1 22 polytope#Related complex polyhedron

The 1₂₂ polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E₆.

• Pentacontatetrapeton (Acronym: mo) - 54-facetted polypeton (Jonathan Bowers)Klitzing, (o3o3o3o3o *c3x - [http://bendwavy.org/klitzing/incmats/mo.htm mo])

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}.

Removing the node on either of 2-length branches leaves the 5-demicube, 1₃₁, {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0₂₂, {{CDD|node|3|node|3|node_1|3|node|3|node}}.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203

File:Complex polyhedron 3-3-3-4-2.png

The regular complex polyhedron ₃{3}₃{4}₂, {{CDD|3node_1|3|3node|4|node}}, in $\backslash mathbb\{C\}^2$ has a real representation as the 1₂₂ polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is ₃[3]₃[4]₂, order 1296. It has a half-symmetry quasiregular construction as {{CDD|3node|3|3node_1|3|3node}}, as a rectification of the Hessian polyhedron, {{CDD|3node_1|3|3node|3|3node}}. Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47

Along with the semiregular polytope, 2₂₁, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

{{1 k2 polytopes}}

The 1₂₂ is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 1₂₂ in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1₂₂.

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 2₂₂, {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}.

The rectified 1₂₂ polytope (also called 0₂₂₁) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).[http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive|url=https://web.archive.org/web/20160130193811/http://home.digital.net/~pervin/publications/vermont.html |date=2016-01-30 }}, Edward Pervin

• Birectified 2₂₁ polytope
• Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers)Klitzing, (o3o3x3o3o *c3o - [http://bendwavy.org/klitzing/incmats/ram.htm ram])

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: {{CDD|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}.

Removing the ring on the short branch leaves the birectified 5-simplex, {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea}}.

Removing the ring on either of 2-length branches leaves the birectified 5-orthoplex in its alternated form: t₂(2₁₁), {{CDD|nodea|3a|branch_10|3a|nodea|3a|nodea}}.

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, {{CDD|node|3|node_1|2|node_1|2|node_1|3|node}}.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.{{r|ram}}

• Truncated 1₂₂ polytope (Acronym: tim)Klitzing, (o3o3x3o3o *c3x - [http://bendwavy.org/klitzing/incmats/tim.htm tim])

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: {{CDD|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}.

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

• Bicantellated 2₂₁
• Birectified pentacontatetrapeton (barm) (Jonathan Bowers)Klitzing, (o3x3o3x3o *c3o - [http://bendwavy.org/klitzing/incmats/scram.htm barm])

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

• Tricantellated 2₂₁
• Trirectified pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)Klitzing, (x3o3o3o3x *c3o - [http://bendwavy.org/klitzing/incmats/cacam.htm trim])

• List of E6 polytopes

{{reflist}}

• {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
• 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1₂₂)
• {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim

{{Polytopes}}

Category:6-polytopes