# Pentakis dodecahedron

Pentakis dodecahedron

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 Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 156° 43' 7"
$\arccos ( -\frac{80 + 9\sqrt{5}}{109} )$
Properties convex, face-transitive

Truncated icosahedron
(dual polyhedron)

Net

In geometry, a pentakis dodecahedron or kisdodecahedron is a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron. This interpretation is expressed in its name. [1] There are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include:

• As the heights of the pentagonal pyramids are raised, at a certain point adjoining pairs of triangular faces merge to become rhombi, and the shape becomes a rhombic triacontahedron.
• As the height is raised further, the shape becomes non-convex. In particular, an equilateral or deltahedron version of the pentakis dodecahedron, which has sixty equilateral triangular faces as shown in the adjoining figure, is slightly non-convex due to its taller pyramids (note, for example, the negative dihedral angle at the upper left of the figure).

Other more non-convex geometric variants include:

If one affixes pentagrammic pyramids into Wenninger's third stellation of icosahedron one obtains the great icosahedron.

## Chemistry

The pentakis dodecahedron in a model of buckminsterfullerene: each surface segment represents a carbon atom. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.

## Biology

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.

## Orthogonal projections

The pentakis dodecahedron has three symmetry positions, two on vertices, and one on a midedge:

Projective Image Dual symmetry image [2] [6] [10]

## Related polyhedra

Spherical pentakis dodecahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

## References

1. ^ Conway, Symmetries of things, p.284
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Sellars, Peter (2005). "Doctor Atomic Libretto". Boosey & Hawkes. We surround the plutonium core from thirty two points spaced equally around its surface, the thirty-two points are the centers of the twenty triangular faces of an icosahedron interwoven with the twelve pentagonal faces of a dodecahedron.
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 730208. (The thirteen semiregular convex polyhedra and their duals, Page 18, Pentakisdodecahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [2] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Pentakis dodecahedron )