5-cube

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
4-faces	10	tesseracts
Cells	40	cubes
Faces	80	squares
Edges	80
Vertices	32
Vertex figure	5-cell
Coxeter group	BC5, [4,3^3], 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 < x_i < 1.

Images

n-cube Coxeter plane projections in the B_k 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.

B5 polytopes 

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

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
    • (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

• Weisstein, Eric W., "Hypercube", MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Multi-dimensional Glossary: hypercube Garrett Jones