Transposing an Octogonal Cupola of Second Sort Into a Bionic Polyspherical Form

Abstract

In the paper there is presented a method of transposing a polyhedral structure, composed of two regular polygons: octagon and hexadecagon as the bases, and the envelope formed of equilateral triangular strip, which is named octagonal concave cupola of second sort, into a bionic form of curved surfaces. Cupola is term taken from Johnson’s solids (J3, J4, J5), but in extended sense, for the lack of subsistent name, though its geometry is closest to the geometry of those Johnson’s solids. Geometry of octagonal concave cupola of second sort has subserved as a model for congenial geometrical form, which includes spherical segments instead of equilateral triangles and base polygons, by ablation of which we obtain a close spatial form. The methods of Constructive geometry are applied in the paper. Since the coordinates of the points and the parameters of octagonal concave cupola of second sort are known, they have been used for defining the spherical radii, which segments form a spatial structure, whereat the new polispherical shape, with its new qualities would be obtained, instead of polyhedral. Four different spherical segments, would substitute a spatial hexahedral layout, consisted of six equilateral triangles, and the fifth would link them into a convergent ensemble, while the base polygon would be replaced with new spherical polygonal calotte. This form preserves altitudes, distances and symmetries of original polyhedron, by which transposing it is obtained. Although the majority of geometrical parameters are invariant to the original geometrical matrix, the new geometrical configurations are found, similar to the ones from nature, as the result of such a substitution of surfaces. They bring additional aesthetic, applicable and static properties, interesting for further researches. This shows that in the future implementation in constructive systems, the plane panels of equilateral triangles can be replaced by spherical shells. There would exist two variants of such a transposing of polyhedral surface to a polyspherical, whereat in the first case the spherical shells would be concave, and in second they would be convex, which refer to two variants of forming the envelope of concave cupola of second sort. The structure derived in this manner, can find its application in architecture, civil engineering, landscape architecture and design.