Disdyakis triacontahedron
The Hexakisikosaeder ( from Greek ἑξάκις hexakis " six times " and icosahedron " icosahedron " ) or Disdyakistriakontaeder ( δίς dis " twice," δυάκις dyakis " twice " and Triakontaeder " Dreißigflächner ") is a convex polyhedron composed of 120 irregular triangles and the bodies the Catalan counts. It is dual to the Ikosidodekaederstumpf and has 62 vertices and 180 edges.
- 2.1 Regular
- 2.2 Rhombisch 2.2.1 General
- 2.2.2 Specially
Formation
Rhombentriakontaeder as a basis
Will the 30 boundary surfaces of a Rhombentriakontaeders ( edge length ) pyramids with edge lengths and placed, creating a Hexakisikosaeder, if the following condition is satisfied:
- For the aforementioned minimum value of the placed pyramids have depth 0, so that only the Rhombentriakontaeder remains with the edge length.
- The special Hexakisikosaeder with the same dihedral angles at the edges and arises when is.
- If b the maximum value previously mentioned, the Hexakisikosaeder degenerates into a Deltoidalhexakontaeder with the edge lengths and.
- Exceeds the maximum value, the polyhedron is not convex.
Ikosidodekaederstumpf as a basis
By connecting the midpoints of three edges that meet at each corner of the room of the truncated icosidodecahedron, creating a triangle whose perimeter inscribed circle of the triangle, the boundary surface of the Hexakisikosaeders is, at the same time. In this particular type, all dihedral angles equal ( ≈ 165 ° ), and there is a uniform sphere radius edges.
Is the edge length of the Ikosidodekaederstumpfs, the resulting side lengths of the triangle are determined by
Formulas
In the following denote the longest edge of each Hexakisikosaeders ().
Regular
The basis is the truncated icosahedron ( dual Archimedean bodies).
Rhomboidal
The basis is the Rhombentriakontaeder ( edge length ).