2₄₁ 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, 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) 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: .

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, .

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-face	f_k	f₀	f₁	f₂	f₃	f₄		f₅		f₆		f₇		k-figure	notes
D₇	( )	f₀	2160	64	672	2240	560	2240	280	1344	84	448	14	64	h{4,3,3,3,3,3}	E₈/D₇ = 192*10!/64/7! = 2160
A₆A₁	{ }	f₁	2	69120	21	105	35	140	35	105	21	42	7	7	r{3,3,3,3,3}	E₈/A₆A₁ = 192*10!/7!/2 = 69120
A₄A₂A₁	{3}	f₂	3	3	483840	10	5	20	10	20	10	10	5	2	{}x{3,3,3}	E₈/A₄A₂A₁ = 192*10!/5!/3!/2 = 483840
A₃A₃	{3,3}	f₃	4	6	4	1209600	1	4	4	6	6	4	4	1	{3,3}V( )	E₈/A₃A₃ = 192*10!/4!/4! = 1209600
A₄A₃	{3,3,3}	f₄	5	10	10	5	241920	*	4	0	6	0	4	0	{3,3}	E₈/A₄A₃ = 192*10!/5!/4! = 241920
A₄A₂	{3,3,3}	f₄	5	10	10	5	*	967680	1	3	3	3	3	1	{3}V( )	E₈/A₄A₂ = 192*10!/5!/3! = 967680
D₅A₂	{3,3,3^1,1}	f₅	10	40	80	80	16	16	60480	*	3	0	3	0	{3}	E₈/D₅A₂ = 192*10!/16/5!/2 = 40480
A₅A₁	{3,3,3,3}	f₅	6	15	20	15	0	6	*	483840	1	2	2	1	{ }V( )	E₈/A₅A₁ = 192*10!/6!/2 = 483840
E₆A₁	{3,3,3^2,1}	f₆	27	216	720	1080	216	432	27	72	6720	*	2	0	{ }	E₈/E₆A₁ = 192*10!/72/6! = 6720
A₆	{3,3,3,3,3}	f₆	7	21	35	35	0	21	0	7	*	138240	1	1	{ }	E₈/A₆ = 192*10!/7! = 138240
E₇	{3,3,3^3,1}	f₇	126	2016	10080	20160	4032	12096	756	4032	56	576	240	*	( )	E₈/E₇ = 192*10!/72!/8! = 240
A₇	{3,3,3,3,3,3}	f₇	8	28	56	70	0	56	0	28	0	8	*	17280	( )	E₈/A₇ = 192*10!/8! = 17280

Visualizations

The projection of 2₄₁ to the E₈ Coxeter plane (aka. the Petrie projection) with polytope radius

2{\sqrt {2}}

and 69120 edges of length

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)

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 normed distance from the origin, and the number of vertices in the group.

The 2160 projected 2₄₁ polytope projected to 3D (as above) with each normed 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 [30]	[20]	[24]
(1)
E7 [18]	E6 [12]	[6]
	(1,8,24,32)

Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). 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.

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]	[[3^1,2,1]]	[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