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A hexeract is a name for a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 penteract 5-faces.
The name hexeract is derived from combining the name tesseract (the 4-cube) with hex for six (dimensions) in Greek.
It can also be called a regular dodeca-6-tope or dodecapeton, being made of 12 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of a penteract can be called a hexacross, and is a part of the infinite family of cross-polytopes.
Applying an alternation operation, deleting alternating vertices of the hexeract, creates another uniform polytope, called a demihexeract, (part of an infinite family called demihypercubes), which has 12 demipenteractic and 32 hexateronic facets.
Cartesian coordinates
Cartesian coordinates for the vertices of a hexeract centered at the origin and edge length 2 are
- (±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with -1 < xi < 1.
Projections
This hypercube 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 1:6:15:20:15:6:1. |
Another orthogonal projection |
See also
- Other Regular 6-polytopes:
- 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}
- ...
References
External links
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