10-demicube

Last updated
Demidekeract
(10-demicube)
Demidekeract ortho petrie.svg
Petrie polygon projection
TypeUniform 10-polytope
Family demihypercube
Coxeter symbol 171
Schläfli symbol {31,7,1}
h{4,38}
s{21,1,1,1,1,1,1,1,1}
Coxeter diagram CDel nodes 10ru.pngCDel split2.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.png = CDel node h1.pngCDel 4.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 3.pngCDel node.png
CDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.png
9-faces53220 {31,6,1} Demienneract ortho petrie.svg
512 {38} 9-simplex t0.svg
8-faces5300180 {31,5,1} Demiocteract ortho petrie.svg
5120 {37} 8-simplex t0.svg
7-faces24000960 {31,4,1} Demihepteract ortho petrie.svg
23040 {36} 7-simplex t0.svg
6-faces648003360 {31,3,1} Demihexeract ortho petrie.svg
61440 {35} 6-simplex t0.svg
5-faces1155848064 {31,2,1} Demipenteract graph ortho.svg
107520 {34} 5-simplex t0.svg
4-faces14246413440 {31,1,1} Cross graph 4.svg
129024 {33} 4-simplex t0.svg
Cells12288015360 {31,0,1} 3-simplex t0.svg
107520 {3,3} 3-simplex t0.svg
Faces61440 {3} 2-simplex t0.svg
Edges11520
Vertices512
Vertex figure Rectified 9-simplex
Rectified 9-simplex.png
Symmetry group D10, [37,1,1] = [1+,4,38]
[29]+
Dual?
Properties convex

In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Contents

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM10 for a ten-dimensional half measure polytope.

Coxeter named this polytope as 171 from its Coxeter diagram, with a ring on one of the 1-length branches, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png and Schläfli symbol or {3,37,1}.

Cartesian coordinates

Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract:

(±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images

10-demicube graph.png
B10 coxeter plane
10-demicube.svg
D10 coxeter plane
(Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

A regular dodecahedron can be embedded as a regular skew polyhedron within the vertices in the 10-demicube, possessing the same symmetries as the 3-dimensional dodecahedron. [1]

References

  1. Deza, Michael; Shtogrin, Mikhael (1998). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices" . Advanced Studies in Pure Mathematics. Arrangements – Tokyo 1998: 77. doi: 10.2969/aspm/02710073 . ISBN   978-4-931469-77-8 . Retrieved 4 April 2020.
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