# 5-cube

5-cube
Penteract

Orthogonal projection
inside Petrie polygon
Central orange vertex is doubled
Type Regular 5-polytope
Family hypercube
Schläfli symbols {4,3,3,3}
{4,3,3}×{ }
{4,3}×{4}
{4,3}×{ }×{ }
{4}×{4}×{ }
{4}×{ }×{ }×{ }
{ }×{ }×{ }×{ }×{ }
Coxeter-Dynkin diagrams

4-faces 10 tesseracts
Cells 40 cubes
Faces 80 squares
Edges 80
Vertices 32
Vertex figure
5-cell
Petrie polygon decagon
Coxeter group BC5, [3,3,3,4]
Dual 5-orthoplex
Properties convex

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 {4,33}, 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.

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## Cartesian coordinates

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

(±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1.

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## Images

n-cube Coxeter plane projections in the BkCoxeter 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]
 Wireframe skew direction B5 Coxeter plane
 Vertex-edge graph.
 A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D. Sorry, your browser either has JavaScript disabled or does not have any supported player. You can download the clip or download a player to play the clip in your browser. Animation of a 5D rotation of a 5-cube perspective projection to 3D.
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## Related polytopes

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

 β5 t1β5 t2γ5 t1γ5 γ5 t0,1β5 t0,2β5 t1,2β5 t0,3β5 t1,3γ5 t1,2γ5 t0,4γ5 t0,3γ5 t0,2γ5 t0,1γ5 t0,1,2β5 t0,1,3β5 t0,2,3β5 t1,2,3γ5 t0,1,4β5 t0,2,4γ5 t0,2,3γ5 t0,1,4γ5 t0,1,3γ5 t0,1,2γ5 t0,1,2,3β5 t0,1,2,4β5 t0,1,3,4γ5 t0,1,2,4γ5 t0,1,2,3γ5 t0,1,2,3,4γ5
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## References

• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Richard Klitzing, 5D uniform polytopes (polytera), o3o3o3o4x - pent
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