5-cube

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-faces10tesseracts
Cells40cubes
Faces80squares
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)
Circumradius sqrt(5)/2 = 1.118034
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.

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.

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane Other B2 A3
Graph
Dihedral symmetry [2] [4] [4]
More orthographic projections

Wireframe skew direction
B5 Coxeter plane
Graph

Vertex-edge graph.
perspective projections

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.
Net

4D net of the 5-cube, perspective projected into 3D.

Related polytopes

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

References

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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