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Demiocteract (8-demicube)
Petrie polygon projection
Type	Uniform 8-polytope
Family	demihypercube
Coxeter symbol	151
Schläfli symbols	{3,3} = h{4,3} s{2}
Coxeter diagrams	=
7-faces	144: 16 {3} 128 {3}
6-faces	112 {3} 1024 {3}
5-faces	448 {3} 3584 {3}
4-faces	1120 {3} 7168 {3,3,3}
Cells	10752: 1792 {3} 8960 {3,3}
Faces	7168 {3}
Edges	1792
Vertices	128
Vertex figure	Rectified 7-simplex
Symmetry group	D8, = A1,
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, and Schläfli symbol ${\displaystyle \left\{3{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}}$ or {3,3}.

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:

Images

orthographic projections

Coxeter plane	B8	D8	D7	D6	D5
Graph
Dihedral symmetry
Coxeter plane	D4	D3	A7	A5	A3
Graph
Dihedral symmetry

References

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 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, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)

External links

• Olshevsky, George. "Demiocteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary

• 8-polytopes