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

Net

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 known as the C₆₀₀, hexacosichoron^[1] and hexacosihedroid.^[2] It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.

The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex.^[a] Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell.

Geometry[]

The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^[b] It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell,^[4] as the 24-cell can be deconstructed into three overlapping instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two overlapping instances of its predecessor the 16-cell.^[5]

The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.^[c] The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius. The 600-cell's radius and edge length are in the golden ratio.

Regular convex 4-polytopes
Symmetry group	A4	B4		F4	H4
Name	5-cell Hyper- tetrahedron	16-cell Hyper- octahedron	8-cell Hyper- cube	24-cell	600-cell Hyper- icosahedron	120-cell Hyper- dodecahedron
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter diagram
Graph
Vertices	5	8	16	24	120	600
Edges	10	24	32	96	720	1200
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 tetrahedra	16 tetrahedra	8 cubes	24 octahedra	600 tetrahedra	120 dodecahedra
Tori	1 5-tetrahedron	2 8-tetrahedron	2 4-cube	4 6-octahedron	20 30-tetrahedron	12 10-dodecahedron
Inscribed	120 in 120-cell	1 16-cell	2 16-cells	3 8-cells	5 24-cells x 5	5 600-cells x 2
Great polygons		2