1 42 polytope

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

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

142
Type	Uniform 8-polytope
Family	1k2 polytope
Schläfli symbol	{3,34,2}
Coxeter symbol	142
Coxeter diagrams
7-faces	2400: 240 132 2160 141
6-faces	106080: 6720 122 30240 131 69120 {35}
5-faces	725760: 60480 112 181440 121 483840 {34}
4-faces	2298240: 241920 102 604800 111 1451520 {33}
Cells	3628800: 1209600 101 2419200 {32}
Faces	2419200 {3}
Edges	483840
Vertices	17280
Vertex figure	t2{36}
Petrie polygon	30-gon
Coxeter group	E8, [34,2,1]
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. ^[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]

		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

Projections

The projection of 1₄₂ to the E₈ Coxeter plane (aka. the Petrie projection) with polytope radius ${\displaystyle 4{\sqrt {2}}}$ is shown below with 483,840 edges of length ${\displaystyle 2{\sqrt {2}}}$ culled 53% on the interior to only 226,444:

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 17280 projected 1₄₂ polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. Notice the last two outer hulls are a combination of two overlapped Dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).

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)

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.

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]

Related polytopes and honeycombs

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

Rectified 1₄₂ polytope

Rectified 142
Type	Uniform 8-polytope
Schläfli symbol	t1{3,34,2}
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	{3,3,3}×{3}×{}
Coxeter group	E8, [34,2,1]
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) ^[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. ^[3]

		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

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.)

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
D6 / B5 / A4 [10]	D7 / B6 [12]	[6]
A5 [6]	A7 [8]	[20]

See also

• List of E8 polytopes

Notes

1. Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
2. Klitzing, (o3o3o3x *c3o3o3o3o - bif)
3. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
4. Klitzing, (o3o3o3x *c3o3o3o3o - buffy)

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, ISBN   978-0-471-01003-6 Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Klitzing, Richard. "8D Uniform polyzetta". o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds