1 22 polytope
122 |
Rectified 122 |
Birectified 122 |
221 |
Rectified 221 | |
Coxeter plane
|
---|
In 6-dimensional
Its
These polytopes are from a family of 39 convex
122 polytope
122 polytope | |
---|---|
Type | Uniform 6-polytope |
Family | 1k2 polytope |
Schläfli symbol | {3,32,2} |
Coxeter symbol | 122 |
Coxeter-Dynkin diagram |
or |
5-faces | 54: 27 |
4-faces | 702: 270 |
Cells | 2160: 1080 110 1080 {3,3} |
Faces | 2160 {3} |
Edges | 720 |
Vertices | 72 |
Vertex figure | |
Petrie polygon | Dodecagon |
Coxeter group | E6, [[3,32,2]], order 103680 |
Properties | isotopic
|
The 122 polytope contains 72 vertices, and 54
Alternate names
- Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)[2]
Images
E6 [12] |
D5 [8] |
D4 / A2 [6] | |
---|---|---|---|
(1,2) |
(1,3) |
(1,9,12) | |
B6 [12/2] |
A5 [6] |
A4 [[5]] = [10] |
A3 / D3 [4] |
(1,2) |
(2,3,6) |
(1,2) |
(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
Removing the node on either of 2-length branches leaves the 5-demicube, 131, .
The
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-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A5 | ( ) | f0 | 72 | 20 | 90 | 60 | 60 | 15 | 15 | 30 | 6 | 6 | r{3,3,3} |
E6/A5 = 72*6!/6! = 72 | |
A2A2A1 | { } | f1 | 2 | 720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}×{3} | E6/A2A2A1 = 72*6!/3!/3!/2 = 720 | |
A2A1A1 | {3} | f2 | 3 | 3 | 2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | s{2,4} |
E6/A2A1A1 = 72*6!/3!/2/2 = 2160 | |
A3A1 | {3,3} | f3 | 4 | 6 | 4 | 1080 | * | 1 | 0 | 2 | 2 | 1 | { }∨( ) | E6/A3A1 = 72*6!/4!/2 = 1080 | |
4 | 6 | 4 | * | 1080 | 0 | 1 | 2 | 1 | 2 | ||||||
A4A1 | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 0 | 216 | * | * | 2 | 0 | { } | E6/A4A1 = 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 |
Related complex polyhedron
The
Related polytopes and honeycomb
Along with the semiregular polytope,
1k2 figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
n | 3 | 4 | 5 | 6 | 7 | 8 | 9
|
10
| |||
Coxeter group |
E3=A2A1 | E4=A4 | E5=D5 | E6 | E7 | E8 | E9 = = E8+ | E10 = = E8++ | |||
Coxeter diagram |
|||||||||||
Symmetry (order) |
[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
E6/F4 Coxeter planes | |
---|---|
122 |
24-cell |
D4/B4 Coxeter planes | |
122 |
24-cell |
Tessellations
This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 222, .
Rectified 122 polytope
Rectified 122 | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | 2r{3,3,32,1} r{3,32,2} |
Coxeter symbol |
0221 |
Coxeter-Dynkin diagram |
or |
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
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.
E6 [12] |
D5 [8] |
D4 / A2 [6] |
B6 [12/2] |
---|---|---|---|
A5 [6] |
A4 [5] |
A3 / D3 [4] | |
Construction
Its construction is based on the
Removing the ring on the short branch leaves the
Removing the ring on the either 2-length branch leaves the
The
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-figure | notes | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A2A2A1 | ( ) | f0 | 720 | 18 | 18 | 18 | 9 | 6 | 18 | 9 | 6 | 9 | 6 | 3 | 6 | 9 | 3 | 2 | 3 | 3 | {3}×{3}×{ } |
E6/A2A2A1 = 72*6!/3!/3!/2 = 720 | |
A1A1A1 | { } | f1 | 2 | 6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }∨{ }∨( ) |
E6/A1A1A1 = 72*6!/2/2/2 = 6480 | |
A2A1 | {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 | |
3 | 3 | * | 4320 | * | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | ||||||
A2A1A1 | 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 | {3,3} | f3 | 4 | 6 | 4 | 0 | 0 | 1080 | * | * | * | * | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }∨( ) | E6/A2A1 = 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 | ||
A3A1 | 6 | 12 | 4 | 0 | 4 | * | * | 1080 | * | * | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | { }∨( ) | E6/A3A1 = 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 | |
A4A1 | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 216 | * | * | * | 0 | 2 | 0 | E6/A4A1 = 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 | |||
A4A1 | 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 | 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
Truncated 122 | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | t{3,32,2} |
Coxeter symbol |
t(122) |
Coxeter-Dynkin diagram |
or |
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
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
E6 [12] |
D5 [8] |
D4 / A2 [6] |
B6 [12/2] |
---|---|---|---|
A5 [6] |
A4 [5] |
A3 / D3 [4] | |
Birectified 122 polytope
Birectified 122 polytope | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | 2r{3,32,2} |
Coxeter symbol | 2r(122) |
Coxeter-Dynkin diagram |
or |
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.
E6 [12] |
D5 [8] |
D4 / A2 [6] |
B6 [12/2] |
---|---|---|---|
A5 [6] |
A4 [5] |
A3 / D3 [4] | |
Trirectified 122 polytope
Trirectified 122 polytope | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | 3r{3,32,2} |
Coxeter symbol | 3r(122) |
Coxeter-Dynkin diagram |
or |
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 or cacam) (Jonathan Bowers)[10]
See also
- List of E6 polytopes
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, 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
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
| |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron
|
Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: List of regular polytopes and compounds
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