120-cell
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| 120-cell | |
|---|---|
| Image:Cell120-4dpolytope.gif | |
| Type | Regular polychoron |
| Cells | 120 (5.5.5) |
| Faces | 720 {5} |
| Edges | 1200 |
| Vertices | 600 |
| Vertex configuration | 4 (5.5.5) (tetrahedron) |
| Schläfli symbol | {5,3,3} |
| Symmetry group | group [3,3,5] |
| Dual | 600-cell |
| Properties | convex |
Image:Tetrahedron.jpg In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron.
The boundary of 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.
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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.
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External links
- 120-cell – some nice projections of the 120-cell to 2-dimensions.
- Polytopes – very nice hidden-detail-removed projection of the 120-cell to 3 dimensions, midway through the page.
Template:4D regular polytopeses:Hecatonicosacoron ja:正百二十胞体 zh:正一百二十胞体