10-cube

10-cube
Dekeract

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and central yellow one has four
TypeRegular 10-polytope
Familyhypercube
Schläfli symbol {4,38}
Coxeter-Dynkin diagram
9-faces20 {4,37}
8-faces180 {4,36}
7-faces960 {4,35}
6-faces3360 {4,34}
5-faces8064 {4,33}
4-faces13440 {4,3,3}
Cells15360 {4,3}
Faces11520 squares
Edges5120
Vertices1024
Vertex figure9-simplex
Petrie polygonicosagon
Coxeter groupC10, [38,4]
Dual10-orthoplex
Propertiesconvex

In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

It can be named by its Schläfli symbol {4,38}, being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the 4-cube) and deka- for ten (dimensions) in Greek, It can also be called an icosaxennon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates

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

(±1,±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, x9) with −1 < xi < 1.

Other images


This 10-cube graph is an orthogonal projection. This orientation 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 1:10:45:120:210:252:210:120:45:10:1.

Petrie polygon, skew orthogonal projection
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]

Derived polytopes

Applying an alternation operation, deleting alternating vertices of the dekeract, creates another uniform polytope, called a 10-demicube, (part of an infinite family called demihypercubes), which has 20 demienneractic and 512 enneazettonic facets.

References

External links

This article is issued from Wikipedia - version of the Saturday, October 24, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.