1 22 polytope

Last updated
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]

Contents

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, constructed 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 symbol122
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-faces54:
27 121 Demipenteract graph ortho.svg
27 121 Demipenteract graph ortho.svg
4-faces702:
270 111 Cross graph 4.svg
432 120 4-simplex t0.svg
Cells2160:
1080 110 3-simplex t0.svg
1080 {3,3} 3-simplex t0.svg
Faces2160 {3} 2-simplex t0.svg
Edges720
Vertices72
Vertex figure Birectified 5-simplex:
022 5-simplex t2.svg
Petrie polygon Dodecagon
Coxeter group E6, [[3,3<sup>2,2</sup>]], 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

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]

E6CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngk-facefkf0f1f2f3f4f5k-figurenotes
A5CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png( )f0722090606015153066 r{3,3,3} E6/A5 = 72*6!/6! = 72
A2A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x1.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png{ }f1272099933933 {3}×{3} E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A2A1A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3} f23321602211422 s{2,4} E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A3A1CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3} f34641080*10221 { }∨( ) 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.png464*108001212
A4A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3,3,3} f45101050216**20{ }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.png5101005*216*02
D4CDel 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} 8243288**27011E6/D4 = 72*6!/8/4! = 270
D5CDel 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} f5168016080401601027*( )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.png1680160408001610*27
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. Complex polyhedron 3-3-3-4-2.png
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]

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.

1k2 figures in n dimensions
SpaceFiniteEuclideanHyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1E4=A4E5=D5 E6 E7 E8 E9 = = E8+E10 = = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry
(order)
[3−1,2,1][30,2,1][31,2,1][[3<sup>2,2,1</sup>]][33,2,1][34,2,1][35,2,1][36,2,1]
Order 121201,920103,6802,903,040696,729,600
Graph Trigonal hosohedron.png 4-simplex t0.svg Demipenteract graph ortho.svg Up 1 22 t0 E6.svg Up2 1 32 t0 E7.svg Gosset 1 42 polytope petrie.svg --
Name 1−1,2 102 112 122 132 142 152 162

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-faces126
4-faces1566
Cells6480
Faces6480
Edges6480
Vertices720
Vertex figure 3-3 duoprism prism
Petrie polygon Dodecagon
Coxeter group E6, [[3,3<sup>2,2</sup>]], 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

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]

E6CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngk-facefkf0f1f2f3f4f5k-figurenotes
A2A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png( )f0720181818961896963693233 {3}×{3}×{ } E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A1A1A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png{ }f1264802211421221241122 { }∨{ }∨( ) E6/A1A1A1 = 72*6!/2/2/2 = 6480
A2A1CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3} f2334320**1210021120121 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.png33*4320*0201110221112
A2A1A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png33**21600020201041022 { }∨{ } E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3} f3464001080****21000120 { }∨( ) E6/A2A1 = 72*6!/3!/2 = 1080
A3CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png r{3,3} 612440*2160***10110111 {3} E6/A3 = 72*6!/4! = 2160
A3A1CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png612404**1080**01020021 { }∨( ) 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} 46040***1080*00201102
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} 612044****108000021012
A4CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png r{3,3,3} f410302010055000432****110{ }E6/A4 = 72*6!/5! = 432
A4A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png10302001050500*216***020E6/A4A1 = 72*6!/5!/2 = 216
A4CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png10301020005050**432**101E6/A4 = 72*6!/5! = 432
D4CDel 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} 249632323208808***270*011E6/D4 = 72*6!/8/4! = 270
A4A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png r{3,3,3} 10300201000055****216002E6/A4A1 = 72*6!/5!/2 = 216
A5CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 2r{3,3,3,3} f5209060600153001506060072**( )E6/A5 = 72*6!/6! = 72
D5CDel 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} 8048032016016080808004016160100*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.png8048016032016008040808000161016**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-faces72+27+27
4-faces32+216+432+270+216
Cells1080+2160+1080+1080+1080
Faces4320+4320+2160
Edges6480+720
Vertices1440
Vertex figure ( )v{3}x{3}
Petrie polygon Dodecagon
Coxeter group E6, [[3,3<sup>2,2</sup>]], order 103680
Properties convex

Alternate names

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 symbol2r(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-faces126
4-faces2286
Cells10800
Faces19440
Edges12960
Vertices2160
Vertex figure
Coxeter group E6, [[3,3<sup>2,2</sup>]], order 103680
Properties convex

Alternate names

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 symbol3r(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-faces558
4-faces4608
Cells8640
Faces6480
Edges2160
Vertices270
Vertex figure
Coxeter group E6, [[3,3<sup>2,2</sup>]], order 103680
Properties convex

Alternate names


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 Archived 2016-01-30 at the Wayback Machine , 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 - cacam

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In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

Gosset–Elte figures

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

Rectified 5-simplexes

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

2<sub> 31</sub> polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

1<sub> 32</sub> polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

1 <sub>42</sub> polytope

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

2<sub> 41</sub> polytope

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

2<sub> 21</sub> polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

3<sub> 21</sub> polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

4<sub> 21</sub> polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

Pentellated 6-simplexes

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

Rectified 5-orthoplexes

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

Rectified 5-cubes

In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

Rectified 7-simplexes Convex uniform 7-polytope in seven-dimensional geometry

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

Rectified 8-simplexes

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

References

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 polychoron Pentachoron 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