1₃₂ polytope

In 7-dimensional geometry, 1₃₂ is a uniform polytope, constructed from the E7 group.

321		231		132
Rectified 321			birectified 321
Rectified 231			Rectified 132
Orthogonal projections in E7 Coxeter plane

Its Coxeter symbol is 1₃₂, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The rectified 1₃₂ is constructed by points at the mid-edges of the 1₃₂.

These polytopes are part of a family of 127 (2⁷-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

1_32 polytope

132
Type	Uniform 7-polytope
Family	1k2 polytope
Schläfli symbol	{3,3^3,2}
Coxeter symbol	132
Coxeter diagram
6-faces	182: 56 122 126 131
5-faces	4284: 756 121 1512 121 2016 {3^4}
4-faces	23688: 4032 {3^3} 7560 111 12096 {3^3}
Cells	50400: 20160 {3^2} 30240 {3^2}
Faces	40320 {3}
Edges	10080
Vertices	576
Vertex figure	t2{3^5}
Petrie polygon	Octadecagon
Coxeter group	E7, [3^3,2,1], order 2903040
Properties	convex

This polytope can tessellate 7-dimensional space, with symbol 1₃₃, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E₇^* lattice.[1]

Alternate names

• Emanuel Lodewijk Elte named it V₅₇₆ (for its 576 vertices) in his 1912 listing of semiregular polytopes.[2]
• Coxeter called it 1₃₂ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
• Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)[3]

Images

Coxeter plane projections

E7	E6 / F4	B7 / A6
[18]	[12]	[7x2]
A5	D7 / B6	D6 / B5
[6]	[12/2]	[10]
D5 / B4 / A4	D4 / B3 / A2 / G2	D3 / B2 / A3
[8]	[6]	[4]

Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-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 6-demicube, 1₃₁,

Removing the node on the end of the 3-length branch leaves the 1₂₂,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 0₃₂,

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[4]

E7		k-face	fk	f0	f1	f2	f3		f4			f5			f6		k-figures	notes
A6		( )	f0	576	35	210	140	210	35	105	105	21	42	21	7	7	2r{3,3,3,3,3}	E7/A6 = 72*8!/7! = 576
A3A2A1		{ }	f1	2	10080	12	12	18	4	12	12	6	12	3	4	3	{3,3}x{3}	E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A2A1		{3}	f2	3	3	40320	2	3	1	6	3	3	6	1	3	2	{ }∨{3}	E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A3A2		{3,3}	f3	4	6	4	20160	*	1	3	0	3	3	0	3	1	{3}∨( )	E7/A3A2 = 72*8!/4!/3! = 20160
A3A1A1				4	6	4	*	30240	0	2	2	1	4	1	2	2	Phyllic disphenoid	E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A4A2		{3,3,3}	f4	5	10	10	5	0	4032	*	*	3	0	0	3	0	{3}	E7/A4A2 = 72*8!/5!/3! = 4032
D4A1		{3,3,4}		8	24	32	8	8	*	7560	*	1	2	0	2	1	{ }∨( )	E7/D4A1 = 72*8!/8/4!/2 = 7560
A4A1		{3,3,3}		5	10	10	0	5	*	*	12096	0	2	1	1	2		E7/A4A1 = 72*8!/5!/2 = 12096
D5A1		h{4,3,3,3}	f5	16	80	160	80	40	16	10	0	756	*	*	2	0	{ }	E7/D5A1 = 72*8!/16/5!/2 = 756
D5				16	80	160	40	80	0	10	16	*	1512	*	1	1		E7/D5 = 72*8!/16/5! = 1512
A5A1		{3,3,3,3,3}		6	15	20	0	15	0	0	6	*	*	2016	0	2		E7/A5A1 = 72*8!/6!/2 = 2016
E6		{3,3^2,2}	f6	72	720	2160	1080	1080	216	270	216	27	27	0	56	*	( )	E7/E6 = 72*8!/72/6! = 56
D6		h{4,3,3,3,3}		32	240	640	160	480	0	60	192	0	12	32	*	126		E7/D6 = 72*8!/32/6! = 126

Related polytopes and honeycombs

The 1₃₂ is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The next figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k dimensional figures

Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A3A1	A5	D6	E7	${\displaystyle {\tilde {E}}_{7}}$=E7^+	${\displaystyle {\bar {T}}_{8}}$=E7^++
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[3^1,3,1]	[3^2,3,1]	[[3<sup>3,3,1</sup>]]	[3^4,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	13,-1	130	131	132	133	134

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]	[3^0,2,1]	[3^1,2,1]	[[3<sup>2,2,1</sup>]]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
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 1_32 polytope

Rectified 132
Type	Uniform 7-polytope
Schläfli symbol	t1{3,3^3,2}
Coxeter symbol	0321
Coxeter-Dynkin diagram
6-faces	758
5-faces	12348
4-faces	72072
Cells	191520
Faces	241920
Edges	120960
Vertices	10080
Vertex figure	{3,3}×{3}×{}
Coxeter group	E7, [3^3,2,1], order 2903040
Properties	convex

The rectified 1₃₂ (also called 0₃₂₁) is a rectification of the 1₃₂ polytope, creating new vertices on the center of edge of the 1₃₂. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.

Alternate names

• Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)[5]

Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 1₂₂ polytope,

Removing the node on the end of the 2-length branch leaves the demihexeract, 1₃₁,

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[6]

E7		k-face	fk	f0	f1	f2			f3					f4						f5					f6			k-figures	notes
A3A2A1		( )	f0	10080	24	24	12	36	8	12	36	18	24	4	12	18	24	12	6	6	8	12	6	3	4	2	3	{3,3}x{3}x{ }	E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A1A1		{ }	f1	2	120960	2	1	3	1	2	6	3	3	1	3	6	6	3	1	3	3	6	2	1	3	1	2	( )v{3}v{ }	E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A2A2		01	f2	3	3	80640	*	*	1	1	3	0	0	1	3	3	3	0	0	3	3	3	1	0	3	1	1	{3}v( )v( )	E7/A2A2 = 72*8!/3!/3! = 80640
A2A2A1				3	3	*	40320	*	0	2	0	3	0	1	0	6	0	3	0	3	0	6	0	1	3	0	2	{3}v{ }	E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A2A1A1				3	3	*	*	120960	0	0	2	1	2	0	1	2	4	2	1	1	2	4	2	1	2	1	2	{ }v{ }v( )	E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A3A2		02	f3	4	6	4	0	0	20160	*	*	*	*	1	3	0	0	0	0	3	3	0	0	0	3	1	0	{3}v( )	E7/A3A2 = 72*8!/4!/3! = 20160
		011		6	12	4	4	0	*	20160	*	*	*	1	0	3	0	0	0	3	0	3	0	0	3	0	1
A3A1				6	12	4	0	4	*	*	60480	*	*	0	1	1	2	0	0	1	2	2	1	0	2	1	1	Sphenoid	E7/A3A1 = 72*8!/4!/2 = 60480
A3A1A1				6	12	0	4	4	*	*	*	30240	*	0	0	2	0	2	0	1	0	4	0	1	2	0	2	{ }v{ }	E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A3A1		02		4	6	0	0	4	*	*	*	*	60480	0	0	0	2	1	1	0	1	2	2	1	1	1	2	Sphenoid	E7/A3A1 = 72*8!/4!/2 = 60480
A4A2		021	f4	10	30	20	10	0	5	5	0	0	0	4032	*	*	*	*	*	3	0	0	0	0	3	0	0	{3}	E7/A4A2 = 72*8!/5!/3! = 4032
A4A1				10	30	20	0	10	5	0	5	0	0	*	12096	*	*	*	*	1	2	0	0	0	2	1	0	{ }v()	E7/A4A1 = 72*8!/5!/2 = 12096
D4A1		0111		24	96	32	32	32	0	8	8	8	0	*	*	7560	*	*	*	1	0	2	0	0	2	0	1		E7/D4A1 = 72*8!/8/4!/2 = 7560
A4		021		10	30	10	0	20	0	0	5	0	5	*	*	*	24192	*	*	0	1	1	1	0	1	1	1	( )v( )v( )	E7/A4 = 72*8!/5! = 34192
A4A1				10	30	0	10	20	0	0	0	5	5	*	*	*	*	12096	*	0	0	2	0	1	1	0	2	{ }v()	E7/A4A1 = 72*8!/5!/2 = 12096
		03		5	10	0	0	10	0	0	0	0	5	*	*	*	*	*	12096	0	0	0	2	1	0	1	2
D5A1		0211	f5	80	480	320	160	160	80	80	80	40	0	16	16	10	0	0	0	756	*	*	*	*	2	0	0	{ }	E7/D5A1 = 72*8!/16/5!/2 = 756
A5		022		20	90	60	0	60	15	0	30	0	15	0	6	0	6	0	0	*	4032	*	*	*	1	1	0		E7/A5 = 72*8!/6! = 4032
D5		0211		80	480	160	160	320	0	40	80	80	80	0	0	10	16	16	0	*	*	1512	*	*	1	0	1		E7/D5 = 72*8!/16/5! = 1512
A5		031		15	60	20	0	60	0	0	15	0	30	0	0	0	6	0	6	*	*	*	4032	*	0	1	1		E7/A5 = 72*8!/6! = 4032
A5A1				15	60	0	20	60	0	0	0	15	30	0	0	0	0	6	6	*	*	*	*	2016	0	0	2		E7/A5A1 = 72*8!/6!/2 = 2016
E6		0221	f6	720	6480	4320	2160	4320	1080	1080	2160	1080	1080	216	432	270	432	216	0	27	72	27	0	0	56	*	*	( )	E7/E6 = 72*8!/72/6! = 56
A6		032		35	210	140	0	210	35	0	105	0	105	0	21	0	42	0	21	0	7	0	7	0	*	576	*		E7/A6 = 72*8!/7! = 576
D6		0311		240	1920	640	640	1920	0	160	480	480	960	0	0	60	192	192	192	0	0	12	32	32	*	*	126		E7/D6 = 72*8!/32/6! = 126

Images

Coxeter plane projections

E7	E6 / F4	B7 / A6
[18]	[12]	[14]
A5	D7 / B6	D6 / B5
[6]	[12/2]	[10]
D5 / B4 / A4	D4 / B3 / A2 / G2	D3 / B2 / A3
[8]	[6]	[4]

See also

• List of E7 polytopes

Notes

1. The Voronoi Cells of the E₆^* and E₇^* Lattices, Edward Pervin
2. Elte, 1912
3. Klitzing, (o3o3o3x *c3o3o3o - lin)
4. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
5. Klitzing, (o3o3x3o *c3o3o3o - rolin)
6. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

References

• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
• 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
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin

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 4-polytope	5-cell	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