120-cell

120-cell
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Vertices	600
Edges	1200
Faces	720 {5}
Cell	120 (5.5.5)
Vertex figure	tetrahedron
Schläfli symbol	{5,3,3}
Coxeter-Dynkin diagram
Petrie polygon	30-gon
Coxeter group	H4, [3,3,5]
Dual	600-cell
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	31 32 33

Vertex figure: tetrahedron
formed by 4 dodecahedral cells meeting at each vertex:

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

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

It can be thought of as the 4-dimensional analog of the dodecahedron and has been called a dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, and polydodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedrons, with 3 around each edge.

Elements

• There are 120 cells, 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.

• The dual polytope of the 120-cell is the 600-cell.
• The vertex figure of the 120-cell is a tetrahedron.

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.

Projections

Projections into 2D

2D orthographic projections
Vertex-centered	Skew projection inside 30-gonal Petrie polygon	Centered on pentagon

Projections into 3D

Comparison with regular dodecahedron
Projection	Dodecahedron	Dodecaplex
Schlegel diagram	12 pentagon faces in the plane	120 dodecahedral cells in 3-space
Stereographic projection		With transparent faces

Perspective projection
	Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied: Nearest dodecahedron to the 4D viewpoint rendered in yellow The 12 dodecahedra immediately adjoining it rendered in cyan; The remaining dodecahedra rendered in green; Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image.
	Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements: Four cells surrounding nearest vertex shown in 4 colors Nearest vertex shown in white (center of image where 4 cells meet) Remaining cells shown in transparent green Cells facing away from 4D viewpoint culled for clarity

See also

• Uniform polychora family with [5,3,3] symmetry

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
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [2]

External links

• Weisstein, Eric W., "120-Cell" from MathWorld.
• Olshevsky, George, Hecatonicosachoron at Glossary for Hyperspace.
  • Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell) - Model 32, George Olshevsky.
• Der 600-Zeller (600-cell) Marco Möller's Regular polytopes in R⁴ (German)
• 120-cell explorer – A free interactive program that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL.