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Hexicated 7-simplexes

{{Short description|Type of 7-polytope}}

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.

There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.

The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called an omnitruncated 7-simplex with all of the nodes ringed.

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|120px
7-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

|120px
Hexicated 7-simplex
{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}

|120px
Hexitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}

|120px
Hexicantellated 7-simplex
{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}

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|120px
Hexiruncinated 7-simplex
{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}

|120px
Hexicantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}

|120px
Hexiruncitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}

|120px
Hexiruncicantellated 7-simplex
{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}

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|120px
Hexisteritruncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}

|120px
Hexistericantellated 7-simplex
{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}

|120px
Hexipentitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}

|120px
Hexiruncicantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}

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|120px
Hexistericantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}

|120px
Hexisteriruncitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}

|120px
Hexisteriruncicantellated 7-simplex
{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}

|120px
Hexipenticantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}

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|120px
Hexipentiruncitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}

|120px
Hexisteriruncicantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}

|120px
Hexipentiruncicantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}

|120px
Hexipentistericantitruncated 7-simplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}

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|colspan=4|480px
Hexipentisteriruncicantitruncated 7-simplex
(Omnitruncated 7-simplex)
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}

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!colspan=4|Orthogonal projections in A7 Coxeter plane

Hexicated 7-simplex

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!bgcolor=#e7dcc3 colspan=2|Hexicated 7-simplex

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,6{36}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}
bgcolor=#e7dcc3|6-faces254:
8+8 {35} 40px
28+28 {}x{34}
56+56 {3}x{3,3,3}
70 {3,3}x{3,3}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges336
bgcolor=#e7dcc3|Vertices56
bgcolor=#e7dcc3|Vertex figure5-simplex antiprism
bgcolor=#e7dcc3|Coxeter groupA7×2, 36, order 80640
bgcolor=#e7dcc3|Propertiesconvex

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.

File:Ammann-Beenker_tiling_example.png.]]

= Root vectors =

Its 56 vertices represent the root vectors of the simple Lie group A7.

= Alternate names =

  • Expanded 7-simplex
  • Small petated hexadecaexon (Acronym: suph) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/suph.htm (x3o3o3o3o3o3x - suph)]}}

= Coordinates =

The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, {{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}.

A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:

: (1,-1,0,0,0,0,0,0)

= Images =

{{7-simplex2 Coxeter plane graphs|t06|150}}

Hexitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|hexitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges1848
style="background:#e7dcc3;"|Vertices336
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petitruncated octaexon (Acronym: puto) (Jonathan Bowers)Klitzing, (x3x3o3o3o3o3x- puto)

= Coordinates =

The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t016|150}}

Hexicantellated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexicantellated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges5880
style="background:#e7dcc3;"|Vertices840
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petirhombated octaexon (Acronym: puro) (Jonathan Bowers)Klitzing, (x3o3x3o3o3o3x - puro)

= Coordinates =

The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, {{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t026|150}}

Hexiruncinated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexiruncinated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,3,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges8400
style="background:#e7dcc3;"|Vertices1120
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petaprismated hexadecaexon (Acronym: puph) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/puph.htm (x3o3o3x3o3o3x - puph)]}}

= Coordinates =

The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, {{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t036|150}}

Hexicantitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexicantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges8400
style="background:#e7dcc3;"|Vertices1680
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petigreatorhombated octaexon (Acronym: pugro) (Jonathan Bowers)Klitzing, (x3o3o3o3x3o3x - pugro)

= Coordinates =

The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t0126|150}}

Hexiruncitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexiruncitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges20160
style="background:#e7dcc3;"|Vertices3360
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petiprismatotruncated octaexon (Acronym: pupato) (Jonathan Bowers)Klitzing, (x3x3x3o3o3o3x - pupato)

= Coordinates =

The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t0136|150}}

Hexiruncicantellated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexiruncicantellated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,3,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges16800
style="background:#e7dcc3;"|Vertices3360
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.

= Alternate names =

  • Petiprismatorhombated octaexon (Acronym: pupro) (Jonathan Bowers)Klitzing, (x3o3x3x3o3o3x - pupro)

= Coordinates =

The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, {{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t0236|150}}

Hexisteritruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|hexisteritruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,4,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges20160
style="background:#e7dcc3;"|Vertices3360
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Peticellitruncated octaexon (Acronym: pucto) (Jonathan Bowers)Klitzing, (x3x3o3o3x3o3x - pucto)

= Coordinates =

The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t0146|150}}

Hexistericantellated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|hexistericantellated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,4,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}
style="background:#e7dcc3;"|6-facest0,2,4{3,3,3,3,3}

{}xt0,2,4{3,3,3,3}

{3}xt0,2{3,3,3}

t0,2{3,3}xt0,2{3,3}

style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges30240
style="background:#e7dcc3;"|Vertices5040
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Peticellirhombihexadecaexon (Acronym: pucroh) (Jonathan Bowers)Klitzing, (x3o3x3o3x3o3x - pucroh)

= Coordinates =

The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, {{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t0246|150}}

Hexipentitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexipentitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,5,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges8400
style="background:#e7dcc3;"|Vertices1680
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petiteritruncated hexadecaexon (Acronym: putath) (Jonathan Bowers)Klitzing, (x3x3o3o3o3x3x - putath)

= Coordinates =

The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t0156|150}}

Hexiruncicantitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexiruncicantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges30240
style="background:#e7dcc3;"|Vertices6720
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petigreatoprismated octaexon (Acronym: pugopo) (Jonathan Bowers)Klitzing, (x3x3x3x3o3o3x - pugopo)

= Coordinates =

The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t01236|150}}

Hexistericantitruncated 7-simplex

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! style="background:#e7dcc3;" colspan="2"|Hexistericantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,4,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges50400
style="background:#e7dcc3;"|Vertices10080
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Peticelligreatorhombated octaexon (Acronym: pucagro) (Jonathan Bowers)Klitzing, (x3x3x3o3x3o3x - pucagro)

= Coordinates =

The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t01246|150}}

Hexisteriruncitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|Hexisteriruncitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,4,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges45360
style="background:#e7dcc3;"|Vertices10080
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Peticelliprismatotruncated octaexon (Acronym: pucpato) (Jonathan Bowers)Klitzing, (x3x3o3x3x3o3x - pucpato)

= Coordinates =

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t01346|150}}

Hexisteriruncicantellated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|Hexisteriruncicantellated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,3,4,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges45360
style="background:#e7dcc3;"|Vertices10080
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Peticelliprismatorhombihexadecaexon (Acronym: pucproh) (Jonathan Bowers)Klitzing, (x3o3x3x3x3o3x - pucproh)

= Coordinates =

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t02346|150}}

Hexipenticantitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|hexipenticantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,5,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges30240
style="background:#e7dcc3;"|Vertices6720
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petiterigreatorhombated octaexon (Acronym: putagro) (Jonathan Bowers)Klitzing, (x3x3x3o3o3x3x - putagro)

= Coordinates =

The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}.

= Images =

{{7-simplex Coxeter plane graphs|t01256|150}}

Hexipentiruncitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|Hexipentiruncitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3,5,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges
style="background:#e7dcc3;"|Vertices10080
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petiteriprismatotruncated hexadecaexon (Acronym: putpath) (Jonathan Bowers)Klitzing, (x3x3o3x3o3x3x - putpath)

= Coordinates =

The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t01356|150}}

Hexisteriruncicantitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|Hexisteriruncicantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,4,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges80640
style="background:#e7dcc3;"|Vertices20160
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petigreatocellated octaexon (Acronym: pugaco) (Jonathan Bowers)Klitzing, (x3x3x3x3x3o3x - pugaco)

= Coordinates =

The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t012346|150}}

Hexipentiruncicantitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|Hexipentiruncicantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,5,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges80640
style="background:#e7dcc3;"|Vertices20160
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7, [36], order 40320
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petiterigreatoprismated octaexon (Acronym: putgapo) (Jonathan Bowers)Klitzing, (x3x3x3x3o3x3x - putgapo)

= Coordinates =

The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t012356|150}}

Hexipentistericantitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:250px"

! style="background:#e7dcc3;" colspan="2"|Hexipentistericantitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,4,5,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges80640
style="background:#e7dcc3;"|Vertices20160
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Petitericelligreatorhombihexadecaexon (Acronym: putcagroh) (Jonathan Bowers)Klitzing, (x3x3x3o3x3x3x - putcagroh)

= Coordinates =

The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t012456|150}}

Omnitruncated 7-simplex

class="wikitable" style="float:right; margin-left:8px; width:280px"

! style="background:#e7dcc3;" colspan="2"|Omnitruncated 7-simplex

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3,4,5,6{36}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
style="background:#e7dcc3;"|6-faces254
style="background:#e7dcc3;"|5-faces5796
style="background:#e7dcc3;"|4-faces40824
style="background:#e7dcc3;"|Cells126000
style="background:#e7dcc3;"|Faces191520
style="background:#e7dcc3;"|Edges141120
style="background:#e7dcc3;"|Vertices40320
style="background:#e7dcc3;"|Vertex figureIrr. 6-simplex
style="background:#e7dcc3;"|Coxeter groupA7×2, 36, order 80640
style="background:#e7dcc3;"|Propertiesconvex

The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A7 symmetry of the regular 7-simplex. It can also be called the hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

= Permutohedron and related tessellation =

The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of {{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|split2|node_1}}.

= Alternate names =

  • Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/guph.htm (x3x3x3x3x3x3x - guph)]}}

= Coordinates =

The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,4,5,6,7). This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t0,1,2,3,4,5,6{36,4}, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}.

= Images =

{{7-simplex2 Coxeter plane graphs|t0123456|150}}

Related polytopes

The 20 polytopes presented in this article are a part of 71 uniform 7-polytopes with A7 symmetry shown in the table below.

{{Octaexon family}}

Notes

{{reflist}}

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 Ivic Weiss, Wiley-Interscience Publication, 1995, [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 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, PhD (1966)
  • {{KlitzingPolytopes|polyexa.htm|7D uniform polytopes (polyexa) with acronyms}} x3o3o3o3o3o3x - suph, x3x3o3o3o3o3x - puto, x3o3x3o3o3o3x - puro, x3o3o3x3o3o3x - puph, x3o3o3o3x3o3x - pugro, x3x3x3o3o3o3x - pupato, x3o3x3x3o3o3x - pupro, x3x3o3o3x3o3x - pucto, x3o3x3o3x3o3x - pucroh, x3x3o3o3o3x3x - putath, x3x3x3x3o3o3x - pugopo, x3x3x3o3x3o3x - pucagro, x3x3o3x3x3o3x - pucpato, x3o3x3x3x3o3x - pucproh, x3x3x3o3o3x3x - putagro, x3x3x3x3o3x3x - putpath, x3x3x3x3x3o3x - pugaco, x3x3x3x3o3x3x - putgapo, x3x3x3o3x3x3x - putcagroh, x3x3x3x3x3x3x - guph

{{sfn whitelist|CITEREFKlitzing}}

External links

  • [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:7-polytopes