Pentagonal polytope
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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).
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.
| n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Elements | ||||
|---|---|---|---|---|---|---|---|---|---|
| Vertices | Edges | Faces | Cells | 4-faces | |||||
| 1 | H 1 {\displaystyle H_{1}} [ ] (order 2) | Line segment {} | 2 vertices | 2 | |||||
| 2 | H 2 {\displaystyle H_{2}} [5] (order 10) | Pentagon {5} | 5 edges | 5 | 5 | ||||
| 3 | H 3 {\displaystyle H_{3}} [5,3] (order 120) | Dodecahedron {5, 3} | 12 pentagons | 20 | 30 | 12 | |||
| 4 | H 4 {\displaystyle H_{4}} [5,3,3] (order 14400) | 120-cell {5, 3, 3} | 120 dodecahedra | 600 | 1200 | 720 | 120 | ||
| 5 | H ¯ 4 {\displaystyle {\bar {H}}_{4}} [5,3,3,3] (order ∞) | 120-cell honeycomb {5, 3, 3, 3} | ∞ 120-cells | ∞ | ∞ | ∞ | ∞ | ∞ |
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.
| n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Elements | ||||
|---|---|---|---|---|---|---|---|---|---|
| Vertices | Edges | Faces | Cells | 4-faces | |||||
| 1 | H 1 {\displaystyle H_{1}} [ ] (order 2) | Line segment {} | 2 vertices | 2 | |||||
| 2 | H 2 {\displaystyle H_{2}} [5] (order 10) | Pentagon {5} | 5 Edges | 5 | 5 | ||||
| 3 | H 3 {\displaystyle H_{3}} [5,3] (order 120) | Icosahedron {3, 5} | 20 equilateral triangles | 12 | 30 | 20 | |||
| 4 | H 4 {\displaystyle H_{4}} [5,3,3] (order 14400) | 600-cell {3, 3, 5} | 600 tetrahedra | 120 | 720 | 1200 | 600 | ||
| 5 | H ¯ 4 {\displaystyle {\bar {H}}_{4}} [5,3,3,3] (order ∞) | Order-5 5-cell honeycomb {3, 3, 3, 5} | ∞ 5-cells | ∞ | ∞ | ∞ | ∞ | ∞ |
Related star polytopes and honeycombs
The pentagonal polytopes can be stellated to form new star regular polytopes:
- In two dimensions, we obtain the pentagram {5/2},
- In three dimensions, this forms the four Kepler–Poinsot polyhedra, {3,5/2}, {5/2,3}, {5,5/2}, and {5/2,5}.
- In four dimensions, this forms the ten Schläfli–Hess polychora: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5,3,5/2}, {5/2,3,5}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, {3,3,5/2}, and {5/2,3,3}.
- In four-dimensional hyperbolic space there are four regular star-honeycombs: {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.
In some cases, the star pentagonal polytopes are themselves counted among the pentagonal polytopes.
Like other polytopes, regular stars can be combined with their duals to form compounds;
- In two dimensions, a decagrammic star figure {10/2} is formed,
- In three dimensions, we obtain the compound of dodecahedron and icosahedron,
- In four dimensions, we obtain the compound of 120-cell and 600-cell.
Star polytopes can also be combined.
Notes
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN978-0-471-01003-6 2016-07-11 at theWayback Machine (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36]
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q, r} in four dimensions, pp.292–293)
| vteFundamental convex regular and uniform polytopes in dimensions 2–10 | |||||
|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn |
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon |
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | |
| Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell |
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | ||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | |
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | |
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | |
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | ||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | ||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope |
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations |