3₂₁ polytope

• 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-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)[3]

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)

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

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 dimensional
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]	[[3<sup>1,3,1</sup>]] = [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 hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)[5]

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, .

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

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, .