600-cell

In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope, or polychoron, with Schläfli symbol {3,3,5}. Its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons. Each vertex of the 600-cell is a vertex of six such decagons.

The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° =, 60°=, 72° =, 90° =, 108° =, 120° =, 144° =, and 180° = . Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an icosadodecahedron, and finally at 180° the antipodal vertex of V. References: S.L. van Oss (1899); F. Buekenhout and M. Parker (1998).

The 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex (abbreviated from "tetrahedral complex") and polytetrahedron, being bounded by tetrahedral cells.

Its vertex figure is an icosahedron, and its dual polytope is the 120-cell.

Each cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.