Demidekeract(10-demicube)Petrie polygon projectionTypeUniform 10-polytopeFamilydemihypercubeCoxeter symbol 171Schläfli symbol{31,7,1}h{4,38}s{21,1,1,1,1,1,1,1,1}Coxeter diagram = 9-faces53220 {31,6,1} 512 {38} 8-faces5300180 {31,5,1} 5120 {37} 7-faces24000960 {31,4,1} 23040 {36} 6-faces648003360 {31,3,1} 61440 {35} 5-faces1155848064 {31,2,1} 107520 {34} 4-faces14246413440 {31,1,1} 129024 {33} Cells122880107520 {3,3} Faces61440{3} Edges11520Vertices512Vertex figureRectified 9-simplex Symmetry groupD10, [37,1,1] = [1+,4,38]+Dual?Propertiesconvex 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 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 B10 Coxeter planeD10 Coxeter plane.Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8.
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), 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, (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, Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Multi-dimensional Glossary