# .

Vertex-edge graph. (*)

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 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.

* 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)

Geometry