2 41 polytope

Orthogonal projections in E₆ Coxeter plane
4₂₁	1₄₂	2₄₁
Rectified 4₂₁	Rectified 1₄₂	Rectified 2₄₁
Birectified 4₂₁	Trirectified 4₂₁

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	2_k1 polytope
Schläfli symbol	{3,3,3^4,1}
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, [3^4,2,1]
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.^[1]
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)^[2]

Coordinates

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

2_k1 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]	[3^0,2,1]	[[3<sup>1,2,1</sup>]]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	384	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	2_-1,1	2₀₁	2₁₁	2₂₁	2₃₁	2₄₁	2₅₁	2₆₁