Cantic 7-cubeTruncated 7-demicubeD7 Coxeter plane projectionTypeuniform 7-polytopeSchläfli symbol t{3,34,1}h2{4,3,3,3,3,3}Coxeter diagram6-faces14 truncated 6-demicubes64 truncated 6-simplexes64 rectified 6-simplexes5-faces84 truncated 5-demicubes448 rectified 5-simplexes448 5-simplexes4-faces280 truncated 16-cells1344 truncated 5-cells1344 rectified 5-cells2688 5-cellsCells560 truncated tetrahedra2240 truncated tetrahedra2240 octahedra6720 tetrahedraFaces2240 hexagons2240 triangles8960 trianglesEdges672 segments6720 segmentsVertices1344Vertex figure( )v{ }x{3,3,3}Coxeter groupsD7, [34,1,1]Propertiesconvex In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.
A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes representing active mirrors. A demihypercube is an alternation of a hypercube.
Its 3-dimensional analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube.
Alternate names Truncated demihepteract Truncated hemihepteract (acronym: thesa) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6 are coordinate permutations: (±1,±1,±3,±3,±3,±3,±3) with an odd number of plus signs.
Images It can be visualized as a 2-dimensional orthogonal projections, for example the a D7 Coxeter plane, containing 12-gonal symmetry. Most visualizations in symmetric projections will contain overlapping vertices, so the colors of the vertices are changed based on how many vertices are at each projective position, here shown with red color for no overlaps.
Related polytopes
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
Notes
References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973 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] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. x3x3o *b3o3o3o3o – thesa
External links
Polytopes of Various Dimensions Multi-dimensional Glossary