Alternate names

• E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V₁₇₂₈₀ for its 17280 vertices.^[1]
• Coxeter named it 1₄₂ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
• Diacositetracont-dischiliahectohexaconta-zetton (acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers)^[2]

Coordinates

The 17280 vertices can be defined as sign and location permutations of:

All sign combinations (32): (280×32=8960 vertices)

(4, 2, 2, 2, 2, 0, 0, 0)

Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)

(2, 2, 2, 2, 2, 2, 2, 2)

(5, 1, 1, 1, 1, 1, 1, 1)

(3, 3, 3, 1, 1, 1, 1, 1)

The edge length is 2√2 in this coordinate set, and the polytope radius is 4√2.

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

Removing the node on the end of the 4-length branch leaves the 1₃₂, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 0₄₂, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[3]

More information Configuration matrix, E8 ...

		Configuration matrix
E8		k-face	fk	f0	f1	f2	f3		f4			f5			f6			f7		k-figure	notes
A7		( )	f0	17280	56	420	280	560	70	280	420	56	168	168	28	56	28	8	8	2r{36}	E8/A7 = 192*10!/8! = 17280
A4A2A1		{ }	f1	2	483840	15	15	30	5	30	30	10	30	15	10	15	3	5	3	{3}x{3,3,3}	E8/A4A2A1 = 192*10!/5!/2/2 = 483840
A3A2A1		{3}	f2	3	3	2419200	2	4	1	8	6	4	12	4	6	8	1	4	2	{3.3}v{ }	E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
A3A3		110	f3	4	6	4	1209600	*	1	4	0	4	6	0	6	4	0	4	1	{3,3}v( )	E8/A3A3 = 192*10!/4!/4! = 1209600
A3A2A1				4	6	4	*	2419200	0	2	3	1	6	3	3	6	1	3	2	{3}v{ }	E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
A4A3		120	f4	5	10	10	5	0	241920	*	*	4	0	0	6	0	0	4	0	{3,3}	E8/A4A3 = 192*10!/4!/4! = 241920
D4A2		111		8	24	32	8	8	*	604800	*	1	3	0	3	3	0	3	1	{3}v( )	E8/D4A2 = 192*10!/8/4!/3! = 604800
A4A1A1		120		5	10	10	0	5	*	*	1451520	0	2	2	1	4	1	2	2	{ }v{ }	E8/A4A1A1 = 192*10!/5!/2/2 = 1451520
D5A2		121	f5	16	80	160	80	40	16	10	0	60480	*	*	3	0	0	3	0	{3}	E8/D5A2 = 192*10!/16/5!/3! = 40480
D5A1				16	80	160	40	80	0	10	16	*	181440	*	1	2	0	2	1	{ }v( )	E8/D5A1 = 192*10!/16/5!/2 = 181440
A5A1		130		6	15	20	0	15	0	0	6	*	*	483840	0	2	1	1	2		E8/A5A1 = 192*10!/6!/2 = 483840
E6A1		122	f6	72	720	2160	1080	1080	216	270	216	27	27	0	6720	*	*	2	0	{ }	E8/E6A1 = 192*10!/72/6!/2 = 6720
D6		131		32	240	640	160	480	0	60	192	0	12	32	*	30240	*	1	1		E8/D6 = 192*10!/32/6! = 30240
A6A1		140		7	21	35	0	35	0	0	21	0	0	7	*	*	69120	0	2		E8/A6A1 = 192*10!/7!/2 = 69120
E7		132	f7	576	10080	40320	20160	30240	4032	7560	12096	756	1512	2016	56	126	0	240	*	( )	E8/E7 = 192*10!/72/8! = 240
D7		141		64	672	2240	560	2240	0	280	1344	0	84	448	0	14	64	*	2160		E8/D7 = 192*10!/64/7! = 2160

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Projections

More information E8 [30], E7 [18] ...

E8 [30]	E7 [18]	E6 [12]
(1)	(1,3,6)	(8,16,24,32,48,64,96)
[20]	[24]	[6]
		(1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20)

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Orthographic projections are shown for the sub-symmetries of E₈: E₇, E₆, B₈, B₇, B₆, B₅, B₄, B₃, B₂, A₇, and A₅ Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

More information D3 / B2 / A3 [4], D4 / B3 / A2 [6] ...

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
(32,160,192,240,480,512,832,960)	(72,216,432,720,864,1080)	(8,16,24,32,48,64,96)
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
B8 [16/2]	A5 [6]	A7 [8]

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Related polytopes and honeycombs

More information = ...

1k2 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 (order)	[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	1,920	103,680	2,903,040	696,729,600	∞
Graph							-	-
Name	1−1,2	102	112	122	132	142	152	162

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Alternate names

• 0₄₂₁ polytope
• Birectified 2₄₁ polytope
• Quadrirectified 4₂₁ polytope
• Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym buffy) (Jonathan Bowers)^[4]

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 end of the 1-length branch leaves the birectified 7-simplex,

Removing the node on the end of the 2-length branch leaves the birectified 7-cube, .

Removing the node on the end of the 3-length branch leaves the rectified 1₃₂, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[5]

More information Configuration matrix, E8 ...

		Configuration matrix
E8		k-face	fk	f0	f1	f2			f3					f4						f5						f6					f7			k-figure
A4A2A1		( )	f0	483840	30	30	15	60	10	15	60	30	60	5	20	30	60	30	30	10	20	30	30	15	6	10	10	15	6	3	5	2	3	{3,3,3}x{3,3}x{}
A3A1A1		{ }	f1	2	7257600	2	1	4	1	2	8	4	6	1	4	8	12	6	4	4	6	12	8	4	1	6	4	8	2	1	4	1	2
A3A2		{3}	f2	3	3	4838400	*	*	1	1	4	0	0	1	4	4	6	0	0	4	6	6	4	0	0	6	4	4	1	0	4	1	1
A3A2A1				3	3	*	2419200	*	0	2	0	4	0	1	0	8	0	6	0	4	0	12	0	4	0	6	0	8	0	1	4	0	2
A2A2A1				3	3	*	*	9676800	0	0	2	1	3	0	1	2	6	3	3	1	3	6	6	3	1	3	3	6	2	1	3	1	2
A3A3		0200	f3	4	6	4	0	0	1209600	*	*	*	*	1	4	0	0	0	0	4	6	0	0	0	0	6	4	0	0	0	4	1	0
		0110		6	12	4	4	0	*	1209600	*	*	*	1	0	4	0	0	0	4	0	6	0	0	0	6	0	4	0	0	4	0	1
A3A2				6	12	4	0	4	*	*	4838400	*	*	0	1	1	3	0	0	1	3	3	3	0	0	3	3	3	1	0	3	1	1
A3A2A1				6	12	0	4	4	*	*	*	2419200	*	0	0	2	0	3	0	1	0	6	0	3	0	3	0	6	0	1	3	0	2
A3A1A1		0200		4	6	0	0	4	*	*	*	*	7257600	0	0	0	2	1	2	0	1	2	4	2	1	1	2	4	2	1	2	1	2
A4A3		0210	f4	10	30	20	10	0	5	5	0	0	0	241920	*	*	*	*	*	4	0	0	0	0	0	6	0	0	0	0	4	0	0
A4A2				10	30	20	0	10	5	0	5	0	0	*	967680	*	*	*	*	1	3	0	0	0	0	3	3	0	0	0	3	1	0
D4A2		0111		24	96	32	32	32	0	8	8	8	0	*	*	604800	*	*	*	1	0	3	0	0	0	3	0	3	0	0	3	0	1
A4A1		0210		10	30	10	0	20	0	0	5	0	5	*	*	*	2903040	*	*	0	1	1	2	0	0	1	2	2	1	0	2	1	1
A4A1A1				10	30	0	10	20	0	0	0	5	5	*	*	*	*	1451520	*	0	0	2	0	2	0	1	0	4	0	1	2	0	2
A4A1		0300		5	10	0	0	10	0	0	0	0	5	*	*	*	*	*	2903040	0	0	0	2	1	1	0	1	2	2	1	1	1	2
D5A2		0211	f5	80	480	320	160	160	80	80	80	40	0	16	16	10	0	0	0	60480	*	*	*	*	*	3	0	0	0	0	3	0	0	{3}
A5A1		0220		20	90	60	0	60	15	0	30	0	15	0	6	0	6	0	0	*	483840	*	*	*	*	1	2	0	0	0	2	1	0	{ }v()
D5A1		0211		80	480	160	160	320	0	40	80	80	80	0	0	10	16	16	0	*	*	181440	*	*	*	1	0	2	0	0	2	0	1
A5		0310		15	60	20	0	60	0	0	15	0	30	0	0	0	6	0	6	*	*	*	967680	*	*	0	1	1	1	0	1	1	1	( )v( )v()
A5A1				15	60	0	20	60	0	0	0	15	30	0	0	0	0	6	6	*	*	*	*	483840	*	0	0	2	0	1	1	0	2	{ }v()
		0400		6	15	0	0	20	0	0	0	0	15	0	0	0	0	0	6	*	*	*	*	*	483840	0	0	0	2	1	0	1	2
E6A1		0221	f6	720	6480	4320	2160	4320	1080	1080	2160	1080	1080	216	432	270	432	216	0	27	72	27	0	0	0	6720	*	*	*	*	2	0	0	{ }
A6		0320		35	210	140	0	210	35	0	105	0	105	0	21	0	42	0	21	0	7	0	7	0	0	*	138240	*	*	*	1	1	0
D6		0311		240	1920	640	640	1920	0	160	480	480	960	0	0	60	192	192	192	0	0	12	32	32	0	*	*	30240	*	*	1	0	1
A6		0410		21	105	35	0	140	0	0	35	0	105	0	0	0	21	0	42	0	0	0	7	0	7	*	*	*	138240	*	0	1	1
A6A1				21	105	0	35	140	0	0	0	35	105	0	0	0	0	21	42	0	0	0	0	7	7	*	*	*	*	69120	0	0	2
E7		0321	f7	10080	120960	80640	40320	120960	20160	20160	60480	30240	60480	4032	12096	7560	24192	12096	12096	756	4032	1512	4032	2016	0	56	576	126	0	0	240	*	*	( )
A7		0420		56	420	280	0	560	70	0	280	0	420	0	56	0	168	0	168	0	28	0	56	0	28	0	8	0	8	0	*	17280	*
D7		0411		672	6720	2240	2240	8960	0	560	2240	2240	6720	0	0	280	1344	1344	2688	0	0	84	448	448	448	0	0	14	64	64	*	*	2160

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Projections

Orthographic projections are shown for the sub-symmetries of B₆, B₅, B₄, B₃, B₂, A₇, and A₅ Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane.

(Planes for E₈: E₇, E₆, B₈, B₇, [24] are not shown for being too large to display.)