8cube Octeract  

Orthogonal projection inside Petrie polygon  
Type  Regular 8polytope 
Family  hypercube 
Schläfli symbol  {4,3^{6}} 
CoxeterDynkin diagrams 

7faces  16 {4,3^{5}} 
6faces  112 {4,3^{4}} 
5faces  448 {4,3^{3}} 
4faces  1120 {4,3^{2}} 
Cells  1792 {4,3} 
Faces  1792 {4} 
Edges  1024 
Vertices  256 
Vertex figure  7simplex 
Petrie polygon  hexadecagon 
Coxeter group  C_{8}, [3^{6},4] 
Dual  8orthoplex 
Properties  convex, Hanner polytope 
In geometry, an 8cube is an eightdimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4faces, 448 5cube 5faces, 112 6cube 6faces, and 16 7cube 7faces.
It is represented by Schläfli symbol {4,3^{6}}, being composed of 3 7cubes around each 6face. It is called an octeract, a portmanteau of tesseract (the 4cube) and oct for eight (dimensions) in Greek. It can also be called a regular hexdeca8tope or hexadecazetton, being an 8dimensional polytope constructed from 16 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8cube can be called an 8orthoplex and is a part of the infinite family of crosspolytopes.
Cartesian coordinates for the vertices of an 8cube 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}) with 1 < x_{i} < 1.
This configuration matrix represents the 8cube. The rows and columns correspond to vertices, edges, faces, cells, 4faces, 5faces, 6faces, and 7faces. The diagonal numbers say how many of each element occur in the whole 8cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
The diagonal fvector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.^{[3]}
B_{8}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  f_{4}  f_{5}  f_{6}  f_{7}  kfigure  notes  

A_{7}  ( )  f_{0}  256  8  28  56  70  56  28  8  {3,3,3,3,3,3}  B_{8}/A_{7} = 2^8*8!/8! = 256  
A_{6}A_{1}  { }  f_{1}  2  1024  7  21  35  35  21  7  {3,3,3,3,3}  B_{8}/A_{6}A_{1} = 2^8*8!/7!/2 = 1024  
A_{5}B_{2}  {4}  f_{2}  4  4  1792  6  15  20  15  6  {3,3,3,3}  B_{8}/A_{5}B_{2} = 2^8*8!/6!/4/2 = 1792  
A_{4}B_{3}  {4,3}  f_{3}  8  12  6  1792  5  10  10  5  {3,3,3}  B_{8}/A_{4}B_{3} = 2^8*8!/5!/8/3! = 1792  
A_{3}B_{4}  {4,3,3}  f_{4}  16  32  24  8  1120  4  6  4  {3,3}  B_{8}/A_{3}B_{4} = 2^8*8!/4!/2^4/4! = 1120  
A_{2}B_{5}  {4,3,3,3}  f_{5}  32  80  80  40  10  448  3  3  {3}  B_{8}/A_{2}B_{5} = 2^8*8!/3!/2^5/5! = 448  
A_{1}B_{6}  {4,3,3,3,3}  f_{6}  64  192  240  160  60  12  112  2  { }  B_{8}/A_{1}B_{6} = 2^8*8!/2/2^6/6!= 112  
B_{7}  {4,3,3,3,3,3}  f_{7}  128  448  672  560  280  84  14  16  ( )  B_{8}/B_{7} = 2^8*8!/2^7/7! = 16 
This 8cube graph is an orthogonal projection. This orientation 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 1:8:28:56:70:56:28:8:1. 
B_{8}  B_{7}  

[16]  [14]  
B_{6}  B_{5}  
[12]  [10]  
B_{4}  B_{3}  B_{2}  
[8]  [6]  [4]  
A_{7}  A_{5}  A_{3}  
[8]  [6]  [4] 
Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8simplex facets.
The 8cube is 8th in an infinite series of hypercube:
Line segment  Square  Cube  4cube  5cube  6cube  7cube  8cube  9cube  10cube 