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120-cell
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the 120-cell (or hecatonicosachoron) is the convex regular polychoron
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
 (4-polytope) with Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 .

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
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 and has been called a dodecaplex (short for "dodecahedral complex") and polydodecahedron.






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Tetrahedron
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the 120-cell (or hecatonicosachoron) is the convex regular polychoron
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
 (4-polytope) with Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 .

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
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 and has been called a dodecaplex (short for "dodecahedral complex") and polydodecahedron. Just as a dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
s, with 3 around each edge.

Elements


  • There are 120 cells, 720 pentagon
    Pentagon

    In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
    al 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.


  • The dual polytope of the 120-cell is the 600-cell
    600-cell

    In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
    .
  • The vertex figure
    Vertex figure

    In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
     of the 120-cell is a tetrahedron
    Tetrahedron

    A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
    .


Cartesian coordinates

The 600 vertices of the 120-cell include all permutation
Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
s of
(0, 0, ±2, ±2)
(±1, ±1, ±1, ±v5)
(±f-2, ±f, ±f, ±f)
(±f-1, ±f-1, ±f-1, ±f2)


and all even permutations of
(0, ±f-2, ±1, ±f2)
(0, ±f-1, ±f, ±v5)
(±f-1, ±1, ±f, ±2)


where f (also called t) is the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
, (1+v5)/2.

Projections


Projections into 2D


2D orthographic projection
Orthographic projection (geometry)

In Euclidean geometry, an orthographic projection is an orthogonal projection. In particular, in 3D it is an affine transformation, parallel Projection of an object onto a perpendicular plane ....
s
Cell120 4dpolytope

Vertex-centered
Skew projection inside 30-gonal Petrie polygon
Petrie polygon

In geometry, a Petrie polygon is a skew polygon such that every two consecutive Edge belong to a Face of a regular polyhedron.This definition extends to higher regular polytopes....

Centered on pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....


Projections into 3D


Comparison with regular dodecahedron
ProjectionDodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
Dodecaplex
Schlegel diagram
Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from into through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in that is combinatorially equivalent to the original polytope....

12 pentagon faces in the plane
120 dodecahedral cells in 3-space
Stereographic projection
Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
Stereographic Polytope 120cell Faces

With transparent faces


Perspective projection
Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied:
  • Nearest dodecahedron to the 4D viewpoint rendered in yellow
  • The 12 dodecahedra immediately adjoining it rendered in cyan;
  • The remaining dodecahedra rendered in green;
  • Cells facing away from the 4D viewpoint (those lying on the "far side" of the 120-cell) culled to minimize clutter in the final image.
Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements:
  • Four cells surrounding nearest vertex shown in 4 colors
  • Nearest vertex shown in white (center of image where 4 cells meet)
  • Remaining cells shown in transparent green
  • Cells facing away from 4D viewpoint culled for clarity


    • See also

    • Uniform polychora family with [5,3,3] symmetry
      Uniform polychoron

      In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....


    External links

    • Marco Möller's Regular polytopes in R4 (German)
    • – 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.