# 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′07″
arccos(−80 + 95/109)
Properties convex, face-transitive

Truncated icosahedron
(dual polyhedron)

Net

In geometry, a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan Solid, meaning that it is a dual of an Archimedean Solid, in this case, the Truncated Icosahedron.

3d model of a pentakis dodecahedron

## Cartesian coordinates

Let ${\displaystyle \phi }$  be the golden ratio. The 12 points given by ${\displaystyle (0,\pm 1,\pm \phi )}$  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 ${\displaystyle (\pm 1,\pm 1,\pm 1)}$  together with the points ${\displaystyle (\pm \phi ,\pm 1/\phi ,0)}$  and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of ${\displaystyle (3\phi +12)/19\approx 0.887\,057\,998\,22}$  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 ${\displaystyle 2/\phi }$ . Its faces are acute isosceles triangles with one angle of ${\displaystyle \arccos((-8+9\phi )/18)\approx 68.618\,720\,931\,19^{\circ }}$  and two of ${\displaystyle \arccos((5-\phi )/6)\approx 55.690\,639\,534\,41^{\circ }}$ . The length ratio between the long and short edges of these triangles equals ${\displaystyle (5-\phi )/3\approx 1.127\,322\,003\,75}$ .

## 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:

Projectivesymmetry Image Dualimage [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
*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures

Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures

Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6

## References

• 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 0730208. (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 )