1 32 polytope

Last updated
Up2 3 21 t0 E7.svg
321
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t0 E7.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Up2 1 32 t0 E7.svg
132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 3 21 t1 E7.svg
Rectified 321
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 3 21 t2 E7.svg
birectified 321
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t1 E7.svg
Rectified 231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Up2 1 32 t1 E7.svg
Rectified 132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Orthogonal projections in E7 Coxeter plane

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

Contents

Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The rectified 132 is constructed by points at the mid-edges of the 132.

These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-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.pngCDel 3a.pngCDel nodea.png.

1_32 polytope

132
Type Uniform 7-polytope
Family 1k2 polytope
Schläfli symbol {3,33,2}
Coxeter symbol 132
Coxeter diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
6-faces182:
56 122 Gosset 1 22 polytope.svg
126 131 Demihexeract ortho petrie.svg
5-faces4284:
756 121 Demipenteract graph ortho.svg
1512 121 Demipenteract graph ortho.svg
2016 {34} 5-simplex t0.svg
4-faces23688:
4032 {33} 4-simplex t0.svg
7560 111 Cross graph 4.svg
12096 {33} 4-simplex t0.svg
Cells50400:
20160 {32} 3-simplex t0.svg
30240 {32} 3-simplex t0.svg
Faces40320 {3} 2-simplex t0.svg
Edges10080
Vertices576
Vertex figure t2{35} 6-simplex t2.svg
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png. It is the Voronoi cell of the dual E7* lattice. [1]

Alternate names

  • Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes. [2]
  • Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
  • Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers) [3]

Images

Coxeter plane projections
E7E6 / F4B7 / A6
Up2 1 32 t0 E7.svg
[18]
Up2 1 32 t0 E6.svg
[12]
Up2 1 32 t0 A6.svg
[7x2]
A5D7 / B6D6 / B5
Up2 1 32 t0 A5.svg
[6]
Up2 1 32 t0 D7.svg
[12/2]
Up2 1 32 t0 D6.svg
[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3
Up2 1 32 t0 D5.svg
[8]
Up2 1 32 t0 D4.svg
[6]
Up2 1 32 t0 D3.svg
[4]

Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-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.pngCDel 3a.pngCDel nodea.png

Removing the node on the end of the 2-length branch leaves the 6-demicube, 131, CDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Removing the node on the end of the 3-length branch leaves the 122, CDel nodea.pngCDel 3a.pngCDel 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 6-simplex, 032, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. [4]

E7CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngk-facefkf0f1f2f3f4f5f6 k-figures notes
A6CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png( )f0576352101402103510510521422177 2r{3,3,3,3,3} E7/A6 = 72*8!/7! = 576
A3A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x1.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png{ }f121008012121841212612343{3,3}x{3}E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3} f233403202316336132{ }∨{3}E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A3A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png {3,3} f346420160*13033031 {3}∨( ) E7/A3A2 = 72*8!/4!/3! = 20160
A3A1A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png464*3024002214122 Phyllic disphenoid E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A4A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 01r.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3,3,3} f451010504032**30030 {3} E7/A4A2 = 72*8!/5!/3! = 4032
D4A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png {3,3,4} 8243288*7560*12021 { }∨( ) E7/D4A1 = 72*8!/8/4!/2 = 7560
A4A1CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3,3} 5101005**1209602112E7/A4A1 = 72*8!/5!/2 = 12096
D5A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png h{4,3,3,3} f51680160804016100756**20{ }E7/D5A1 = 72*8!/16/5!/2 = 756
D5CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png1680160408001016*1512*11E7/D5 = 72*8!/16/5! = 1512
A5A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3,3,3,3} 61520015006**201602E7/A5A1 = 72*8!/6!/2 = 2016
E6CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3,32,2} f6727202160108010802162702162727056*( )E7/E6 = 72*8!/72/6! = 56
D6CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png h{4,3,3,3,3} 3224064016048006019201232*126E7/D6 = 72*8!/32/6! = 126

The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.

13k dimensional figures
SpaceFiniteEuclideanHyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1A5D6 E7 =E7+=E7++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel nodea.pngCDel 3a.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 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 nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry [3−1,3,1][30,3,1][31,3,1][32,3,1][[3<sup>3,3,1</sup>]][34,3,1]
Order 4872023,0402,903,040
Graph 5-simplex t0.svg Demihexeract ortho petrie.svg Up2 1 32 t0 E7.svg --
Name 13,-1 130 131 132 133 134
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

Rectified 1_32 polytope

Rectified 132
Type Uniform 7-polytope
Schläfli symbol t1{3,33,2}
Coxeter symbol 0321
Coxeter-Dynkin diagram CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
6-faces758
5-faces12348
4-faces72072
Cells191520
Faces241920
Edges120960
Vertices10080
Vertex figure {3,3}×{3}×{}
Coxeter group E7, [33,2,1], order 2903040
Properties convex

The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.

Alternate names

Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png, and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 122 polytope, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Removing the node on the end of the 2-length branch leaves the demihexeract, 131, CDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.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 tetrahedron-triangle duoprism prism, {3,3}×{3}×{}, CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 2.pngCDel nodea 1.pngCDel 2.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. [6]

E7CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngk-facefkf0f1f2f3f4f5f6 k-figures notes
A3A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png( )f010080242412368123618244121824126681263423{3,3}x{3}x{ }E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A1A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png{ }f121209602131263313663133621312( )v{3}v{ }E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A2A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 01 f23380640**1130013330033310311{3}v( )v( )E7/A2A2 = 72*8!/3!/3! = 80640
A2A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png33*40320*0203010603030601302{3}v{ }E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A2A1A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png33**1209600021201242112421212{ }v{ }v( )E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A3A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 02 f34640020160****13000033000310 {3}v( ) E7/A3A2 = 72*8!/4!/3! = 20160
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 011 612440*20160***10300030300301
A3A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png612404**60480**01120012210211 Sphenoid E7/A3A1 = 72*8!/4!/2 = 60480
A3A1A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png612044***30240*00202010401202 { }v{ } E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A3A1CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png0246004****6048000021101221112SphenoidE7/A3A1 = 72*8!/4!/2 = 60480
A4A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 021 f4103020100550004032*****30000300 {3} E7/A4A2 = 72*8!/5!/3! = 4032
A4A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png10302001050500*12096****12000210 { }v() E7/A4A1 = 72*8!/5!/2 = 12096
D4A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 0111 249632323208880**7560***10200201E7/D4A1 = 72*8!/8/4!/2 = 7560
A4CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png02110301002000505***24192**01110111 ( )v( )v( ) E7/A4 = 72*8!/5! = 34192
A4A1CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png10300102000055****12096*00201102 { }v() E7/A4A1 = 72*8!/5!/2 = 12096
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png03510001000005*****1209600021012
D5A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 0211 f580480320160160808080400161610000756****200{ }E7/D5A1 = 72*8!/16/5!/2 = 756
A5CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 022 20906006015030015060600*4032***110E7/A5 = 72*8!/6! = 4032
D5CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png021180480160160320040808080001016160**1512**101E7/D5 = 72*8!/16/5! = 1512
A5CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png0311560200600015030000606***4032*011E7/A5 = 72*8!/6! = 4032
A5A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png1560020600001530000066****2016002E7/A5A1 = 72*8!/6!/2 = 2016
E6CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 0221 f672064804320216043201080108021601080108021643227043221602772270056**( )E7/E6 = 72*8!/72/6! = 56
A6CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 1x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 032 352101400210350105010502104202107070*576*E7/A6 = 72*8!/7! = 576
D6CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 0311 240192064064019200160480480960006019219219200123232**126E7/D6 = 72*8!/32/6! = 126

Images

Coxeter plane projections
E7E6 / F4B7 / A6
Up2 1 32 t1 E7.svg
[18]
Up2 1 32 t1 E6.svg
[12]
Up2 1 32 t1 A6.svg
[14]
A5D7 / B6D6 / B5
Up2 1 32 t1 A5.svg
[6]
Up2 1 32 t1 D7.svg
[12/2]
Up2 1 32 t1 D6.svg
[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3
Up2 1 32 t1 D5.svg
[8]
Up2 1 32 t1 D4.svg
[6]
Up2 1 32 t1 D3.svg
[4]

See also

Notes

  1. The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine , Edward Pervin
  2. Elte, 1912
  3. Klitzing, (o3o3o3x *c3o3o3o - lin)
  4. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  5. Klitzing, (o3o3x3o *c3o3o3o - rolin)
  6. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

Related Research Articles

Uniform 4-polytope

In geometry, a uniform polychoron is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

Uniform 6-polytope

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> 22</sub> polytope

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 <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 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2k1 family.

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, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

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

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.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

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.

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

Simplectic honeycomb

In geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-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