 Pentakis Dodecahedron
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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(-)
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. This interpretation is expressed in its name. There are in fact several topologically equivalent but geometrically distinct kinds of pentakis dodecahedron, depending on the height of the pentagonal pyramids. These include:

$h={\frac {\sqrt {65+22{\sqrt {5}}}}{19{\sqrt {5}}}}\approx 0.2515$ At this size, the dihedral angle between all neighbouring triangular faces is equal to the value in the table above. Flatter pyramids have higher intra-pyramid dihedrals and taller pyramids have higher inter-pyramid dihedrals.
• 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 an excavated dodecahedron one obtains the great icosahedron.

If one keeps the center dodecahedron, one get the net of a Dodecahedral pyramid.

## Cartesian coordinates

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

## 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: