Pentagonal polytope

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In geometry, a pentagonal polytope 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).

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

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
[ ]
(order 2)
Cross graph 1.svg Line segment
CDel node 1.png
{ }
2 vertices 2
2 H_2
[5]
(order 10)
Regular polygon 5.svg Pentagon
CDel node 1.pngCDel 5.pngCDel node.png
{5}
5 edges 5 5
3 H_3
[5,3]
(order 120)
Dodecahedron t0 H3.png Dodecahedron
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
{5, 3}
12 pentagons
Regular polygon 5.svg
20 30 12
4 H_4
[5,3,3]
(order 14400)
120-cell graph H4.svg 120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5, 3, 3}
120 dodecahedra
Dodecahedron t0 H3.png
600 1200 720 120
5 {\bar{H}}_4
[5,3,3,3]
(order ∞)
120-cell honeycomb
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5, 3, 3, 3}
120-cells
120-cell graph H4.svg

Icosahedral

The complete family of icosahedral pentagonal polytopes are:

  1. Line segment, { }
  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.

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
[ ]
(order 2)
Cross graph 1.svg Line segment
CDel node 1.png
{ }
2 vertices 2
2 H_2
[5]
(order 10)
Regular polygon 5.svg Pentagon
CDel node 1.pngCDel 5.pngCDel node.png
{5}
5 Edges 5 5
3 H_3
[5,3]
(order 120)
Icosahedron t0 H3.png Icosahedron
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{3, 5}
20 equilateral triangles
Regular polygon 3.svg
12 30 20
4 H_4
[5,3,3]
(order 14400)
600-cell graph H4.svg 600-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{3, 3, 5}
600 tetrahedra
3-simplex t0.svg
120 720 1200 600
5 {\bar{H}}_4
[5,3,3,3]
(order ∞)
Order-5 5-cell honeycomb
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{3, 3, 3, 5}
5-cells
4-simplex t0.svg

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}.

Notes

References

  • 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, ISBN 978-0-471-01003-6 [1]
    • (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. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q,r} in four dimensions, pp. 292–293)







Creative Commons License