2 41 polytope

421	142	241
Rectified 421	Rectified 142	Rectified 241
Birectified 421	Trirectified 421
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.

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 and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2₄₁ polytope

241 polytope
Type	Uniform 8-polytope
Family	2k1 polytope
Schläfli symbol	{3,3,34,1}
Coxeter symbol	241
Coxeter diagram
7-faces	17520: 240 231 17280 {36}
6-faces	144960: 6720 221 138240 {35}
5-faces	544320: 60480 211 483840 {34}
4-faces	1209600: 241920 {201 967680 {33}
Cells	1209600 {32}
Faces	483840 {3}
Edges	69120
Vertices	2160
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:

Alternate names

• E. L. Elte named it V₂₁₆₀ (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
• It is named 2₄₁ by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
• Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)

The 2160 vertices can be defined as follows:

16 permutations of (±4,0,0,0,0,0,0,0)

1120 permutations of (±2,±2,±2,±2,0,0,0,0)

1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an even 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: .

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

Removing the node on the end of the 4-length branch leaves the 2₃₁, . 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, .

Images

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]
(1)
E7 [18]	E6 [12]	[6]
	(1,8,24,32)

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
	(1,3,9,12,18,21,36)
B8 [16/2]	A5 [6]	A7 [8]

Related polytopes and honeycombs

2k1 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 = ${\tilde{E}}_{8}$ = E8+	E10 = ${\bar{T}}_8$ = E8++
Coxeter diagram
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	384	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	2-1,1	201	211	221	231	241	251	261

Rectified 2_41 polytope

Rectified 241 polytope
Type	Uniform 8-polytope
Schläfli symbol	t1{3,3,34,1}
Coxeter symbol	t1(241)
Coxeter diagram
7-faces	19680 total: 240 t1(221) 17280 t1{36} 2160 141
6-faces	313440
5-faces	1693440
4-faces	4717440
Cells	7257600
Faces	5322240
Edges	19680
Vertices	69120
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

• Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (acronym robay) (Jonathan Bowers)

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: .

Removing the node on the short branch leaves the rectified 7-simplex: .

Removing the node on the end of the 4-length branch leaves the rectified 2₃₁, .

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1₄₁.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .

Images

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]
(1)
E7 [18]	E6 [12]	[6]
	(1,8,24,32)

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
	(1,3,9,12,18,21,36)
B8 [16/2]	A5 [6]	A7 [8]

See also

• List of E8 polytopes

References

• 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]
• Richard Klitzing, 8D, Uniform polyzetta x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay