2 41 polytope

Last updated
4 21 t0 E6.svg
421
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
1 42 polytope E6 Coxeter plane.svg
142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t0 E6.svg
241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
4 21 t1 E6.svg
Rectified 421
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4 21 t4 E6.svg
Rectified 142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
2 41 t1 E6.svg
Rectified 241
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
4 21 t2 E6.svg
Birectified 421
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4 21 t3 E6.svg
Trirectified 421
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel 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 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

Contents

Its Coxeter symbol is 241, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.

The rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the triangle face centers of the 241, and is the same as the rectified 142.

These polytopes are part of a family of 255 (28  1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, 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.pngCDel 3a.pngCDel nodea.png.

241 polytope

241 polytope
TypeUniform 8-polytope
Family 2k1 polytope
Schläfli symbol {3,3,34,1}
Coxeter symbol 241
Coxeter diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7-faces17520:
240 231 Gosset 2 31 polytope.svg
17280 {36} 7-simplex t0.svg
6-faces144960:
6720 221 E6 graph.svg
138240 {35} 6-simplex t0.svg
5-faces544320:
60480 211 Cross graph 5.svg
483840 {34} 5-simplex t0.svg
4-faces1209600:
241920 {201 4-simplex t0.svg
967680 {33} 4-simplex t0.svg
Cells1209600 {32} 3-simplex t0.svg
Faces483840 {3} 2-simplex t0.svg
Edges69120
Vertices2160
Vertex figure 141
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

The 241 is composed of 17,520 facets (240 231 polytopes and 17,280 7-simplices), 144,960 6-faces (6,720 221 polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 211 and 483,840 5-simplices), 1,209,600 4-faces (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.

This polytope is a facet in the uniform tessellation, 251 with Coxeter-Dynkin diagram:

CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Alternate names

Coordinates

The 2160 vertices can be defined as follows:

16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)
1120 permutations of (±2,±2,±2,±2,0,0,0,0) of (trirectified 8-orthoplex)
1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an odd number of minus-signs

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: CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the short branch leaves the 7-simplex: CDel nodea 1.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. There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 231, CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png. There are 240 of these facets. They are centered at the positions of the 240 vertices in the 421 polytope.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 141, CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.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. [3]

E8CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png k-face fkf0f1f2f3f4f5f6f7 k-figure notes
D7CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png( )f0216064672224056022402801344844481464 h{4,3,3,3,3,3} E8/D7 = 192*10!/64/7! = 2160
A6A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea 1.png{ }f1269120211053514035105214277 r{3,3,3,3,3} E8/A6A1 = 192*10!/7!/2 = 69120
A4A2A1CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3} f233483840105201020101052 {}x{3,3,3} E8/A4A2A1 = 192*10!/5!/3!/2 = 483840
A3A3CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3} f3464120960014466441 {3,3}V( ) E8/A3A3 = 192*10!/4!/4! = 1209600
A4A3CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3} f4510105241920*406040 {3,3} E8/A4A3 = 192*10!/5!/4! = 241920
A4A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png510105*967680133331 {3}V( ) E8/A4A2 = 192*10!/5!/3! = 967680
D5A2CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,31,1} f510408080161660480*3030 {3} E8/D5A2 = 192*10!/16/5!/2 = 40480
A5A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3,3} 615201506*4838401221 { }V( ) E8/A5A1 = 192*10!/6!/2 = 483840
E6A1CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,32,1} f627216720108021643227726720*20{ }E8/E6A1 = 192*10!/72/6! = 6720
A6CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3,3,3} 721353502107*13824011E8/A6 = 192*10!/7! = 138240
E7CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,33,1} f712620161008020160403212096756403256576240*( )E8/E7 = 192*10!/72!/8! = 240
A7CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3,3,3,3} 828567005602808*17280E8/A7 = 192*10!/8! = 17280

Images

Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:
u = (1, ph, 0, -1, ph, 0,0,0)
v = (ph, 0, 1, ph, 0, -1,0,0)
w = (0, 1, ph, 0, -1, ph,0,0)
The 2160 projected 241 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. The overlapping vertices are color coded by overlap count. Also shown is a list of each hull group, the Norm'd distance from the origin, and the number of vertices in the group. E8 241-3D.png
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)
The 2160 projected 241 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. The overlapping vertices are color coded by overlap count. Also shown is a list of each hull group, the Norm'd distance from the origin, and the number of vertices in the group.
The 2160 projected 241 polytope projected to 3D (as above) with each Norm'd hull group listed individually with vertex counts. Notice the last two outer hulls are a combination of two overlapped Icosahedrons (24) and a Icosidodecahedron (30). E8 241-3D Concentric Hulls List.png
The 2160 projected 241 polytope projected to 3D (as above) with each Norm'd hull group listed individually with vertex counts. Notice the last two outer hulls are a combination of two overlapped Icosahedrons (24) and a Icosidodecahedron (30).

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8
[30]
[20][24]
2 41 t0 E8.svg
(1)
2 41 t0 p20.svg 2 41 t0 p24.svg
E7
[18]
E6
[12]
[6]
2 41 t0 E7.svg 2 41 t0 E6.svg
(1,8,24,32)
2 41 t0 mox.svg
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
2 41 t0 B2.svg 2 41 t0 B3.svg 2 41 t0 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
2 41 t0 B5.svg 2 41 t0 B6.svg
(1,3,9,12,18,21,36)
2 41 t0 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
2 41 t0 B8.svg 2 41 t0 A5.svg 2 41 t0 A7.svg
2k1 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 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.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 [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 1212038451,8402,903,040696,729,600
Graph Trigonal dihedron.png 4-simplex t0.svg 5-cube t4.svg Up 2 21 t0 E6.svg Up2 2 31 t0 E7.svg 2 41 t0 E8.svg --
Name 2−1,1 201 211 221 231 241 251 261

Rectified 2_41 polytope

Rectified 241 polytope
TypeUniform 8-polytope
Schläfli symbol t1{3,3,34,1}
Coxeter symbol t1(241)
Coxeter diagram CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
7-faces19680 total:

240 t1(221)
17280 t1{36}
2160 141

6-faces313440
5-faces1693440
4-faces4717440
Cells7257600
Faces5322240
Edges19680
Vertices69120
Vertex figure rectified 6-simplex prism
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

The rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241.

Alternate names

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E8 Coxeter group.

The facet information can be extracted from its Coxeter-Dynkin diagram: CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the short branch leaves the rectified 7-simplex: CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 4-length branch leaves the rectified 231, CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.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 7-demicube, 141CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.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 rectified 6-simplex prism, CDel nodea 1.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Visualizations

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8
[30]
[20][24]
2 41 t1 E8.svg
(1)
2 41 t1 p20.svg 2 41 t1 p24.svg
E7
[18]
E6
[12]
[6]
2 41 t1 E7.svg 2 41 t1 E6.svg
(1,8,24,32)
2 41 t1 mox.svg
D3 / B2 / A3
[4]
D4 / B3 / A2
[6]
D5 / B4
[8]
2 41 t1 B2.svg 2 41 t1 B3.svg 2 41 t1 B4.svg
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
2 41 t1 B5.svg 2 41 t1 B6.svg
(1,3,9,12,18,21,36)
2 41 t1 B7.svg
B8
[16/2]
A5
[6]
A7
[8]
2 41 t1 B8.svg 2 41 t1 A5.svg 2 41 t1 A7.svg

See also

Notes

  1. Elte, 1912
  2. Klitzing, (x3o3o3o *c3o3o3o3o - bay)
  3. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  4. Jonathan Bowers
  5. Klitzing, (o3x3o3o *c3o3o3o3o - robay)

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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