with angles of 72° and 108°, and supplemented with five rhombuses with angles of 36° and 144° to form a regular decagon, then the resulting rhombuses can fill the plane without overlaps and gaps. It is assumed that, in addition to the presented method of tiling the plane with the rhombuses pointed out above, there are other ways of tiling the plane without overlaps and gaps with polygons whose angles are multiples of 36°. This explains why the challenge of tiling a plane with regular pentagons attracts the attention of not only geometers, but also designers who create new types of ornaments. Moreover, the regular pentagon among other types of regular polygons has the highest aesthetic qualities, and parquets made up of polygons, the angles of which are multiples of 36°, surpass other types of parquet in beauty and perfection. Therefore, the working out of methods for tiling a plane without overlaps and gaps with polygons, the angles of which are multiples of 36°, is an actual challenge for both geometers and designers creating new types of ornaments. For the first time, two variants of the parquet, composed of rhombuses forming five-pointed and ten-pointed stars, were worked out. If in the first variant the center of the parquet is a five-pointed star, then in the second variant it is a ten-pointed star. Moreover, if in the first variant the parquet does not have a single plane of symmetry, then in the second variant the parquet has twenty planes of symmetry. Another difference is that if in the first variant the parquet has a rotational symmetry with a 5th order symmetry axis, then in the second variant it has a rotational symmetry with a 10th order symmetry axis. Common to both variants of parquet is that they belong to non-periodic parquets, that is, they are new, previously unexplored types of Penrose mosaics. It is assumed that our further research will be directed to the invention of parquet, which has neither translation nor rotation symmetry, and at the same time maintains order in the arrangement of tiles.