Enneract 9-cube
Type	Regular 9-polytope
Family	hypercube
Schläfli symbol	{4,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
8-faces	18 octeracts
7-faces	144 hepteracts
6-faces	672 hexeracts
5-faces	2016 penteracts
4-faces	4032 tesseracts
Cells	5376 cubes
Faces	4608 squares
Edges	2304
Vertices	512
Vertex figure	8-simplex
Symmetry group	B9, [3,3,3,3,3,3,3,4]
Dual	Enneacross
Properties	convex

An enneract is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 penteract 5-faces, 672 hexeract 6-faces, 144 hepteract 7-faces, and 18 octeract 8-faces.

The name enneract is derived from combining the name tesseract (the 4-cube) with enne for nine (dimensions) in Greek.

It can also be called a regular octadeca-9-tope or octadecayotton, being made of 18 regular facets.

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

Cartesian coordinates

Cartesian coordinates for the vertices of a penteract centered at the origin and edge length 2 are

(±1,±1,±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7, x8) with −1 < xi < 1.

Projections

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

Derived polytopes

Applying an alternation operation, deleting alternating vertices of the enneract, creates another uniform polytope, called a demienneract, (part of an infinite family called demihypercubes), which has 18 demiocteractic and 256 enneazettonic facets.

See also

* Hypercubes family

o square - {4}
o Cube - {4,3}
o Tesseract - {4,3,3}
o Penteract - {4,3,3,3}
o Hexeract - {4,3,3,3,3}
o Hepteract - {4,3,3,3,3,3}
o Octeract - {4,3,3,3,3,3,3}

o ...

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)

Links

Geometry