6-orthoplexHexacrossOrthogonal projectioninside Petrie polygonTypeRegular 6-polytopeFamilyorthoplexSchläfli symbols {3,3,3,3,4}{3,3,3,31,1}Coxeter-Dynkin diagrams = 5-faces64 {34} 4-faces192 {33} Cells240 {3,3} Faces160 {3} Edges60Vertices12Vertex figure5-orthoplexPetrie polygondodecagonCoxeter groupsB6, [4,34]D6, [33,1,1]Dual6-cubePropertiesconvex, Hanner polytope In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.
It has two constructed forms, the first being regular with Schläfli symbol {34,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,31,1} or Coxeter symbol 311.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.
Alternate names Hexacross, derived from combining the family name cross polytope with hex for six (dimensions) in Greek. Hexacontatetrapeton as a 64-facetted 6-polytope. Acronym: gee (Jonathan Bowers)
As a configuration This configuration matrix represents the 6-orthoplex. 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-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Coxeter, Regular Polytopes, sec 1.8 Configurations
Construction There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.
NameCoxeterSchläfliSymmetryOrderRegular 6-orthoplex{3,3,3,3,4}[4,3,3,3,3]46080Quasiregular 6-orthoplex{3,3,3,31,1}[3,3,3,31,1]230406-fusil{3,3,3,4}+{}[4,3,3,3,3]7680{3,3,4}+{4}[4,3,3,2,4]30722{3,4}[4,3,2,4,3]2304{3,3,4}+2{}[4,3,3,2,2]1536{3,4}+{4}+{}[4,3,2,4,2]7683{4}[4,2,4,2,4]512{3,4}+3{}[4,3,2,2,2]3842{4}+2{}[4,2,4,2,2]256{4}+4{}[4,2,2,2,2]1286{}[2,2,2,2,2]64
Cartesian coordinates Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are (±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
Images
Related polytopes The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.Quasicrystals and Geometry, Marjorie Senechal, 1996, Cambridge University Press, p. 64. 2.7.1 The I6 crystal 2D3DIcosahedron{3,5} = H3 Coxeter plane6-orthoplex{3,3,3,31,1} = D6 Coxeter planeIcosahedron6-orthoplexThis construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. This represents a geometric folding of the D6 to H3 Coxeter groups: : to . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron.
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including also the regular 6-cube.
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. 1966 x3o3o3o3o4o - gee
External links
Polytopes of Various Dimensions Multi-dimensional Glossary