Pentagonal polytope

In geometry, a pentagonal polytope[1] is a regular polytope in n dimensions constructed from the Hn 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, 3n − 1} (dodecahedral) or {3n − 1, 5} (icosahedral).

Contents

Related star polytopes and honeycombs

The pentagonal polytopes can be stellated to form new star regular polytopes. In two dimensions, this forms 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}; and 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}.

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:

  1. Edge, {}
  2. Pentagon, {5}
  3. Dodecahedron, {5, 3} (12 pentagonal faces)
  4. 120-cell, {5, 3, 3} (120 dodecahedral cells)
  5. 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.

Elements

Dodecahedral pentagonal polytopes
n Coxeter group Petrie polygon
projection		 Name
Coxeter-Dynkin diagram
Schläfli symbol		 Facets Elements
Vertices Edges Faces Cells 4-faces
1 H_1 Line segment

{}		 2 vertices 2
2 H_2 Pentagon

{5}		 5 edges 5 5
3 H_3 Dodecahedron

{5, 3}		 12 pentagons
 20 30 12
4 H_4 120-cell

{5, 3, 3}		 120 dodecahedra
 600 1200 720 120
5 {\bar{H}}_4 120-cell honeycomb

{5, 3, 3, 3}		 ∞ 120-cells
 ∞ ∞ ∞ ∞ ∞

Icosahedral

The complete family of icosahedral pentagonal polytopes are:

  1. Edge, {}
  2. Pentagon, {5}
  3. Icosahedron, {3, 5} (20 triangular faces)
  4. 600-cell, {3, 3, 5} (120 tetrahedron cells)
  5. 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.

Elements

Icosahedral pentagonal polytopes
n Coxeter group Petrie polygon
projection		 Name
Coxeter-Dynkin diagram
Schläfli symbol		 Facets Elements
Vertices Edges Faces Cells 4-faces
1 H_1 Line segments

{}		 2 vertices 2
2 H_2 Pentagon

{5}		 5 Edges 5 5
3 H_3 Icosahedron

{3, 5}		 20 equilateral triangles
 12 30 20
4 H_4 600-cell

{3, 3, 5}		 600 tetrahedra
 120 720 1200 600
5 {\bar{H}}_4 Order-5 5-cell honeycomb

{3, 3, 3, 5}		 ∞ 5-cells
 ∞ ∞ ∞ ∞ ∞

Nots

  1. ^ Olshevsky, George, [1] at Glossary for Hyperspace.

References