The pentakis dodecahedron is a convex polyhedron , which is composed of 60 isosceles triangles and belongs to the Catalan solids . It is dual to the truncated icosahedron and has 32 vertices and 90 edges. The name is from the Greek words πεντάκις ( pentakis , five times) and δωδεκάεδρον ( dodekaedron , dodecahedron) together.
The basic body is the dodecahedron with side length , on the 12 boundary surfaces of which a pyramid with a pentagonal base and the side length is placed. A pentakis dodecahedron arises from this construction if and only if the following condition is met:
For the aforementioned minimum value of , the pyramids on top have the height 0, so that only the dodecahedron with the edge length remains.
The special pentakis dodecahedron with equal face angles is created when is.
Takes the above maximum value, the degenerate pentakis dodecahedron to a rhombic triacontahedron with an edge length .
If the maximum value is exceeded , the polyhedron is no longer convex and finally degenerates into a dodecahedron star .
Formulas
General
Sizes of a pentakis dodecahedron with edge lengths a , b