Pentakis dodecahedron  

(Click here for rotating model)  
Type  Catalan solid 
Coxeter diagram  
Conway notation  kD 
Face type  V5.6.6 isosceles triangle 
Faces  60 
Edges  90 
Vertices  32 
Vertices by type  20{6}+12{5} 
Symmetry group  I_{h}, H_{3}, [5,3], (*532) 
Rotation group  I, [5,3]^{+}, (532) 
Dihedral angle  156°43′07″ arccos(−80 + 9√5/109) 
Properties  convex, facetransitive 
Truncated icosahedron (dual polyhedron) 
Net 
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.
Let be the golden ratio. The 12 points given by 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 together with the points and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of gives a slightly smaller icosahedron. The 12 vertices of this icosahedron, together with the vertices of the dodecahedron, are the vertices of a pentakis dodecahedron centered at the origin. The length of its long edges equals . Its faces are acute isosceles triangles with one angle of and two 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.
The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses, such as Adenoassociated virus. 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:
Projective symmetry 
[2]  [6]  [10] 

Image  
Dual image 
A concave pentakis dodecahedron replaces the pentagonal faces of a dodecahedron with inverted pyramids.
Related polyhedra[edit]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 nonconvex deltahedron. Any such elevated dodecahedron has the same combinatorial structure as a pentakis dodecahedron, i.e., the same Schlegel diagram.
See also[edit]Cultural references[edit]
References[edit]
External links[edit]
