120-cell

From Wikipedia, the free encyclopedia

120-cell
Schlegel diagram
Type	Regular polychoron
Cells	120 (5.5.5)
Faces	720 {5}
Edges	1200
Vertices	600
Vertex figure	4 (5.5.5) (tetrahedron)
Schläfli symbol	{5,3,3}
Symmetry group	H4, [3,3,5]
Dual	600-cell
Properties	convex

In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}.

It can be thought of as the 4-dimensional analog of the dodecahedron and has been called a dodecaplex for being constructed of dodecahedron cells.

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

• Together they have 720 pentagonal faces, 1200 edges, and 600 vertices.
• There are 4 dodecahedra, 6 pentagons, and 4 edges meeting at every vertex.
• There are 3 dodecahedra and 3 pentagons meeting every edge.

Related polytopes:

• The dual polytope of the 120-cell is the 600-cell.
• The vertex figure of the 120-cell is a tetrahedron.

[edit] Cartesian coordinates

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

(0, 0, ±2, ±2)

(±1, ±1, ±1, ±√5)

(±τ^-2, ±τ, ±τ, ±τ)

(±τ^-1, ±τ^-1, ±τ^-1, ±τ²)

and all even permutations of

(0, ±τ^-2, ±1, ±τ²)

(0, ±τ^-1, ±τ, ±√5)

(±τ^-1, ±1, ±τ, ±2)

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

Convex regular 4-polytopes
pentachoron	tesseract	16-cell	24-cell	120-cell	600-cell
{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}

[edit] External links

• 120-cell – some nice projections of the 120-cell to 2 dimensions.
• 120-cell explorer – A free interactive program that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL.
• Polytopes – very nice hidden-detail-removed projection of the 120-cell to 3 dimensions, midway through the page.