120-cell

120-cell
Schlegel diagram (vertices and edges)
Type	Convex regular 4-polytope
Schläfli symbol	{5,3,3}
Coxeter diagram
Cells	120 {5,3}
Faces	720 {5}
Edges	1200
Vertices	600
Vertex figure	tetrahedron
Petrie polygon	30-gon
Coxeter group	H4, [3,3,5]
Dual	600-cell
Properties	convex, isogonal, isotoxal, isohedral
Uniform index	32

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is also called a C₁₂₀, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.

The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex.

It can be thought of as the 4-dimensional analog of the dodecahedron and has been called a dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.

The Davis 120-cell, introduced by Davis (1985), is a compact 4-dimensional hyperbolic manifold obtained by identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4-dimensional hyperbolic space.

The 600 vertices of the 120-cell include all permutations of:

and all even permutations of

where ϕ (also called τ) is the golden ratio, (1+√5)/2.

The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24-cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.