Concave Regular Faced Cupolae of Second Sort

Abstract

Cupola is a polyhedron which is consisted of two regular polygons: n-gon and 2n-gon in parallel planes, connected by alternating sequence of squares and equilateral triangles, i.e. Johnson’s solids J3, J4 and J5. However, it is possible to form a polyhedron with the analogously chosen regular n-gon and 2n-gon in parallel planes, in that manner to have a base polygon with n ≥ 6, and whose envelope would be formed of series of equilateral triangles, creating a concave polyhedron, similar to the Johnson’s cupolae, and furthermore to the Johnson’s rotundae. In a lack of an adequate name, we deemed to engender the meaning of the term cupola to a concave polyhedron that includes regular faces polygons in its geometry, whereat two of them are n-gon and 2n-gon in parallel planes. The method of forming such a cupola is based on wrinkling the net of equilateral triangles, which produce a twofold strip, by folding of which we obtain a deltahedral envelope surface. Such manner of creating a polyhedron, gives a solution to a problem of creating a regular faced solid which includes even ‘unconstructable’ polygons, as heptagon and nonagon. In this paper, there are described concave regular faced cupolae originated by wrinkling the envelope net consisted of two rows [(2x3+1)n] of equilateral triangles; therefore they are named: the cupolae of second sort. The cupolae originated by using envelope net made of tree rows of equilateral triangles would thus be named: concave regular faced cupolae of third sort, and so on. Concave regular faced cupolae of second sort can have the starting bases from n=4 to n=10. Hendecagon can not be used for the start base polygon (n-gon), because the distance from its double sided counterpart polygon in the parallel plane, would exceed the double value of equilateral triangle’s altitude, the width of the envelope strip. The main parameters of these solids can be found by determining the trajectory of the envelope strip’s elementary cell’s vertices, consisted of six equilateral triangles, which will move around the edge of 2n-gon, behaving as mechanism. The shape of trajectory would show the curve of higher order, therefore there would be two ways to assemble the envelope, so there would exist two possible altitudes of such obtained polyhedrons. There are fourteen solids that would be classified as the concave regular faced cupolae of second sort.