Pentakis dodecahedron
The Pentakisdodekaeder is a convex polyhedron composed of 60 equilateral triangles and one of the the Catalan bodies. It is dual to the truncated icosahedron and has 32 vertices and 90 edges. The name is derived from the Greek words πεντάκις ( pentakis, five times) and δωδεκάεδρον ( dodecahedron, Zwölfflächner ).
Formation
The basic body is virtually the dodecahedron with side length, is placed on the 12 boundary faces each have a pyramid with a pentagonal base and the edge length. A Pentakisdodekaeder arises from this construction if and only if the following condition is met:
- For the aforementioned minimum value of the placed pyramids have depth 0, so that only the dodecahedron with an edge length remains.
- The special Pentakisdodekaeder with the same dihedral angles formed when is.
- Assuming the above-mentioned maximum value, the Pentakisdodekaeder degenerates into a Rhombentriakontaeder with the edge length.
- Exceeds the maximum value, the polyhedron is not convex and finally degenerate for the Dodekaederstern.