120-cell

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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
Symmetry group	H₄, [3,3,5]
Dual	600-cell
Properties	convex

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 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:

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

1 Cartesian coordinates
2 Images
3 References
4 External links

[edit] Cartesian coordinates

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

(0, 0, ±2, ±2)

(±1, ±1, ±1, ±√5)

(±τ^-2, ±τ, ±τ, ±τ)

(±τ^-1, ±τ^-1, ±τ^-1, ±τ²)

and all even permutations of

(0, ±τ^-2, ±1, ±τ²)

(0, ±τ^-1, ±τ, ±√5)

(±τ^-1, ±1, ±τ, ±2)

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

[edit] Images

Stereographic projection

Orthographic projection

[edit] References

H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.

[edit] External links

Eric W. Weisstein, 120-Cell at MathWorld.
Olshevsky, George, Hecatonicosachoron at Glossary for Hyperspace.
120-cell – some nice projections of the 120-cell to 2 dimensions.
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.
Polytopes – very nice hidden-detail-removed projection of the 120-cell to 3 dimensions, midway through the page.