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	fk	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				5	10	10	5	*	967680	1	3	3	3	3	1	{3}V( )	E8/A4A2 = 192*10!/5!/3! = 967680
D5A2		{3,3,31,1}	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}		6	15	20	15	0	6	*	483840	1	2	2	1	{ }V( )	E8/A5A1 = 192*10!/6!/2 = 483840
E6A1		{3,3,32,1}	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}		7	21	35	35	0	21	0	7	*	138240	1	1		E8/A6 = 192*10!/7! = 138240
E7		{3,3,33,1}	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}		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 ${\displaystyle 2{\sqrt {2}}}$ and 69120 edges of length ${\displaystyle 2{\sqrt {2}}}$

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)

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

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 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = 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

Alternate names

• Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)^[4][5]

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, .

Visualizations

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.

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]