1 32 polytope: Difference between revisions

Latest revision as of 06:46, 27 June 2023

Orthogonal projections in E₇ Coxeter plane
3₂₁ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	2₃₁ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination		1₃₂ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
Rectified 3₂₁ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination		birectified 3₂₁ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
Rectified 2₃₁ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination		Rectified 1₃₂ Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination

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

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:

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.

1_32 polytope

1₃₂
Type	Uniform 7-polytope
Family	1_k2 polytope
Schläfli symbol	{3,3^3,2}
Coxeter symbol	1₃₂
Coxeter diagram	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
6-faces	182: 56 1₂₂ 126 1₃₁
5-faces	4284: 756 1₂₁25px 1512 1₂₁25px 2016 {3⁴}
4-faces	23688: 4032 {3³} 7560 1₁₁25px 12096 {3³}
Cells	50400: 20160 {3²} 30240 {3²}
Faces	40320 {3}
Edges	10080
Vertices	576
Vertex figure	t₂{3⁵}
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,

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. 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,

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Removing the node on the end of the 2-length branch leaves the 6-demicube, 1₃₁,

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

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

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

E₇	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	k-face	f_k	f₀	f₁	f₂	f₃		f₄			f₅			f₆		k-figures	notes
A₆	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	( )	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	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{ }	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	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{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	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{3,3}	f₃	4	6	4	20160	*	1	3	0	3	3	0	3	1	{3}∨( )	E₇/A₃A₂ = 72*8!/4!/3! = 20160
A₃A₁A₁	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{3,3}	f₃	4	6	4	*	30240	0	2	2	1	4	1	2	2	Phyllic disphenoid	E₇/A₃A₁A₁ = 72*8!/4!/2/2 = 30240
A₄A₂	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{3,3,3}	f₄	5	10	10	5	0	4032	*	*	3	0	0	3	0	{3}	E₇/A₄A₂ = 72*8!/5!/3! = 4032
D₄A₁	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{3,3,4}		8	24	32	8	8	*	7560	*	1	2	0	2	1	{ }∨( )	E₇/D₄A₁ = 72*8!/8/4!/2 = 7560
A₄A₁	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{3,3,3}		5	10	10	0	5	*	*	12096	0	2	1	1	2	{ }∨( )	E₇/A₄A₁ = 72*8!/5!/2 = 12096
D₅A₁	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	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	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	h{4,3,3,3}		16	80	160	40	80	0	10	16	*	1512	*	1	1		E₇/D₅ = 72*8!/16/5! = 1512
A₅A₁	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{3,3,3,3,3}		6	15	20	0	15	0	0	6	*	*	2016	0	2		E₇/A₅A₁ = 72*8!/6!/2 = 2016
E₆	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	{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	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination	h{4,3,3,3,3}	f₆	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.

Rectified 1_32 polytope

Rectified 1₃₂
Type	Uniform 7-polytope
Schläfli symbol	t₁{3,3^3,2}
Coxeter symbol	0₃₂₁
Coxeter-Dynkin diagram	Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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}×{}.