Pentakis dodecahedron

In geometry, a pentakis dodecahedron or kisdodecahedron is a polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron.

Specifically, the term typically refers to a particular Catalan solid, namely the dual of a truncated icosahedron.

and cyclic permutations of these coordinates are the vertices of a regular icosahedron.

Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points

Multiplying all coordinates of the icosahedron by a factor of

The length of its long edges equals

Its faces are acute isosceles triangles with one angle of

The length ratio between the long and short edges of these triangles equals

The pentakis dodecahedron in a model of buckminsterfullerene: each (spherical) surface segment represents a carbon atom, and if all are replaced with planar faces, a pentakis dodecahedron is produced.

Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.

These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron.

The pentakis dodecahedron has three symmetry positions, two on vertices, and one on a midedge: A concave pentakis dodecahedron replaces the pentagonal faces of a dodecahedron with inverted pyramids.

The faces of a regular dodecahedron may be replaced (or augmented with) any regular pentagonal pyramid to produce what is in general referred to as an elevated dodecahedron.

For example, if pentagonal pyramids with equilateral triangles are used, the result is a non-convex deltahedron.