Rectified 10-orthoplexes#Quadrirectified 10-orthoplex

In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex.

The rectified 10-orthoplex is the vertex figure of the demidekeractic honeycomb.

: {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

• Rectified decacross (Acronym: rake) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rake.htm (o3x3o3o3o3o3o3o3o4o - rake)]}}

There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C₁₀ or [4,3⁸] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D₁₀ or [3^7,1,1] Coxeter group.

Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length $\backslash sqrt\{2\}$ are all permutations of:

: (±1,±1,0,0,0,0,0,0,0,0)

Its 180 vertices represent the root vectors of the simple Lie group D₁₀. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B₁₀.

{{B10 Coxeter plane graphs|t8|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

• Birectified decacross (Acronym: brake) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/brake.htm (o3o3x3o3o3o3o3o3o4o - brake)]}}

Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length $\backslash sqrt\{2\}$ are all permutations of:

: (±1,±1,±1,0,0,0,0,0,0,0)

{{B10 Coxeter plane graphs|t7|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

• Trirectified decacross (Acronym: trake) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o3o3o3o4o - trake)

Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length $\backslash sqrt\{2\}$ are all permutations of:

: (±1,±1,±1,±1,0,0,0,0,0,0)

{{B10 Coxeter plane graphs|t6|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

• Quadrirectified decacross (Acronym: terake) (Jonthan Bowers)Klitzing, (o3o3o3o3x3o3o3o3o4o - terake)

Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length $\backslash sqrt\{2\}$ are all permutations of:

: (±1,±1,±1,±1,±1,0,0,0,0,0)

{{B10 Coxeter plane graphs|t6|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

{{reflist}}

• 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|polyxenna.htm|10D uniform polytopes (polyxenna) with acronyms}} x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker {{sfn whitelist| CITEREFKlitzing}}

• [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
• [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:10-polytopes