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.

Cartesian coordinates[edit]

Let 

ϕ

 be the golden ratio. The 12 points given by 

(0,±1,±ϕ)

 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 

(±1,±1,±1)

 together with the points 

(±ϕ,±1/ϕ,0)

 and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of 

(3ϕ+12)/19≈0.88705799822

 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 

2/ϕ

. Its faces are acute isosceles triangles with one angle of 

arccos⁡((−8+9ϕ)/18)≈68.61872093119∘

 and two of 

arccos⁡((5−ϕ)/6)≈55.69063953441∘

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

(5−ϕ)/3≈1.12732200375

.

Chemistry[edit]

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.

Biology[edit]

The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses, such as Adeno-associated virus. These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron

https://en.wikipedia.org/wiki/Pentakis_dodecahedron