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.
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, and Schläfli symbol




{

3




3
,
3
,
3
,
3
,
3
,
3
,
3




3





}



{\displaystyle \left\{3{\begin{array}{l}3,3,3,3,3,3,3\\3\end{array}}\right\}}

or {3,37,1}.Acronym: hede

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

Related polytopes
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.

References

H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, pp. 294–295, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), ISBN 0-486-61480-8
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
(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: 1n1, ISBN 978-1-56881-220-5
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
Klitzing, Richard. "10D uniform polytopes (polyxenna) x3o3o *b3o3o3o3o3o3o3o - hede".

External links
Olshevsky, George. "Demienneract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Multi-dimensional Glossary