ПАРКЕТИ НА ОСНОВІ ФРАКТАЛЬНОГО РОЗШИРЕННЯ ПРАВИЛЬНОГО П’ЯТИКУТНИКА

Анотація

Consider the development of a dodecahedron - a regular polyhedron, the surface of which consists of twelve regular pentagons. Let’s represent the deployment of the dodecahedron in the form of two groups of polygons, consisting of 6 regular pentagons, one of which is located in the center of the group, while the others are adjacent to its sides. However, the most remarkable thing is that each group of 6 regular pentagons forms a figure that inscribes into a regular pentagon. It follows that each group of 6 regular pentagons can be considered as the result of a fractal expansion of the regular pentagon located in its center. Moreover, if we attach the same figure to each side of the figure into which a group of 6 regular pentagons inscribes, we will get another figure similar to the original regular pentagon. Repeat the previous steps infinitely many times and get a figure similar to a regular pentagon and completely filling the plane.

For the first time, a fractal extension of a regular pentagon was applied to the tiling of a plane, the gaps of which are eliminated by figures that are combinations of a rhombus with angles of 36° and 144°, a rhombus with angles of 72° and 108°, and a regular five-pointed star with an angle of 36°. It is shown that the variety of polygons used to eliminate gaps provides parquets in the form of a fractal extension of a regular pentagon with a higher artistic value compared to parquets constructed by means of a fractal extension of a square. It is presented a parquet variant, in which gaps between regular pentagons, formed after the first iteration, are filled with rhombuses with angles of 36° and 144°. It is shown that the figures that fill the gaps that appear after each iteration are similar to a rhombus with angles of 36° and 144°, and the parquet obtained after the fourth iteration and truncated by a regular pentagon is similar to a figure consisting of four rhombuses with angles of 72° and 108° and one regular five-pointed star with an angle of 36° and inscribed in a regular pentagon. Additionally, parquet variants are presented, in which the gaps between regular pentagons, formed after the first iteration, are filled with the corners of regular five-pointed stars with an angle of 36° at the top.