Construction of the tangent pentagon on the beveled hexahedron
Left and right handed version of a Pentagonikositetrahedron (cardboard models)
By connecting the centers of five edges that meet in each corner of the beveled hexahedron, a chordal pentagon is created , the circumference of which is at the same time an inscribed circle of the tangent pentagon , the boundary surface of the pentagonikosite tetrahedron. In this special type, all face angles are the same (≈ 136 °) and there is a uniform edge sphere radius .
In the following, the term denotes the cosine of the smaller central angle in the aforementioned chordal pentagon.
Let be the edge length of the beveled hexahedron, then the resulting side lengths of the tangent pentagon are given by
↑ t is the only real solution of the cubic equation 4t 3 + 4t 2 - 1 = 0. If the number 1 is added to twice the value of t , the Tribonacci constant is obtained, which is the limit of the ratio (= 1.83928675521416 ...) represents two consecutive numbers of this sequence.
↑ With a > the longer of the two sides of the Pentagonikositetraeders be designated.
↑ These formulas only apply to the case b = a: (1 + t) or equivalent to a = b (1 + t)
↑ a b c d e f g h i This formula also applies to the pentagon hexacontahedron and the pentagon dodecahedron , provided that the corresponding values are used for b (short side length), n (number of boundary surfaces) and t (cosine of the smaller central angle) and furthermore note that O = n · A and V = 1/3 · O · ρ.