1 22 polytope
Uniform 6-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]
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 :
122 polytope
The 122 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
D5
D4 / A2
B6
A5
A4
A3 / D3
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 ,
Removing the node on either of 2-length branches leaves the 5-demicube , 131 ,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex , 022 ,
Seen in a configuration matrix , the element counts can be derived by mirror removal and ratios of Coxeter group orders.[ 3]
E6
k-face
fk
f0
f1
f2
f3
f4
f5
k -figurenotes
A5
( )
f0
72 20
90
60
60
15
15
30
6
6
r{3,3,3} E6 /A5 = 72*6!/6! = 72
A2 A2 A1
{ }
f1
2
720 9
9
9
3
3
9
3
3
{3}×{3} E6 /A2 A2 A1 = 72*6!/3!/3!/2 = 720
A2 A1 A1
{3}
f2
3
3
2160 2
2
1
1
4
2
2
s{2,4} E6 /A2 A1 A1 = 72*6!/3!/2/2 = 2160
A3 A1
{3,3}
f3
4
6
4
1080 *
1
0
2
2
1
{ }∨( ) E6 /A3 A1 = 72*6!/4!/2 = 1080
4
6
4
*
1080 0
1
2
1
2
A4 A1
{3,3,3}
f4
5
10
10
5
0
216 *
*
2
0
{ }
E6 /A4 A1 = 72*6!/5!/2 = 216
5
10
10
0
5
*
216 *
0
2
D4
h{4,3,3}
8
24
32
8
8
*
*
270 1
1
E6 /D4 = 72*6!/8/4! = 270
D5
h{4,3,3,3}
f5
16
80
160
80
40
16
0
10
27 *
( )
E6 /D5 = 72*6!/16/5! = 27
16
80
160
40
80
0
16
10
*
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. The regular complex polyhedron 3 {3}3 {4}2 ,
C
2
{\displaystyle \mathbb {C} ^{2}}
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 rectification of the Hessian polyhedron , [ 4]
Along with the semiregular polytope, 221 uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures , defined by all permutations of rings in this Coxeter-Dynkin diagram :
1k2 figures in n dimensions
Space
Finite
Euclidean
Hyperbolic
n
3
4
5
6
7
8
9
10
Coxeter
E3 =A2 A1
E4 =A4
E5 =D5
E6
E7
E8
E9 =
E
~ ~ -->
8
{\displaystyle {\tilde {E}}_{8}}
8 +
E10 =
T
¯ ¯ -->
8
{\displaystyle {\bar {T}}_{8}}
8 ++
Coxeter
Symmetry [3−1,2,1 ]
[30,2,1 ]
[31,2,1 ]
[[32,2,1 ]]
[33,2,1 ]
[34,2,1 ]
[35,2,1 ]
[36,2,1 ]
Order
12
120
1,920
103,680
2,903,040
696,729,600
∞
Graph
-
-
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
122
D4/B4 Coxeter planes
122
Tessellations
This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222
Rectified 122 polytope
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.
Construction
Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope:
Removing the ring on the short branch leaves the birectified 5-simplex ,
Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: t2 (211 ) ,
The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism , {3}×{3}×{},
Seen in a configuration matrix , the element counts can be derived by mirror removal and ratios of Coxeter group orders.[ 7] [ 8]
E6
k-face
fk
f0
f1
f2
f3
f4
f5
k -figurenotes
A2 A2 A1
( )
f0
720 18
18
18
9
6
18
9
6
9
6
3
6
9
3
2
3
3
{3}×{3}×{ } E6 /A2 A2 A1 = 72*6!/3!/3!/2 = 720
A1 A1 A1
{ }
f1
2
6480 2
2
1
1
4
2
1
2
2
1
2
4
1
1
2
2
{ }∨{ }∨( ) E6 /A1 A1 A1 = 72*6!/2/2/2 = 6480
A2 A1
{3}
f2
3
3
4320 *
*
1
2
1
0
0
2
1
1
2
0
1
2
1
Sphenoid E6 /A2 A1 = 72*6!/3!/2 = 4320
3
3
*
4320 *
0
2
0
1
1
1
0
2
2
1
1
1
2
A2 A1 A1
3
3
*
*
2160 0
0
2
0
2
0
1
0
4
1
0
2
2
{ }∨{ } E6 /A2 A1 A1 = 72*6!/3!/2/2 = 2160
A2 A1
{3,3}
f3
4
6
4
0
0
1080 *
*
*
*
2
1
0
0
0
1
2
0
{ }∨( ) E6 /A2 A1 = 72*6!/3!/2 = 1080
A3
r{3,3}
6
12
4
4
0
*
2160 *
*
*
1
0
1
1
0
1
1
1
{3} E6 /A3 = 72*6!/4! = 2160
A3 A1
6
12
4
0
4
*
*
1080 *
*
0
1
0
2
0
0
2
1
{ }∨( ) E6 /A3 A1 = 72*6!/4!/2 = 1080
{3,3}
4
6
0
4
0
*
*
*
1080 *
0
0
2
0
1
1
0
2
r{3,3}
6
12
0
4
4
*
*
*
*
1080 0
0
0
2
1
0
1
2
A4
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
A4 A1
10
30
20
0
10
5
0
5
0
0
*
216 *
*
*
0
2
0
E6 /A4 A1 = 72*6!/5!/2 = 216
A4
10
30
10
20
0
0
5
0
5
0
*
*
432 *
*
1
0
1
E6 /A4 = 72*6!/5! = 432
D4
{3,4,3}
24
96
32
32
32
0
8
8
0
8
*
*
*
270 *
0
1
1
E6 /D4 = 72*6!/8/4! = 270
A4 A1
r{3,3,3}
10
30
0
20
10
0
0
0
5
5
*
*
*
*
216 0
0
2
E6 /A4 A1 = 72*6!/5!/2 = 216
A5
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
2r{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
80
480
160
320
160
0
80
40
80
80
0
0
16
10
16
*
*
27
Truncated 122 polytope
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:
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
Birectified 122 polytope
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.
Trirectified 122 polytope
Alternate names
Tricantellated 221
Trirectified pentacontitetrapeton (trim or cacam) (Jonathan Bowers)[ 10]
See also
Notes
^ Elte, 1912
^ Klitzing, (o3o3o3o3o *c3x - mo )
^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
^ Coxeter, H. S. M., Regular Complex Polytopes , second edition, Cambridge University Press, (1991). p.30 and p.47
^ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine , Edward Pervin^ Klitzing, (o3o3x3o3o *c3o - ram )
^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
^ Klitzing, Richard. "6D convex uniform polypeta o3o3x3o3o *c3o - ram" . ^ Klitzing, (o3x3o3x3o *c3o - barm )
^ Klitzing, (x3o3o3o3x *c3o - cacam
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, 1973Kaleidoscopes: 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)" .
