Regular 5-orthoplexPentacrossOrthogonal projectioninside Petrie polygonTypeRegular 5-polytopeFamilyorthoplexSchläfli symbol {3,3,3,4}{3,3,31,1}Coxeter-Dynkin diagrams4-faces32 {3,3,3} Cells80 {3,3} Faces80 {3}Edges40Vertices10Vertex figure16-cellPetrie polygondecagonCoxeter groupsBC5, [3,3,3,4]D5, [32,1,1]Dual5-cubePropertiesconvex, Hanner polytope In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.
Alternate names Pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek. Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron). Acronym: tac
As a configuration This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-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
Cartesian coordinates Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are (±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)
Construction There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
NameCoxeter diagramSchläfli symbolSymmetryOrderVertex figure(s)regular 5-orthoplex{3,3,3,4}[3,3,3,4]3840Quasiregular 5-orthoplex{3,3,31,1}[3,3,31,1]19205-fusil{3,3,3,4}[4,3,3,3]3840{3,3,4}+{}[4,3,3,2]768{3,4}+{4}[4,3,2,4]384{3,4}+2{}[4,3,2,2]1922{4}+{}[4,2,4,2]128{4}+3{}[4,2,2,2]645{}[2,2,2,2]32
Other images
The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.
Related polytopes and honeycombs
This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.
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) x3o3o3o4o - tac
External links
Polytopes of Various Dimensions Multi-dimensional Glossary