600-cell

600-cell
Schlegel diagram, vertex-centered (vertices and edges)
Type	Convex regular 4-polytope
Schläfli symbol	{3,3,5}
Coxeter diagram
Cells	600 (3.3.3)
Faces	1200 {3}
Edges	720
Vertices	120
Vertex figure	icosahedron
Petrie polygon	30-gon
Coxeter group	H4, [3,3,5], order 14400
Dual	120-cell
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	35

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also called a C₆₀₀, hexacosichoron and hexacosidedroid.

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 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° = ${\displaystyle \pi /5}$, 60°= ${\displaystyle \pi /3}$, 72° = ${\displaystyle 2\pi /5}$, 90° = ${\displaystyle \pi /2}$, 108° = ${\displaystyle 3\pi /5}$, 120° = ${\displaystyle 2\pi /3}$, 144° = ${\displaystyle 4\pi /5}$, and 180° = ${\displaystyle \pi }$. 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 icosidodecahedron, and finally at 180° the antipodal vertex of V. References: S.L. van Oss (1899); F. Buekenhout and M. Parker (1998).