Pentakis dodecahedron | |
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(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(-) |
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.^{[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:
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.
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 icosacahedron 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 surface segment represents a carbon atom. 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 Adeno-associated 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:
Family of uniform icosahedral polyhedra | |||||||
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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 | ||||||||||||
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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 |
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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 |
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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 |
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.