5-orthoplex#Cartesian coordinates

• 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{{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/tac.htm (x3o3o3o4o - tac)]}}

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 ConfigurationsCoxeter, Complex Regular Polytopes, p.117

$\backslash begin\{bmatrix\}\backslash begin\{matrix\}$

10 & 8 & 24 & 32 & 16 \\

2 & 40 & 6 & 12 & 8 \\

3 & 3 & 80 & 4 & 4 \\

4 & 6 & 4 & 80 & 2 \\

5 & 10 & 10 & 5 & 32

\end{matrix}\end{bmatrix}

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)

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 [3^2,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

{{5-cube Coxeter plane graphs|t4|150}}

{{2 k1 polytopes}}

This polytope is one of 31 uniform 5-polytopes generated from the B₅ Coxeter plane, including the regular 5-cube and 5-orthoplex.

{{Penteract family}}

• 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 Ivic Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
• (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)
• {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera) with acronyms}} x3o3o3o4o - tac {{sfn whitelist| CITEREFKlitzing}}

• {{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
• [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
• [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:5-polytopes