120-cell

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120-cell
Image:Cell120-4dpolytope.gif
TypeRegular polychoron
Cells120 (5.5.5)
Faces720 {5}
Edges1200
Vertices600
Vertex configuration4 (5.5.5)
(tetrahedron)
Schläfli symbol {5,3,3}
Symmetry groupgroup [3,3,5]
Dual 600-cell
Propertiesconvex

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:

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.

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