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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(−80 + 9√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:
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.
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:
Projective symmetry |
[2] | [6] | [10] |
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Image | |||
Dual image |
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.
Convex polyhedra | |||||
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Platonic solids (regular) | |||||
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Dihedral regular | |||||
Dihedral uniform |
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Dihedral others | |||||
Degenerate polyhedra are in italics. |
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