1 42 polytope
In 8-dimensional geometry , the 142 is a uniform 8-polytope , constructed within the symmetry of the E8 group.
Its Coxeter symbol is 142 , describing its bifurcating Coxeter-Dynkin diagram , with a single ring on the end of the 1-node sequences.
The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241 , and the quadrirectified 421 .
These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex figures , defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram :
142 polytope
The 142 is composed of 2400 facets: 240 132 7-demicubes (141 ). Its vertex figure is a birectified 7-simplex .
This polytope, along with the demiocteract , can tessellate 8-dimensional space, represented by the symbol 152 , and Coxeter-Dynkin diagram:
Alternate names
E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V17280 for its 17280 vertices.[ 1] Coxeter named it 142 for its bifurcating Coxeter-Dynkin diagram , with a single ring on the end of the 1-node branch.Diacositetracont-dischiliahectohexaconta-zetton (acronym bif ) - 240-2160 facetted polyzetton (Jonathan Bowers)[ 2]
Coordinates
The 17280 vertices can be defined as sign and location permutations of:
All sign combinations (32): (280×32=8960 vertices)
(4, 2, 2, 2, 2, 0, 0, 0) Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)
(2, 2, 2, 2, 2, 2, 2, 2)
(5, 1, 1, 1, 1, 1, 1, 1)
(3, 3, 3, 1, 1, 1, 1, 1) The edge length is 2√2 in this coordinate set, and the polytope radius is 4√2 .
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram :
Removing the node on the end of the 2-length branch leaves the 7-demicube , 141 ,
Removing the node on the end of the 4-length branch leaves the 132 ,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex , 042 ,
Seen in a configuration matrix , the element counts can be derived by mirror removal and ratios of Coxeter group orders.[ 3]
Configuration matrix
E8
k -facefk
f0
f1
f2
f3
f4
f5
f6
f7
k -figurenotes
A7
( )
f0
17280 56
420
280
560
70
280
420
56
168
168
28
56
28
8
8
2r{36 } E8 /A7 = 192*10!/8! = 17280
A4 A2 A1
{ }
f1
2
483840 15
15
30
5
30
30
10
30
15
10
15
3
5
3
{3}x{3,3,3}
E8 /A4 A2 A1 = 192*10!/5!/2/2 = 483840
A3 A2 A1
{3}
f2
3
3
2419200 2
4
1
8
6
4
12
4
6
8
1
4
2
{3.3}v{ }
E8 /A3 A2 A1 = 192*10!/4!/3!/2 = 2419200
A3 A3
110
f3
4
6
4
1209600 *
1
4
0
4
6
0
6
4
0
4
1
{3,3}v( ) E8 /A3 A3 = 192*10!/4!/4! = 1209600
A3 A2 A1
4
6
4
*
2419200 0
2
3
1
6
3
3
6
1
3
2
{3}v{ }
E8 /A3 A2 A1 = 192*10!/4!/3!/2 = 2419200
A4 A3
120
f4
5
10
10
5
0
241920 *
*
4
0
0
6
0
0
4
0
{3,3} E8 /A4 A3 = 192*10!/4!/4! = 241920
D4 A2
111
8
24
32
8
8
*
604800 *
1
3
0
3
3
0
3
1
{3}v( ) E8 /D4 A2 = 192*10!/8/4!/3! = 604800
A4 A1 A1
120
5
10
10
0
5
*
*
1451520 0
2
2
1
4
1
2
2
{ }v{ } E8 /A4 A1 A1 = 192*10!/5!/2/2 = 1451520
D5 A2
121
f5
16
80
160
80
40
16
10
0
60480 *
*
3
0
0
3
0
{3} E8 /D5 A2 = 192*10!/16/5!/3! = 40480
D5 A1
16
80
160
40
80
0
10
16
*
181440 *
1
2
0
2
1
{ }v( ) E8 /D5 A1 = 192*10!/16/5!/2 = 181440
A5 A1
130
6
15
20
0
15
0
0
6
*
*
483840 0
2
1
1
2
E8 /A5 A1 = 192*10!/6!/2 = 483840
E6 A1
122
f6
72
720
2160
1080
1080
216
270
216
27
27
0
6720 *
*
2
0
{ }
E8 /E6 A1 = 192*10!/72/6!/2 = 6720
D6
131
32
240
640
160
480
0
60
192
0
12
32
*
30240 *
1
1
E8 /D6 = 192*10!/32/6! = 30240
A6 A1
140
7
21
35
0
35
0
0
21
0
0
7
*
*
69120 0
2
E8 /A6 A1 = 192*10!/7!/2 = 69120
E7
132
f7
576
10080
40320
20160
30240
4032
7560
12096
756
1512
2016
56
126
0
240 *
( )
E8 /E7 = 192*10!/72/8! = 240
D7
141
64
672
2240
560
2240
0
280
1344
0
84
448
0
14
64
*
2160 E8 /D7 = 192*10!/64/7! = 2160
Projections
The projection of 142 to the E8 Coxeter plane (aka. the Petrie projection) with polytope radius
4
2
{\displaystyle 4{\sqrt {2}}}
2
2
{\displaystyle 2{\sqrt {2}}}
Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry: u = (1, φ , 0, −1, φ , 0,0,0) v = (φ , 0, 1, φ , 0, −1,0,0) w = (0, 1, φ , 0, −1, φ ,0,0) 42 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. Notice the last two outer hulls are a combination of two overlapped Dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).
E8
E7
E6
[20]
[24]
[6]
Orthographic projections are shown for the sub-symmetries of E8 : E7 , E6 , B8 , B7 , B6 , B5 , B4 , B3 , B2 , A7 , and A5 Coxeter planes , as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.
D3 / B2 / A3
D4 / B3 / A2
D5 / B4
D6 / B5 / A4
D7 / B6
D8 / B7 / A6
B8
A5
A7
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
Rectified 142 polytope
The rectified 142 is named from being a rectification of the 142 polytope, with vertices positioned at the mid-edges of the 142 . It can also be called a 0421 polytope with the ring at the center of 3 branches of length 4, 2, and 1.
Alternate names
0421 polytope
Birectified 241 polytope
Quadrirectified 421 polytope
Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym buffy ) (Jonathan Bowers)[ 4]
Construction
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram :
Removing the node on the end of the 1-length branch leaves the birectified 7-simplex ,
Removing the node on the end of the 2-length branch leaves the birectified 7-cube ,
Removing the node on the end of the 3-length branch leaves the rectified 132 ,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell -triangle duoprism prism,
Seen in a configuration matrix , the element counts can be derived by mirror removal and ratios of Coxeter group orders.[ 3]
Configuration matrix
E8
k -facefk
f0
f1
f2
f3
f4
f5
f6
f7
k -figure
A4 A2 A1
( )
f0
483840 30
30
15
60
10
15
60
30
60
5
20
30
60
30
30
10
20
30
30
15
6
10
10
15
6
3
5
2
3
{3,3,3}x{3,3}x{}
A3 A1 A1
{ }
f1
2
7257600 2
1
4
1
2
8
4
6
1
4
8
12
6
4
4
6
12
8
4
1
6
4
8
2
1
4
1
2
A3 A2
{3}
f2
3
3
4838400 *
*
1
1
4
0
0
1
4
4
6
0
0
4
6
6
4
0
0
6
4
4
1
0
4
1
1
A3 A2 A1
3
3
*
2419200 *
0
2
0
4
0
1
0
8
0
6
0
4
0
12
0
4
0
6
0
8
0
1
4
0
2
A2 A2 A1
3
3
*
*
9676800 0
0
2
1
3
0
1
2
6
3
3
1
3
6
6
3
1
3
3
6
2
1
3
1
2
A3 A3
0200
f3
4
6
4
0
0
1209600 *
*
*
*
1
4
0
0
0
0
4
6
0
0
0
0
6
4
0
0
0
4
1
0
0110
6
12
4
4
0
*
1209600 *
*
*
1
0
4
0
0
0
4
0
6
0
0
0
6
0
4
0
0
4
0
1
A3 A2
6
12
4
0
4
*
*
4838400 *
*
0
1
1
3
0
0
1
3
3
3
0
0
3
3
3
1
0
3
1
1
A3 A2 A1
6
12
0
4
4
*
*
*
2419200 *
0
0
2
0
3
0
1
0
6
0
3
0
3
0
6
0
1
3
0
2
A3 A1 A1
0200
4
6
0
0
4
*
*
*
*
7257600 0
0
0
2
1
2
0
1
2
4
2
1
1
2
4
2
1
2
1
2
A4 A3
0210
f4
10
30
20
10
0
5
5
0
0
0
241920 *
*
*
*
*
4
0
0
0
0
0
6
0
0
0
0
4
0
0
A4 A2
10
30
20
0
10
5
0
5
0
0
*
967680 *
*
*
*
1
3
0
0
0
0
3
3
0
0
0
3
1
0
D4 A2
0111
24
96
32
32
32
0
8
8
8
0
*
*
604800 *
*
*
1
0
3
0
0
0
3
0
3
0
0
3
0
1
A4 A1
0210
10
30
10
0
20
0
0
5
0
5
*
*
*
2903040 *
*
0
1
1
2
0
0
1
2
2
1
0
2
1
1
A4 A1 A1
10
30
0
10
20
0
0
0
5
5
*
*
*
*
1451520 *
0
0
2
0
2
0
1
0
4
0
1
2
0
2
A4 A1
0300
5
10
0
0
10
0
0
0
0
5
*
*
*
*
*
2903040 0
0
0
2
1
1
0
1
2
2
1
1
1
2
D5 A2
0211
f5
80
480
320
160
160
80
80
80
40
0
16
16
10
0
0
0
60480 *
*
*
*
*
3
0
0
0
0
3
0
0
{3}
A5 A1
0220
20
90
60
0
60
15
0
30
0
15
0
6
0
6
0
0
*
483840 *
*
*
*
1
2
0
0
0
2
1
0
{ }v()
D5 A1
0211
80
480
160
160
320
0
40
80
80
80
0
0
10
16
16
0
*
*
181440 *
*
*
1
0
2
0
0
2
0
1
A5
0310
15
60
20
0
60
0
0
15
0
30
0
0
0
6
0
6
*
*
*
967680 *
*
0
1
1
1
0
1
1
1
( )v( )v()
A5 A1
15
60
0
20
60
0
0
0
15
30
0
0
0
0
6
6
*
*
*
*
483840 *
0
0
2
0
1
1
0
2
{ }v()
0400
6
15
0
0
20
0
0
0
0
15
0
0
0
0
0
6
*
*
*
*
*
483840 0
0
0
2
1
0
1
2
E6 A1
0221
f6
720
6480
4320
2160
4320
1080
1080
2160
1080
1080
216
432
270
432
216
0
27
72
27
0
0
0
6720 *
*
*
*
2
0
0
{ }
A6
0320
35
210
140
0
210
35
0
105
0
105
0
21
0
42
0
21
0
7
0
7
0
0
*
138240 *
*
*
1
1
0
D6
0311
240
1920
640
640
1920
0
160
480
480
960
0
0
60
192
192
192
0
0
12
32
32
0
*
*
30240 *
*
1
0
1
A6
0410
21
105
35
0
140
0
0
35
0
105
0
0
0
21
0
42
0
0
0
7
0
7
*
*
*
138240 *
0
1
1
A6 A1
21
105
0
35
140
0
0
0
35
105
0
0
0
0
21
42
0
0
0
0
7
7
*
*
*
*
69120 0
0
2
E7
0321
f7
10080 120960 80640 40320 120960 20160 20160 60480 30240 60480 4032 12096 7560 24192 12096 12096 756 4032 1512 4032 2016 0 56 576 126 0 0 240 *
*
( )
A7
0420
56
420
280
0
560
70
0
280
0
420
0
56
0
168
0
168
0
28
0
56
0
28
0
8
0
8
0
*
17280 *
D7
0411
672
6720
2240
2240
8960
0
560
2240
2240
6720
0
0
280
1344
1344
2688
0
0
84
448
448
448
0
0
14
64
64
*
*
2160
Projections
Orthographic projections are shown for the sub-symmetries of B6 , B5 , B4 , B3 , B2 , A7 , and A5 Coxeter planes . Vertices are shown as circles, colored by their order of overlap in each projective plane.
(Planes for E8 : E7 , E6 , B8 , B7 , [24] are not shown for being too large to display.)
D3 / B2 / A3
D4 / B3 / A2
D5 / B4
D6 / B5 / A4
D7 / B6
[6]
A5
A7
See also
Notes
^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces , Groningen: University of Groningen ^ Klitzing, (o3o3o3x *c3o3o3o3o - bif)
^ a b Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
^ Klitzing, (o3o3o3x *c3o3o3o3o - buffy)
References
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 Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III , [Math. Zeit. 200 (1988) 3-45] Klitzing, Richard. "8D Uniform polyzetta" .
