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	Template:CDD = Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
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, Template:CDD and Schläfli symbol $\left\{3\begin{array}{l}3,3,3,3,3\\ 3\end{array}\right\}$ or {3,3^5,1}.

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:

Template:CDD

References

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, 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, Template:ISBN [1]
    • (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, Template:ISBN (Chapter 26. pp. 409: Hemicubes: 1_n1)

External links

• Template:GlossaryForHyperspace
• Multi-dimensional Glossary