---
tag: 3RQEK50CP0
kind: object
name: 120-cell
msc: [51M20, 52B15, 52B11]
instance_of: [F60PD912D5]
uses: [MAE6BF4DFK]
sources:
  - url: https://en.wikipedia.org/wiki/120-cell
same_as:
  wikidata: Q736191
  wikipedia: 120-cell
---

## Description

The *120-cell* is the convex [[regular polytope|F60PD912D5]] in $\R^4$ with [[Schläfli symbol|MAE6BF4DFK]] $\set{5, 3, 3}$.
Its cells are $120$ regular dodecahedra, three around each edge, and four cells meet at each vertex.
It has $600$ vertices, $1200$ edges, $720$ pentagonal faces and $120$ cells.

Check of the counts: each dodecahedron has $12$ pentagons, and each pentagon lies in exactly $2$ cells, so there are $120 \cdot 12 / 2 = 720$ faces.
Each pentagon has $5$ edges, and each edge lies in $3$ pentagons, so there are $720 \cdot 5 / 3 = 1200$ edges.
The vertex figure is a regular tetrahedron $\set{3,3}$, so each vertex lies on $4$ edges, and there are $1200 \cdot 2 / 4 = 600$ vertices.
Then $600 - 1200 + 720 - 120 = 0$, as Euler's relation for convex $4$-polytopes requires.

## Remarks

The 120-cell is the dual of the [[600-cell|8MNQDK5X5W]].
Its symmetry group is the reflection group $H_4$, of order $14400$.
It is the four-dimensional analogue of the regular dodecahedron $\set{5, 3}$.
Three dodecahedra fit around an edge with a small angular gap, since the dihedral angle of the dodecahedron is about $116.57^\circ$.

## Explanations

### L1 human

The 120-cell is a four-dimensional solid made of $120$ regular dodecahedra.
It has $600$ corners, and it is the largest and most complicated of the six regular solids of four-dimensional space.
It plays the role in four dimensions that the dodecahedron plays in three.