5-cube

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5-cube
penteract (pent)
Type uniform 5-polytope
Schläfli symbol {4,3,3,3}
{4,3,3}×{ }
{4,3}×{4}
{4,3}×{ }×{ }
{4}×{4}×{ }
{4}×{ }×{ }×{ }
{ }×{ }×{ }×{ }×{ }
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
4-faces 10 tesseracts
Cells 40 cubes
Faces 80 squares
Edges 80
Vertices 32
Vertex figure 5-cube verf.png
5-cell
Coxeter group BC5, [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

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.

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 5-cube t0.svg 4-cube t0.svg 150px
Dihedral symmetry [10] [8] [6]
Coxeter plane Other B2 A3
Graph 150px 150px 150px
Dihedral symmetry [2] [4] [4]
More orthographic projections
240px
Wireframe skew direction
240px
B5 Coxeter plane
Graph
200px
Vertex-edge graph.
perspective projections
Penteract projected.png
A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D.
240px
Animation of a 5D rotation of a 5-cube perspective projection to 3D.
Net
200px
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.

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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, edited 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

External links