3 21 polytope

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

In 7-dimensional geometry, the 3₂₁ polytope is a uniform 7-polytope, constructed within the symmetry of the E₇ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure. ^[1]

Its Coxeter symbol is 3₂₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The rectified 3₂₁ is constructed by points at the mid-edges of the 3₂₁. The birectified 3₂₁ is constructed by points at the triangle face centers of the 3₂₁. The trirectified 3₂₁ is constructed by points at the tetrahedral centers of the 3₂₁, and is the same as the rectified 1₃₂.

These polytopes are part of a family of 127 (2⁷−1) convex uniform polytopes in 7 dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

3₂₁ polytope

321 polytope
Type	Uniform 7-polytope
Family	k21 polytope
Schläfli symbol	{3,3,3,32,1}
Coxeter symbol	321
Coxeter diagram
6-faces	702 total: 126 311 576 {35}
5-faces	6048: 4032 {34} 2016 {34}
4-faces	12096 {33}
Cells	10080 {3,3}
Faces	4032 {3}
Edges	756
Vertices	56
Vertex figure	221 polytope
Petrie polygon	octadecagon
Coxeter group	E7, [33,2,1], order 2903040
Properties	convex

In 7-dimensional geometry, the 3₂₁ polytope is a uniform polytope. It has 56 vertices, and 702 facets: 126 3₁₁ and 576 6-simplexes.

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The 1-skeleton of the 3₂₁ polytope is the Gosset graph.

This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 3₃₁ and Coxeter-Dynkin diagram: .

Alternate names

• It is also called the Hess polytope for Edmund Hess who first discovered it.
• It was enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure. ^[1]
• E. L. Elte named it V₅₆ (for its 56 vertices) in his 1912 listing of semiregular polytopes. ^[2]
• H.S.M. Coxeter called it 3₂₁ due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
• Hecatonicosihexa-pentacosiheptacontahexa-exon (acronym: naq) - 126-576 facetted polyexon (Jonathan Bowers) ^[3]

Coordinates

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:

± (−3, −3, 1, 1, 1, 1, 1, 1)

Construction

Its construction is based on the E7 group. Coxeter named it as 3₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on the short branch leaves the 6-simplex, .

Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 3₁₁, .

Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2₂₁ polytope, .

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
E6		( )	f0	56	27	216	720	1080	432	216	72	27	221	E7/E6 = 72x8!/72x6! = 56
D5A1		{ }	f1	2	756	16	80	160	80	40	16	10	5-demicube	E7/D5A1 = 72x8!/16/5!/2 = 756
A4A2		{3}	f2	3	3	4032	10	30	20	10	5	5	rectified 5-cell	E7/A4A2 = 72x8!/5!/2 = 4032
A3A2A1		{3,3}	f3	4	6	4	10080	6	6	3	2	3	triangular prism	E7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A4A1		{3,3,3}	f4	5	10	10	5	12096	2	1	1	2	isosceles triangle	E7/A4A1 = 72x8!/5!/2 = 12096
A5A1		{3,3,3,3}	f5	6	15	20	15	6	4032	*	1	1	{ }	E7/A5A1 = 72x8!/6!/2 = 4032
A5				6	15	20	15	6	*	2016	0	2		E7/A5 = 72x8!/6! = 2016
A6		{3,3,3,3,3}	f6	7	21	35	35	21	10	0	576	*	( )	E7/A6 = 72x8!/7! = 576
D6		{3,3,3,3,4}		12	60	160	240	192	32	32	*	126		E7/D6 = 72x8!/32/6! = 126

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]

Related polytopes

The 3₂₁ is fifth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

k21 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
En	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	[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	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−121	021	121	221	321	421	521	621

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3_k1 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]	[30,3,1]	[[31,3,1]] = [4,3,3,3,3]	[32,3,1]	[33,3,1]	[34,3,1]
Order	48	720	46,080	2,903,040	∞
Graph					-	-
Name	31,-1	310	311	321	331	341

Rectified 3₂₁ polytope

Rectified 321 polytope
Type	Uniform 7-polytope
Schläfli symbol	t1{3,3,3,32,1}
Coxeter symbol	t1(321)
Coxeter diagram
6-faces	758
5-faces	44352
4-faces	70560
Cells	48384
Faces	11592
Edges	12096
Vertices	756
Vertex figure	5-demicube prism
Petrie polygon	octadecagon
Coxeter group	E7, [33,2,1], order 2903040
Properties	convex

Alternate names

• Rectified hecatonicosihexa-pentacosiheptacontahexa-exon as a rectified 126-576 facetted polyexon (acronym: ranq) (Jonathan Bowers) ^[5]

Construction

Its construction is based on the E7 group. Coxeter named it as 3₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on the short branch leaves the 6-simplex, .

Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t₁3₁₁, .

Removing the node on the end of the 3-length branch leaves the 2₂₁, .

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

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]

Birectified 3₂₁ polytope

Birectified 321 polytope
Type	Uniform 7-polytope
Schläfli symbol	t2{3,3,3,32,1}
Coxeter symbol	t2(321)
Coxeter diagram
6-faces	758
5-faces	12348
4-faces	68040
Cells	161280
Faces	161280
Edges	60480
Vertices	4032
Vertex figure	5-cell-triangle duoprism
Petrie polygon	octadecagon
Coxeter group	E7, [33,2,1], order 2903040
Properties	convex

Alternate names

• Birectified hecatonicosihexa-pentacosiheptacontahexa-exon as a birectified 126-576 facetted polyexon (acronym: branq) (Jonathan Bowers) ^[6]

Construction

Its construction is based on the E7 group. Coxeter named it as 3₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on the short branch leaves the birectified 6-simplex, .

Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: t₂(3₁₁), .

Removing the node on the end of the 3-length branch leaves the rectified 2₂₁ polytope in its alternated form: t₁(2₂₁), .

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

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]

See also

• List of E7 polytopes

Notes

1. Gosset, 1900
2. Elte, 1912
3. Klitzing, (o3o3o3o *c3o3o3x - naq)
4. Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203
5. Klitzing. (o3o3o3o *c3o3x3o - ranq)
6. Klitzing, (o3o3o3o *c3x3o3o - branq)

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• 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, wiley.com, ISBN   978-0-471-01003-6
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] p. 342 (figure 3.7c) by Peter mcMullen: (18-gonal node-edge graph of 3₂₁)
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq

External links

• Gosset’s Polytopes in vZome

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 • Polytope operations