Pentagonal polytope

In geometry, a pentagonal polytope^[1] is a regular polytope in n dimensions constructed from the H_{n Coxeter group}. The family was named by George Olshevsky, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3^{n - 1}} (dodecahedral) or {3^{n - 1}, 5} (icosahedral).

Such polytopes can always be stellated to form new star regular polytopes. In two dimensions, this forms the pentagram; in three dimensions, this forms the Kepler-Poinsot polyhedra; and in four dimensions, this forms the Schläfli-Hess polychora.

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)

Elements

Dodecahedral pentagonal polytopes
n	Petrie polygon projection	Name Coxeter-Dynkin diagram Schläfli symbol	Facets	Elements
n	Petrie polygon projection	Name Coxeter-Dynkin diagram Schläfli symbol	Facets	Vertices	Edges	Faces	Cells	4-faces
1		Line segment {}	2 points	2
2		Pentagon {5}	5 line segments	5	5
3		Dodecahedron {5, 3}	12 pentagons	20	30	12
4		120-cell {5, 3, 3}	120 dodecahedra	600	1200	720	120
5		Order-3 120-cell honeycomb {5, 3, 3, 3}	∞ 120-cells	∞	∞	∞	∞	∞

Icosahedral

References

^ Olshevsky, George. Glossary for Hyperspace https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#{{{anchor}}}. Archived from the original on 4 February 2007. {{cite web}}: Missing or empty |title= (help)

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Olshevsky, George. Glossary for Hyperspace https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#{{{anchor}}}. Archived from the original on 4 February 2007. {{cite web}}: Missing or empty |title= (help)

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