2 21 polytope#Truncated 221 polytope

The 2₂₁ has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube.

For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The Schläfli graph is the 1-skeleton of this polytope.

• E. L. Elte named it V₂₇ (for its 27 vertices) in his 1912 listing of semiregular polytopes.Elte, 1912
• Icosihepta-heptacontadi-peton - 27-72 facetted polypeton (Acronym: jak) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/jak.htm (x3o3o3o3o *c3o - jak)]}}

The 27 vertices can be expressed in 8-space as an edge-figure of the 4₂₁ polytope:

(-2, 0, 0, 0,-2, 0, 0, 0),

( 0,-2, 0, 0,-2, 0, 0, 0),

( 0, 0,-2, 0,-2, 0, 0, 0),

( 0, 0, 0,-2,-2, 0, 0, 0),

( 0, 0, 0, 0,-2, 0, 0,-2),

( 0, 0, 0, 0, 0,-2,-2, 0)

( 2, 0, 0, 0,-2, 0, 0, 0),

( 0, 2, 0, 0,-2, 0, 0, 0),

( 0, 0, 2, 0,-2, 0, 0, 0),

( 0, 0, 0, 2,-2, 0, 0, 0),

( 0, 0, 0, 0,-2, 0, 0, 2)

(-1,-1,-1,-1,-1,-1,-1,-1),

(-1,-1,-1, 1,-1,-1,-1, 1),

(-1,-1, 1,-1,-1,-1,-1, 1),

(-1,-1, 1, 1,-1,-1,-1,-1),

(-1, 1,-1,-1,-1,-1,-1, 1),

(-1, 1,-1, 1,-1,-1,-1,-1),

(-1, 1, 1,-1,-1,-1,-1,-1),

( 1,-1,-1,-1,-1,-1,-1, 1),

( 1,-1, 1,-1,-1,-1,-1,-1),

( 1,-1,-1, 1,-1,-1,-1,-1),

( 1, 1,-1,-1,-1,-1,-1,-1),

(-1, 1, 1, 1,-1,-1,-1, 1),

( 1,-1, 1, 1,-1,-1,-1, 1),

( 1, 1,-1, 1,-1,-1,-1, 1),

( 1, 1, 1,-1,-1,-1,-1, 1),

( 1, 1, 1, 1,-1,-1,-1,-1)

Its construction is based on the E₆ group.

The facet information can be extracted from its Coxeter-Dynkin diagram, {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}.

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

Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (2₁₁), {{CDD|nodea|3a|branch|3a|nodea|3a|nodea_1}}.

Every simplex facet touches a 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube (1₂₁ polytope), {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1}}. The edge-figure is the vertex figure of the vertex figure, a rectified 5-cell, (0₂₁ polytope), {{CDD|nodea|3a|nodea|3a|branch_10}}.

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

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.

The 2₂₁ is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. 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 2₂₁.

This polytope can tessellate Euclidean 6-space, forming the 2₂₂ honeycomb with this Coxeter-Dynkin diagram: {{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}.

The regular complex polygon ₃{3}₃{3}₃, {{CDD|3node_1|3|3node|3|3node}}, in $\backslash mathbb\{C\}^2$ has a real representation as the 2₂₁ polytope, {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}}, in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its complex reflection group is ₃[3]₃[3]₃, order 648.

The 2₂₁ is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

{{k 21 polytopes}}

The 2₂₁ polytope is fourth in dimensional series 2_k2.

{{2 k1 polytopes}}

The 2₂₁ polytope is second in dimensional series 2_2k.

{{2 2k polytopes}}

The rectified 2₂₁ has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure is a rectified 5-cell prism.

• Rectified icosihepta-heptacontadi-peton as a rectified 27-72 facetted polypeton (Acronym: rojak) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rojak.htm (o3x3o3o3o *c3o - rojak)]}}

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|3a|nodea_1|3a|nodea}}.

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

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t₁(2₁₁), {{CDD|nodea|3a|branch|3a|nodea_1|3a|nodea}}.

Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (1₂₁), {{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1}}.

The vertex figure is determined by removing the ringed ring and ringing the neighboring ring. This makes rectified 5-cell prism, t₁{3,3,3}x{}, {{CDD|nodea|3a|nodea|3a|branch_10|2|nodea_1}}.

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

The truncated 2₂₁ has 432 vertices, 2376 edges, 5040 faces, 4320 cells, 1350 4-faces, and 126 5-faces. Its vertex figure is a rectified 5-cell pyramid.

• Truncated icosihepta-heptacontadi-peton as a truncated 27-72 facetted polypeton (Acronym: tojak){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/tojak.htm (x3x3o3o3o *c3o - tojak)]}}

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

• List of E6 polytopes

{{reflist}}

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• {{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}
• 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 17) Coxeter, The Evolution of Coxeter-Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233-248] See figure 1: (p. 232) (Node-edge graph of polytope)
• {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak, x3x3o3o3o *c3o - tojak {{sfn whitelist| CITEREFKlitzing}}

{{Polytopes}}

Category:6-polytopes