Tiling the Lateral Surface of the Concave Cupolae of the Second Sort

Abstract

As architectural structures, concave cupolae of the second sort (CC II) are suitable for prefabrication, thanks to the uniformity of their elementsregular polygons. This paper discusses ways to subdivide the lateral polyhedral triangle (LPT) of CC II into new regular polygons, triangles and hexagons, which can then tile the whole deltahedral lateral surface, forming patterns applicable in architectural design. The number of different arrangements of triangles and hexagons depends on the number of unit hexagonal cells (tiles) that can be placed within the triangular grid of the equilateral triangle. Considering solutions that involve fragments of k-uniform Euclidean tilings, we explored the ones that arise when the edges of the LPT are subdivided into 3b9 (bN) segments. Without attempting to analyse all the possible cases, we focused on the ones with the D-3 symmetry, in order to propose the simplest solutions for the assembly, so it will not matter if the face of the CC II is rotated or flipped. This results in 30 different solutions, excluding the ones consisting of triangles alone. The solutions found are applicable to all the representatives of CC II. We chose several examples of these results for an architectural design proposal.