Pentagon-based radial tiling with triangles and rectangles and its spatial interpretation

Abstract

The paper considers a type of radial pentagon-based tiling consisted of two shapes: triangle and rectangle. The obtained solution has a spatial interpretation in a 3D arrangement of equilateral triangles and squares dictated by the particular array of concave cupolae of the second sort, minor type (CC-II-5.m). These cupolae are arranged so that their decagonal bases partly overlap, making a pentagonal pattern (similar to one of the Penrose tiling). Covering the folds between the faces of such a polyhedral structure with polygons,we use exactly equilateral triangles and squares, thanks to the trigonometric properties of CC-II-5.m. Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles will be projected into congruent isosceles triangles corresponding to those obtained by the radial section of a regular pentagon in 5 parts. The squares are projected into rectangles whose ratio a:b = 1:φ/√(1+φ2), where φ is the golden ratio. These triangles and rectangles form a radial tiling consisting of 5 sectors of the plane, where the established tiles’ patterns are repeated locally periodically. However, with 5-fold rotation of the pattern, the tilling itself is non-periodic. The various tiling solution that can be obtained in this way may serve as inspiration for the geometric design, especially interesting in architecture and applied arts, e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the like