| Penteract (5-cube) |
|
|---|---|
 Orthogonal projection inside Petrie polygon |
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| Type | Regular 5-polytope |
| Family | hypercube |
| Schläfli symbols | {4,3,3,3} {4,3,3}x{} {4}x{4}x{} {}x{}x{}x{}x{} |
| Coxeter-Dynkin diagrams |      |
| Hypercells | 10 tesseracts |
| Cells | 40 cubes |
| Faces | 80 squares |
| Edges | 80 |
| Vertices | 32 |
| Vertex figure |  pentachoron |
| Petrie polygon | decagon |
| Coxeter group | C5, [3,3,3,4] |
| Dual | Pentacross |
| Properties | convex |
In five dimensional geometry, 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.
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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 < xi < 1.
Projections
 A perspective projection 3D to 2D of stereographic projection 4D to 3D of Schlegel diagram 5D to 4D. |
 This penteract graph is an orthogonal projection. This oriention shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:5:10:10:5:1. |
 Wireframe skew orthogonal projection |
 Vertex-edge graph. |
See also
- Other regular 5-polytopes:
- 5-simplex (hexateron) - {3,3,3,3}
- 5-orthoplex (pentacross) - {3,3,3,4}
- Related semiregular 5-polytope:
- 5-demicube (demipenteract) - {31,2,1}
- Others in the hypercube family
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)
External links
- Weisstein, Eric W., "Hypercube" from MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- Multi-dimensional Glossary: hypercube Garrett Jones
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