In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3n − 2} (dodecahedral) or {3n − 2, 5} (icosahedral).
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Family members
The family starts as 1-polytopes and ends with n = 5 as infinite tessellations of 4-dimensional hyperbolic space.
There are two types of pentagonal polytopes; they may be termed the dodecahedral and icosahedral types, by their three-dimensional members. The two types are duals of each other.
Dodecahedral
The complete family of dodecahedral pentagonal polytopes are:
- Line segment, { }
- Pentagon, {5}
- Dodecahedron, {5, 3} (12 pentagonal faces)
- 120-cell, {5, 3, 3} (120 dodecahedral cells)
- Order-3 120-cell honeycomb, {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ 120-cell facets)
The facets of each dodecahedral pentagonal polytope are the dodecahedral pentagonal polytopes of one less dimension. Their vertex figures are the simplices of one less dimension.
Icosahedral
The complete family of icosahedral pentagonal polytopes are:
- Line segment, { }
- Pentagon, {5}
- Icosahedron, {3, 5} (20 triangular faces)
- 600-cell, {3, 3, 5} (600 tetrahedron cells)
- Order-5 5-cell honeycomb, {3, 3, 3, 5} (tessellates hyperbolic 4-space (∞ 5-cell facets)
The facets of each icosahedral pentagonal polytope are the simplices of one less dimension. Their vertex figures are icosahedral pentagonal polytopes of one less dimension.
Related star polytopes and honeycombs
The pentagonal polytopes can be stellated to form new star regular polytopes: