To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

1 22 polytope

From Wikipedia, the free encyclopedia

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

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:
27 121
Demipenteract graph ortho.svg

27 121
Demipenteract graph ortho.svg
4-faces 702:
270 111
Cross graph 4.svg

432 120
4-simplex t0.svg
Cells 2160:
1080 110
3-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

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

Images

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

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

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.
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

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

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

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

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

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

Images

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

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

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 ( )v{3}x{3}
Petrie polygon Dodecagon
Coxeter group E6, [[3,32,2]], order 103680
Properties convex

Alternate names

  • Truncated 122 polytope

Construction

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

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

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

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

Images

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

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

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


See also

Notes

  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[permanent dead link])

References

  • 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
This page was last edited on 22 February 2019, at 22:31
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.