Demihexeract(6-demicube)Petrie polygon projectionTypeUniform 6-polytopeFamilydemihypercubeSchläfli symbol{3,33,1} = h{4,34}s{21,1,1,1,1}Coxeter diagrams = = Coxeter symbol1315-faces4412 {31,2,1} 32 {34} 4-faces25260 {31,1,1} 192 {33} Cells640480 {3,3} Faces640{3} Edges240Vertices32Vertex figureRectified 5-simplexSymmetry groupD6, [33,1,1] = [1+,4,34]+Petrie polygondecagonPropertiesconvex In geometry, a 6-demicube, demihexeract or hemihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. Acronym: hax.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM6 for a 6-dimensional half measure polytope.
Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol or {3,33,1}.
Cartesian coordinates Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: (±1,±1,±1,±1,±1,±1) with an odd number of plus signs.
As a configuration This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Coxeter, Regular Polytopes, p. 12, Section 1.8 Configurations
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
D6k-face fkf0f1f2f3f4f5 k-figure NotesA5 ( )f0 3215602060153066r{3,3,3,3} D6/A5 = 32·6!/6! = 32A3A1A1 { }f1 224084126842{}x{3,3} A3A2 {3}f2 33640133331{3}v( ) D6/A3A2 = 32·6!/4!/3! = 640A3A1 h{4,3}f3 464160*3030{3} D6/A3A1 = 32·6!/4!/2 = 160A3A2 {3,3}464*4801221{}v( ) D6/A3A2 = 32·6!/4!/3! = 480D4A1 h{4,3,3}f4 824328860*20{ } D6/D4A1 = 32·6!/8/4!/2 = 60A4 {3,3,3}5101005*19211 D6/A4 = 32·6!/5! = 192D5 h{4,3,3,3}f5 16801604080101612*( ) D6/D5 = 32·6!/16/5! = 12A5 {3,3,3,3} 6152001506*32 D6/A5 = 32·6!/6! = 32
Images
Related polytopes There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
Skew icosahedron Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.
References
H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 12, Section 1.8 Configurations,
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,
External links
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