Pentakis dodecahedron
The lead section of this article may need to be rewritten. (January 2021) |
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 | Ih, H3, [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 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.
Cartesian coordinates[]
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 icosahedron 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 .
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 symmetry |
[2] | [6] | [10] |
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Image | |||
Dual image |
Concave pentakis dodecahedron[]
A concave pentakis dodecahedron adds inverted pyramids on the pentagonal faces of a dodecahedron.
Related polyhedra[]
See also[]Cultural references[]
References[]
External links[]
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- Catalan solids
- Geodesic polyhedra