Abstract
The paper presents the results of the study of the new set of polihedra, the Concave pyramids of the fourth sort, the construction procedures for generating them and their possible application. Correspondingly to the method of generating the Concave cupolae of fourth sort, the Concave pyramids of fourth sort have the similar logic of origination, and their counterpart in regular faced convex pyramids. They are characterised by the polygonal base, deltahedral surface net, obtained by folding the planar net of unilateral triangles, the polar distribution of the unit space cells with common apex - the top of the Concave pyramid. Polihedral surface of the planar net of Concave pyramids is produced by polar distribution of unit cells, consisting of a spatial sexagon and spatial pentagon - six, or five, unilateral triangles grouped around the common vertex. In the deltahedral surface, the two neighbouring unit cells are joined by means of a unilateral triangle in the zone of the polygonal base and a spatial quadrangle with which they share common sides. The criterion of face regularity is respected, as well as the criterion of multiple axial symmetry. The sort of the Concave pyramids is determined by the number of equilateral triangle rows in thus obtained polyhedron's net. The parameters of the solids were determined constructively by geometric methods.
Highlights
Polyhedral surfaces are widely used in architecture, engineering, and industrial design to form constructive details, façade covers, interior details and spatial struc– tures (Fig. 1)
This study has proven the existence of the Concave pyramids of the fourth sort, type B, and it has shown that the generation of concave pyramids follows the geometrical principles for generating concave cupolae
The unit cell consists of the spatial hexagon ABCDEF (six equilateral triangles grouped around the common vertex O1 and the spatial pentagon EDGHJ, with the common edge ED (Fig. 4)
Summary
Polyhedral surfaces are widely used in architecture, engineering, and industrial design to form constructive details, façade covers, interior details and spatial struc– tures (Fig. 1). The analysis of the forms reveals the main techniques for defining, constructing and implementing the forms suggested for use in designing spatial structures [5] Due to their characteristics, the structures generated by means of triangles stand out in the large set of polyhedral structures [6], and deltahedral surface net is the main characteristic of Concave pyramids. The results of the exploration of the application of these polyhedral structures in architecture and engineering were presented in [16, 17] These polyhedra are characterized by their envelope - developmental deltahedral surface formed above the regular polygonal base. All the unit cells in formed deltahedral net of Concave pyramids of the second sort have a common vertex located on the rotation axis, ort– hogonal to the plane of the polygon’s base. The net CP II-10B and CP II-9mA protrudes the plane of the polygon’s base so that the polyhedron itself cannot be VOL. 49, No 4, 2021 ▪ 1048 formed, but it is possible to generate the net, which has been used [15] to form composite polyhedral structures [21]
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