8-demicube

Demiocteract (8-demicube)
File:Demiocteract ortho petrie.svg Petrie polygon projection
Type	Uniform 8-polytope
Family	demihypercube
Coxeter symbol	151
Schläfli symbols	{3,35,1} = h{4,36} s{21,1,1,1,1,1,1}
Coxeter diagrams	File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.png
7-faces	144: 16 {31,4,1} File:Demihepteract ortho petrie.svg 128 {36} File:7-simplex t0.svg
6-faces	112 {31,3,1} File:Demihexeract ortho petrie.svg 1024 {35} File:6-simplex t0.svg
5-faces	448 {31,2,1} File:Demipenteract graph ortho.svg 3584 {34} File:5-simplex t0.svg
4-faces	1120 {31,1,1} File:Cross graph 4.svg 7168 {3,3,3} File:4-simplex t0.svg
Cells	10752: 1792 {31,0,1} File:3-simplex t0.svg 8960 {3,3} File:3-simplex t0.svg
Faces	7168 {3} File:2-simplex t0.svg
Edges	1792
Vertices	128
Vertex figure	Rectified 7-simplex File:7-simplex t1.svg
Symmetry group	D8, [35,1,1] = [1+,4,36] A18, [27]+
Dual	?
Properties	convex

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₈ for an 8-dimensional half measure polytope.

Coxeter named this polytope as 1₅₁ from its Coxeter diagram, with a ring on one of the 1-length branches, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png and Schläfli symbol $\left\{3\begin{array}{c}3,3,3,3,3\\ 3\end{array}\right\}$ or {3,3^5,1}.

Acronym: hocto (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube:

(±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Related polytopes and honeycombs

This polytope is the vertex figure for the uniform tessellation, 2₅₁ with Coxeter-Dynkin diagram:

File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

Images

orthographic projections

Coxeter plane	B8	D8	D7	D6	D5
Graph	File:8-demicube t0 B8.svg	File:8-demicube t0 D8.svg	File:8-demicube t0 D7.svg	File:8-demicube t0 D6.svg
Dihedral symmetry	[16/2]	[14]	[12]	[10]	[8]
Coxeter plane	D4	D3	A7	A5	A3
Graph		File:8-demicube t0 D3.svg	File:8-demicube t0 A7.svg	File:8-demicube t0 A5.svg	File:8-demicube t0 A3.svg
Dihedral symmetry	[6]	[4]	[8]	[6]	[4]

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5),
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
    • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Chapter 26, p. 409, Hemicubes: 1_n1,
• x3o3o *b3o3o3o3o3o - hocto

External links

• Multi-dimensional Glossary