Last updated
Orthogonal projection
inside Petrie polygon
TypeRegular 10-polytope
Family Orthoplex
Schläfli symbol {38,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
9-faces1024 {38} 9-simplex t0.svg
8-faces5120 {37} 8-simplex t0.svg
7-faces11520 {36} 7-simplex t0.svg
6-faces15360 {35} 6-simplex t0.svg
5-faces13440 {34} 5-simplex t0.svg
4-faces8064 {33} 4-simplex t0.svg
Cells3360 {3,3} 3-simplex t0.svg
Faces960 {3} 2-simplex t0.svg
Vertex figure 9-orthoplex
Petrie polygon Icosagon
Coxeter groups C10, [38,4]
D10, [37,1,1]
Dual 10-cube
Properties Convex, Hanner polytope

In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces.


It has two constructed forms, the first being regular with Schläfli symbol {38,4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {37,31,1} or Coxeter symbol 711.

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.

Alternate names


There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or [4,38] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or [37,1,1] symmetry group.

Cartesian coordinates

Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are

(±1,0,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0,0), (0,0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.


Orthographic projections
10-cube t9.svg 10-cube t9 B9.svg 10-cube t9 B8.svg
10-cube t9 B7.svg 10-cube t9 B6.svg 10-cube t9 B5.svg
10-cube t9 B4.svg 10-cube t9 B3.svg 10-cube t9 B2.svg

Related Research Articles

<span class="mw-page-title-main">Schläfli symbol</span> Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

<span class="mw-page-title-main">5-polytope</span> 5-dimensional geometric object

In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

<span class="mw-page-title-main">5-demicube</span>

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

<span class="mw-page-title-main">5-orthoplex</span>

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

<span class="mw-page-title-main">6-orthoplex</span>

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

<span class="mw-page-title-main">7-cube</span> 7-dimensional hypercube

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

<span class="mw-page-title-main">8-cube</span> 8-dimensional hypercube

In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

<span class="mw-page-title-main">9-cube</span> 9-dimensional hypercube

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

<span class="mw-page-title-main">7-orthoplex</span>

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

<span class="mw-page-title-main">8-orthoplex</span>

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

<span class="mw-page-title-main">9-orthoplex</span>

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

<span class="mw-page-title-main">10-cube</span> 10-dimensional hypercube

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.

<span class="mw-page-title-main">Uniform 10-polytope</span>

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

<span class="mw-page-title-main">Rectified 5-orthoplexes</span>

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

<span class="mw-page-title-main">Rectified 6-orthoplexes</span>

In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

<span class="mw-page-title-main">6-polytope</span>

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.


Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds