Enneract (9cube) 


Orthogonal projection inside Petrie polygon Orange vertices are doubled, green have 4, and the blue center has 8 

Type  Regular 9polytope 
Family  hypercube 
Schläfli symbol  {4,3^{7}} 
CoxeterDynkin diagram  
8faces  18 {4,3^{6}} 
7faces  144 {4,3^{5}} 
6faces  672 {4,3^{4}} 
5faces  2016 {4,3^{3}} 
4faces  4032 {4,3,3} 
Cells  5376 {4,3} 
Faces  4608 {4} 
Edges  2304 
Vertices  512 
Vertex figure  8simplex 
Petrie polygon  octadecagon 
Coxeter group  C_{9}, [3^{7},4] 
Dual  Enneacross 
Properties  convex 
In geometry, an enneract is a ninedimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4faces, 2016 penteract 5faces, 672 hexeract 6faces, 144 hepteract 7faces, and 18 octeract 8faces.
The name enneract is derived from combining the name tesseract (the 4cube) with enne for nine (dimensions) in Greek.
It can also be called a regular octadeca9tope 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 crosspolytopes.
Contents 
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 (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}) with −1 < x_{i} < 1.
This 9cube graph is an orthogonal projection. This oriention shows columns of vertices positioned a vertexedgevertex 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. 
Petrie polygon, skew orthographic projection 
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.
Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
