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From Wikipedia, the free encyclopedia

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, green have 4, and the blue center has 8
Type Regular 9-polytope
Family hypercube
Schläfli symbol {4,37}
Coxeter-Dynkin diagram CDW ring.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
8-faces 18 {4,36}Octeract Petrie polygon.svg
7-faces 144 {4,35}Hepteract ortho petrie.svg
6-faces 672 {4,34}Hexeract ortho petrie.svg
5-faces 2016 {4,33}Penteract ortho petrie.svg
4-faces 4032 {4,3,3}Hypercubestar.svg
Cells 5376 {4,3}Cube graph ortho vcenter.png
Faces 4608 {4}2-cube column graph.svg
Edges 2304
Vertices 512
Vertex figure 8-simplex
Petrie polygon octadecagon
Coxeter group C9, [37,4]
Dual Enneacross
Properties convex

In geometry, 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 an enneract centered at the origin and edge length 2 are


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


9-cube column graph.svg
This 9-cube 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 1:9:36:84:126:126:84:36:9:1.
Enneract ortho petrie.svg
Petrie polygon, skew orthographic projection

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.


  • 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)

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