Three-Dimensional Rosettes Based on the Geometry of Concave Deltahedral Surfaces

Abstract

In this paper concave deltahedral surfaces are applied to link the two concepts of geometric rosette design – the polar distribution of the unit element (circular arc) around the center of the contour circle and the rosettes obtained by means of regular polygons. Forming composite polyhedral structures based on the geometry of concave deltahedral surfaces over a n-sided polygonal base, we have demonstrated one possible method of geometrical generation of three-dimensional rosettes. The concave polyhedral surfaces are lateral surfaces of the concave polyhedrons of the second, fourth and higher sorts, consisting of series of equilateral triangles, grouped into spatial pentahedrons and hexahedrons. Positioned polarly around the central axis of the regular polygon in the polyhedron’s basis and linked by triangles, the spatial pentahedrons and hexahedrons form the deltahedral surface. The sort of the concave polyhedron is determined by the number of equilateral triangle rows in thus obtained polyhedron’s net. In this study, composite polyhedral structures whose surface areas form the three-dimensional rosette are obtained through the combination of concave cupolae of the second sort (CC-II), concave cupolae of the fourth sort (CC-IV), concave antiprisms of the second sort (CA-II) and concave pyramids (CP). By means of elongation, gyro-elongation and augmentation of the listed concave polyhedrons it was possible to generate complex polyhedral structures, which can be used to create three-dimensional rosettes. The parameters of the solids were determined constructively by geometric methods and analytical methods which useiterative numericalprocedures.