# 5-cube

(Redirected from Penteract)
5-cube
penteract (pent)
Type uniform 5-polytope
Schläfli symbol {4,3,3,3}
{4,3,3}×{ }
{4,3}×{4}
{4,3}×{ }2
{4}×{4}×{ }
{4}×{ }3
{ }5
Coxeter diagram                                                               4-faces 10 tesseracts
Cells 40 cubes
Faces 80 squares
Edges 80
Vertices 32
Vertex figure 5-cell
Coxeter group B5, [4,33], order 3840
[4,3,3,2], order 768
[4,3,2,4], order 384
[4,3,2,2], order 192
[4,2,4,2], order 128
[4,2,2,2], order 64
[2,2,2,2], order 32
Dual 5-orthoplex
Base point (1,1,1,1,1,1)
Properties convex, isogonal regular

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

It is represented by Schläfli symbol {4,3,3,3} or {4,33}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a penteract, a portmanteau of tesseract (the 4-cube) and pente for five (dimensions) in Greek. It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets.

## Related polytopes

It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes.

Applying an alternation operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes.

The 5-cube can be seen as an order-3 tesseractic honeycomb on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic honeycomb.

## As a configuration

This configuration matrix represents the 5-cube. The rows and columns correspond to vertices, edges, faces, cells, and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

${\begin{bmatrix}{\begin{matrix}32&5&10&10&5\\2&80&4&6&4\\4&4&80&3&3\\8&12&6&40&2\\16&32&24&8&10\end{matrix}}\end{bmatrix}}$ ## Cartesian coordinates

The Cartesian coordinates of the vertices of a 5-cube centered at the origin and having edge length 2 are

(±1,±1,±1,±1,±1),

while this 5-cube's interior consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1 for all i.

## Images

n-cube Coxeter plane projections in the Bk Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs. A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D. Animation of a 5D rotation of a 5-cube perspective projection to 3D. 4D net of the 5-cube, perspective projected into 3D.

## Projection

The 5-cube can be projected down to 3 dimensions with a rhombic icosahedron envelope. There are 22 exterior vertices, and 10 interior vertices. The 10 interior vertices have the convex hull of a pentagonal antiprism. The 80 edges project into 40 external edges and 40 internal ones. The 40 cubes project into golden rhombohedra which can be used to dissect the rhombic icosahedron. The projection vectors are u = {1, φ, 0, -1, φ}, v = {φ, 0, 1, φ, 0}, w = {0, 1, φ, 0, -1}, where φ is the golden ratio, ${\frac {1+{\sqrt {5}}}{2}}$ .

## Related polytopes

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.