120-cell

From Wikipedia, the free encyclopedia

 All or part of this article may be confusing or unclear.
Please help clarify the article. Suggestions may be on the talk page. (May 2008)
120-cell

Schlegel diagram
Type Regular polychoron
Cells 120 (5.5.5)
Faces 720 {5}
Edges 1200
Vertices 600
Vertex figure (3.3.3)
Schläfli symbol {5,3,3}
Coxeter-Dynkin diagram Image:CDW_ring.pngImage:CDW_5.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_dot.pngImage:CDW_3b.pngImage:CDW_dot.png
Symmetry group H4, [3,3,5]
Dual 600-cell
Properties convex
Vertex figure: tetrahedronformed by 4 dodecahedral cells meeting at each vertex:
Vertex figure: tetrahedron
formed by 4 dodecahedral cells meeting at each vertex:

In geometry, the 120-cell (or hecatonicosachoron) is the convex regular polychoron (4-polytope) with Schläfli symbol {5,3,3}.

It can be thought of as the 4-dimensional analog of the dodecahedron and has been called a dodecaplex and 'polydodecahedron for being constructed of dodecahedron cells.

The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex.

  • Together they have 720 pentagonal faces, 1200 edges, and 600 vertices.
  • There are 4 dodecahedra, 6 pentagons, and 4 edges meeting at every vertex.
  • There are 3 dodecahedra and 3 pentagons meeting every edge.

Related polytopes:

Contents

[edit] Cartesian coordinates

The 600 vertices of the 120-cell include all permutations of

(0, 0, ±2, ±2)
(±1, ±1, ±1, ±√5)
(±φ-2, ±φ, ±φ, ±φ)
(±φ-1, ±φ-1, ±φ-1, ±φ2)

and all even permutations of

(0, ±φ-2, ±1, ±φ2)
(0, ±φ-1, ±φ, ±√5)
(±φ-1, ±1, ±φ, ±2)

where φ (also called τ) is the golden ratio, (1+√5)/2.

[edit] Images


Stereographic projection
Orthographic projection

[edit] See also

[edit] References

  • H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • M. Möller: Definitions and computations to the Platonic and Archimedean polyhedrons, thesis (diploma), University of Hamburg, 2001

[edit] External links

Convex regular 4-polytopes
pentachoron tesseract 16-cell 24-cell 120-cell 600-cell
{3,3,3} {4,3,3} {3,3,4} {3,4,3} {5,3,3} {3,3,5}