Penteract

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Penteract 5-cube
Vertex-edge graph
Type	Regular 5-polytope
Family	hypercube
Schläfli symbol	{4,3,3,3}
Coxeter-Dynkin diagram
Hypercells	10 tesseracts
Cells	40 cubes
Faces	80 squares
Edges	80
Vertices	32
Vertex figure	pentachoron
Symmetry group	B5, [3,3,3,4]
Dual	Pentacross
Properties	convex

A penteract is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells.

The name penteract is derived from combining the name tesseract (the 4-cube) with pente for five (dimensions) in Greek.

It can also be called a regular deca-5-tope or decateron, being made of 10 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a penteract can be called a pentacross, of the infinite family of cross-polytopes.

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

Contents 1 Cartesian coordinates 2 Images 3 See also 4 References 5 External links

[edit] Cartesian coordinates

Cartesian coordinates for the vertices of a penteract 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.

[edit] Images

Wireframe (orthogonal projection)

An orthogonal projection viewed along the axes of two opposite vertices and the average plane of one edge path between.

A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D.

[edit] See also

• Other Regular 5-polytopes:
  • Hexateron - {3,3,3,3}
  • Pentacross - {3,3,3,4}
• Others in the Hypercubes family
  • Square - {4}
  • Cube - {4,3}
  • Tesseract - {4,3,3}
  • Penteract - {4,3,3,3}
  • Hexeract - {4,3,3,3,3}
  • Hepteract - {4,3,3,3,3,3}
  • Octeract - {4,3,3,3,3,3,3}
  • Enneract - {4,3,3,3,3,3,3,3}
  • 10-cube - {4,3,3,3,3,3,3,3,3}
  • ...

[edit] References

• Coxeter, H.S.M. 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)

[edit] External links

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