1 42 polytope explained

In 8-dimensional geometry, the 1₄₂ is a uniform 8-polytope, constructed within the symmetry of the E₈ group.

Its Coxeter symbol is 1₄₂, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.

The rectified 1₄₂ is constructed by points at the mid-edges of the 1₄₂ and is the same as the birectified 2₄₁, and the quadrirectified 4₂₁.

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 non-empty combinations of rings in this Coxeter-Dynkin diagram: .

1₄₂ polytope

bgcolor=#e7dcc3 colspan=2	142
Type	Uniform 8-polytope
Family	1k2 polytope
Schläfli symbol
Coxeter symbol	142
Coxeter diagrams
7-faces	2400: 240 132 2160 141
6-faces	106080: 6720 122 30240 131 69120
5-faces	725760: 60480 112 181440 121 483840
4-faces	2298240: 241920 102 604800 111 1451520
Cells	3628800: 1209600 101 2419200
Faces	2419200
Edges	483840
Vertices	17280
Vertex figure
Petrie polygon	30-gon
Coxeter group	E8, [3<sup>4,2,1</sup>]
Properties	convex

The 1₄₂ is composed of 2400 facets: 240 1₃₂ polytopes, and 2160 7-demicubes (1₄₁). Its vertex figure is a birectified 7-simplex.

This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 1₅₂, and Coxeter-Dynkin diagram: .

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.
• 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)^[1]

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 in this coordinate set, and the polytope radius is 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 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.^[2]

		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		E8/A7 = 192*10!/8	= 17280
A4A2A1			f1	2	483840	15	15	30	5	30	30	10	30	15	10	15	3	5	3	x	E8/A4A2A1 = 192*10!/5	/2/2 = 483840
A3A2A1			f2	3	3	2419200	2	4	1	8	6	4	12	4	6	8	1	4	2	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		E8/A3A3 = 192*10!/4	/4! = 1209600
A3A2A1				4	6	4		2419200	0	2	3	1	6	3	3	6	1	3	2	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		E8/A4A3 = 192*10!/4	/4! = 241920
D4A2		111		8	24	32	8	8		604800		1	3	0	3	3	0	3	1		E8/D4A2 = 192*10!/8/4	/3! = 604800
A4A1A1				5	10	10	0	5			1451520	0	2	2	1	4	1	2	2		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		E8/D5A2 = 192*10!/16/5	/3! = 40480
D5A1				16	80	160	40	80	0	10	16		181440		1	2	0	2	1		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

Projections

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.

Related polytopes and honeycombs

Rectified 1₄₂ polytope

bgcolor=#e7dcc3 colspan=2	Rectified 142
Type	Uniform 8-polytope
Schläfli symbol	t1
Coxeter symbol	0421
Coxeter diagrams
7-faces	19680
6-faces	382560
5-faces	2661120
4-faces	9072000
Cells	16934400
Faces	16934400
Edges	7257600
Vertices	483840
Vertex figure	××
Coxeter group	E8, [3<sup>4,2,1</sup>]
Properties	convex

The rectified 1₄₂ is named from being a rectification of the 1₄₂ polytope, with vertices positioned at the mid-edges of the 1₄₂. It can also be called a 0₄₂₁ polytope with the ring at the center of 3 branches of length 4, 2, and 1.

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)^[3]

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.^[4]

		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	xx
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			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
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	vv
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

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.)

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy

Notes and References

1. Klitzing, (o3o3o3x *c3o3o3o3o - bif)
2. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
3. Klitzing, (o3o3o3x *c3o3o3o3o - buffy)
4. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203