1 22 polytope

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Up 1 22 t0 E6.svg
122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t1 E6.svg
Rectified 122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t2 E6.svg
Birectified 122
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Up 2 21 t0 E6.svg
221
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 2 21 t1 E6.svg
Rectified 221
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
orthogonal projections in E6 Coxeter plane

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).[1]

Its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, construcated by positions points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122.

These polytopes are from 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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

1_22 polytope[edit]

122 polytope
Type Uniform 6-polytope
Family 1k2 polytope
Schläfli symbol {3,32,2}
Coxeter symbol 122
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
5-faces 54:
27 121Demipenteract graph ortho.svg
27 121Demipenteract graph ortho.svg
4-faces 702:
270 111Cross graph 4.svg
432 1204-simplex t0.svg
Cells 2160:
1080 1103-simplex t0.svg
1080 {3,3}3-simplex t0.svg
Faces 2160 {3}2-simplex t0.svg
Edges 720
Vertices 72
Vertex figure Birectified 5-simplex:
022 5-simplex t2.svg
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
Properties convex, isotopic

The 1_22 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.

Alternate names[edit]

  • Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)[2]

Images[edit]

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
Up 1 22 t0 E6.svg
(1,2)
Up 1 22 t0 D5.svg
(1,3)
Up 1 22 t0 D4.svg
(1,9,12)
B6
[12/2]
A5
[6]
A4
[[5]] = [10]
A3 / D3
[4]
Up 1 22 t0 B6.svg
(1,2)
Up 1 22 t0 A5.svg
(2,3,6)
Up 1 22 t0 A4.svg
(1,2)
Up 1 22 t0 D3.svg
(1,6,8,12)

Construction[edit]

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, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on either of 2-length branches leaves the 5-demicube, 131, CDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring node, this makes the birectified 5-simplex, 022, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]

E6 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png k-face fk f0 f1 f2 f3 f4 f5 k-figure notes
A5 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png ( ) f0 72 20 90 60 60 15 15 30 6 6 r{3,3,3} E6/A5 = 72*6!/6! = 72
A2A2A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x1.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png { } f1 2 720 9 9 9 3 3 9 3 3 {3}x{3} E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A2A1A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3} f2 3 3 2160 2 2 1 1 4 2 2 s{2,4} E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A3A1 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3} f3 4 6 4 1080 * 1 0 2 2 1 { }v( ) E6/A3A1 = 72*6!/4!/2 = 1080
CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 4 6 4 * 1080 0 1 2 1 2
A4A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3,3,3} f4 5 10 10 5 0 216 * * 2 0 { } E6/A4A1 = 72*6!/5!/2 = 216
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 5 10 10 0 5 * 216 * 0 2
D4 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png h{4,3,3} 8 24 32 8 8 * * 270 1 1 E6/D4 = 72*6!/8/4! = 270
D5 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png h{4,3,3,3} f5 16 80 160 80 40 16 0 10 27 * ( ) E6/D5 = 72*6!/16/5! = 27
CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 16 80 160 40 80 0 16 10 * 27

Related complex polyhedron[edit]

Orthographic projection in Aut(E6) Coxeter plane with 18-gonal symmetry for complex polyhedron, 3{3}3{4}2. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces.

The regular complex polyhedron 3{3}3{4}2, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png, in 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 CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.png, as a rectification of the Hessian polyhedron, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png.[4]

Related polytopes and honeycomb[edit]

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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Geometric folding[edit]

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.

E6/F4 Coxeter planes
Up 1 22 t0 E6.svg
122
24-cell t3 F4.svg
24-cell
D4/B4 Coxeter planes
Up 1 22 t0 D4.svg
122
24-cell t3 B3.svg
24-cell

Tessellations[edit]

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

Rectified 1_22 polytope[edit]

Rectified 122
Type Uniform 6-polytope
Schläfli symbol 2r{3,3,32,1}
r{3,32,2}
Coxeter symbol 0221
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
or CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
5-faces 126
4-faces 1566
Cells 6480
Faces 6480
Edges 6480
Vertices 720
Vertex figure 3-3 duoprism prism
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

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).[5]

Alternate names[edit]

  • Birectified 221 polytope
  • Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)[6]

Images[edit]

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

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
Up 1 22 t1 E6.svg Up 1 22 t1 D5.svg Up 1 22 t1 D4.svg Up 1 22 t1 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 1 22 t1 A5.svg Up 1 22 t1 A4.svg Up 1 22 t1 D3.svg

Construction[edit]

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the short branch leaves the birectified 5-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: t2(211), CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

The vertex figure is determined by removing the ringed node and ringing the neighboring ring, this makes 3-3 duoprism prism, {3}×{3}×{}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[7][8]

E6 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png k-face fk f0 f1 f2 f3 f4 f5 k-figure notes
A2A2A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png ( ) f0 720 18 18 18 9 6 18 9 6 9 6 3 6 9 3 2 3 3 {3}x{3}x{ } E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A1A1A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png { } f1 2 6480 2 2 1 1 4 2 1 2 2 1 2 4 1 1 2 2 { }v{ }v( ) E6/A1A1A1 = 72*6!/2/2/2 = 6480
A2A1 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3} f2 3 3 4320 * * 1 2 1 0 0 2 1 1 2 0 1 2 1 Sphenoid E6/A2A1 = 72*6!/3!/2 = 4320
CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 3 3 * 4320 * 0 2 0 1 1 1 0 2 2 1 1 1 2
A2A1A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 3 3 * * 2160 0 0 2 0 2 0 1 0 4 1 0 2 2 { }∨{ } E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A2A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3} f3 4 6 4 0 0 1080 * * * * 2 1 0 0 0 1 2 0 { }v( ) E6/A2A1 = 72*6!/3!/2 = 1080
A3 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png r{3,3} 6 12 4 4 0 * 2160 * * * 1 0 1 1 0 1 1 1 {3} E6/A3 = 72*6!/4! = 2160
A3A1 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 6 12 4 0 4 * * 1080 * * 0 1 0 2 0 0 2 1 { }v( ) E6/A3A1 = 72*6!/4!/2 = 1080
CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3,3} 4 6 0 4 0 * * * 1080 * 0 0 2 0 1 1 0 2
CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png r{3,3} 6 12 0 4 4 * * * * 1080 0 0 0 2 1 0 1 2
A4 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png r{3,3,3} f4 10 30 20 10 0 5 5 0 0 0 432 * * * * 1 1 0 { } E6/A4 = 72*6!/5! = 432
A4A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 10 30 20 0 10 5 0 5 0 0 * 216 * * * 0 2 0 E6/A4A1 = 72*6!/5!/2 = 216
A4 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 10 30 10 20 0 0 5 0 5 0 * * 432 * * 1 0 1 E6/A4 = 72*6!/5! = 432
D4 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png h{4,3,3} 24 96 32 32 32 0 8 8 0 8 * * * 270 * 0 1 1 E6/D4 = 72*6!/8/4! = 270
A4A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png r{3,3,3} 10 30 0 20 10 0 0 0 5 5 * * * * 216 0 0 2 E6/A4A1 = 72*6!/5!/2 = 216
A5 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 2r{3,3,3,3} f5 20 90 60 60 0 15 30 0 15 0 6 0 6 0 0 72 * * ( ) E6/A5 = 72*6!/6! = 72
D5 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png rh{4,3,3,3} 80 480 320 160 160 80 80 80 0 40 16 16 0 10 0 * 27 * E6/D5 = 72*6!/16/5! = 27
CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 80 480 160 320 160 0 80 40 80 80 0 0 16 10 16 * * 27

Truncated 1_22 polytope[edit]

Truncated 122
Type Uniform 6-polytope
Schläfli symbol t{3,32,2}
Coxeter symbol t(122)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
or CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
5-faces 72+27+27
4-faces 32+216+432+270+216
Cells 1080+2160+1080+1080+1080
Faces 4320+4320+2160
Edges 6480+720
Vertices 1440
Vertex figure
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate names[edit]

  • Truncated 122 polytope

Construction[edit]

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Images[edit]

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

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
Up 1 22 t01 E6.svg Up 1 22 t01 D5.svg Up 1 22 t01 D4.svg Up 1 22 t01 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 1 22 t01 A5.svg Up 1 22 t01 A4.svg Up 1 22 t01 D3.svg

Birectified 1_22 polytope[edit]

Birectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 2r{3,32,2}
Coxeter symbol 2r(122)
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
or CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png
5-faces 126
4-faces 2286
Cells 10800
Faces 19440
Edges 12960
Vertices 2160
Vertex figure
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate names[edit]

  • Bicantellated 221
  • Birectified pentacontitetrapeton (barm) (Jonathan Bowers)[9]

Images[edit]

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

Coxeter plane orthographic projections
E6
[12]
D5
[8]
D4 / A2
[6]
B6
[12/2]
Up 1 22 t2 E6.svg Up 1 22 t2 D5.svg Up 1 22 t2 D4.svg Up 1 22 t2 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 1 22 t2 A5.svg Up 1 22 t2 A4.svg Up 1 22 t2 D3.svg

Trirectified 1_22 polytope[edit]

Trirectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 3r{3,32,2}
Coxeter symbol 3r(122)
Coxeter-Dynkin diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
or CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png
5-faces 558
4-faces 4608
Cells 8640
Faces 6480
Edges 2160
Vertices 270
Vertex figure
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate names[edit]

  • Tricantellated 221
  • Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)[10]


See also[edit]

Notes[edit]

  1. ^ Elte, 1912
  2. ^ Klitzing, (o3o3o3o3o *c3x - mo)
  3. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  4. ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
  5. ^ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
  6. ^ Klitzing, (o3o3x3o3o *c3o - ram)
  7. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  8. ^ Klitzing, Richard. "6D convex uniform polypeta o3o3x3o3o *c3o - ram". 
  9. ^ Klitzing, (o3x3o3x3o *c3o - barm)
  10. ^ Klitzing, (x3o3o3o3x *c3o - trim)

References[edit]

  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen 
  • 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, ISBN 978-0-471-01003-6 [1]
    • (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)
  • Klitzing, Richard. "6D uniform polytopes (polypeta)".  o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds