4₂₁ polytope

• This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure.[1] It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.
• E. L. Elte named it V₂₄₀ (for its 240 vertices) in his 1912 listing of semiregular polytopes.[2]
• H.S.M. Coxeter called it 4₂₁ because its Coxeter-Dynkin diagram has three branches of length 4, 2, and 1, with a single node on the terminal node of the 4 branch.
• Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton (Acronym Fy) - 2160-17280 facetted polyzetton (Jonathan Bowers)[3]

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The 240 vertices of the 4₂₁ polytope can be constructed in two sets: 112 (2²×⁸C₂) with coordinates obtained from ${\displaystyle (\pm 2,\pm 2,0,0,0,0,0,0)\,}$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2⁷) with coordinates obtained from ${\displaystyle (\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,}$ by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be a multiple of 4).

Each vertex has 56 nearest neighbors; for example, the nearest neighbors of the vertex ${\displaystyle (1,1,1,1,1,1,1,1)}$ are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of ${\displaystyle (2,2,0,0,0,0,0,0)\,}$ and the 28 obtained by permuting the coordinates of ${\displaystyle (1,1,1,1,1,1,-1,-1)}$. These 56 points are the vertices of a 3₂₁ polytope in 7 dimensions.

Each vertex has 126 second nearest neighbors: for example, the nearest neighbors of the vertex ${\displaystyle (1,1,1,1,1,1,1,1)}$ are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of ${\displaystyle (2,-2,0,0,0,0,0,0)\,}$ and the 70 obtained by permuting the coordinates of ${\displaystyle (1,1,1,1,-1,-1,-1,-1)}$. These 126 points are the vertices of a 2₃₁ polytope in 7 dimensions.

Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, and one antipodal vertex, for a total of ${\displaystyle 1+56+126+56+1=240}$ vertices.

Another construction is by taking signed combination of 14 codewords of 8-bit Extended Hamming code(8,4) that give 14 × 2⁴ = 224 vertices and adding trivial signed axis ${\displaystyle (\pm 2,0,0,0,0,0,0,0)}$ for last 16 vertices. In this case, vertices are distance of ${\displaystyle {\sqrt {4}}}$ from origin rather than ${\displaystyle {\sqrt {8}}}$.

  Hamming 8-bit Code
  0  0 0 0 0 0 0 0 0
  1  1 1 1 1 0 0 0 0   ⇒   ± ± ± ± 0 0 0 0
  2  1 1 0 0 1 1 0 0   ⇒   ± ± 0 0 ± ± 0 0
  3  0 0 1 1 1 1 0 0   ⇒   0 0 ± ± ± ± 0 0
  4  1 0 1 0 1 0 1 0   ⇒   ± 0 ± 0 ± 0 ± 0        ±2 0 0 0 0 0 0 0
  5  0 1 0 1 1 0 1 0   ⇒   0 ± 0 ± ± 0 ± 0        0 ±2 0 0 0 0 0 0
  6  0 1 1 0 0 1 1 0   ⇒   0 ± ± 0 0 ± ± 0        0 0 ±2 0 0 0 0 0
  7  1 0 0 1 0 1 1 0   ⇒   ± 0 0 ± 0 ± ± 0        0 0 0 ±2 0 0 0 0
  8  0 1 1 0 1 0 0 1   ⇒   0 ± ± 0 ± 0 0 ±        0 0 0 0 ±2 0 0 0
  9  1 0 0 1 1 0 0 1   ⇒   ± 0 0 ± ± 0 0 ±        0 0 0 0 0 ±2 0 0
  A  1 0 1 0 0 1 0 1   ⇒   ± 0 ± 0 0 ± 0 ±        0 0 0 0 0 0 ±2 0
  B  0 1 0 1 0 1 0 1   ⇒   0 ± 0 ± 0 ± 0 ±        0 0 0 0 0 0 0 ±2
  C  1 1 0 0 0 0 1 1   ⇒   ± ± 0 0 0 0 ± ±
  D  0 0 1 1 0 0 1 1   ⇒   0 0 ± ± 0 0 ± ±
  E  0 0 0 0 1 1 1 1   ⇒   0 0 0 0 ± ± ± ±
  F  1 1 1 1 1 1 1 1
                           ( 224 vertices     +     16 vertices )

Another decomposition gives the 240 points in 9-dimensions as an expanded 8-simplex, and two opposite birectified 8-simplexes, and .

(3,-3,0,0,0,0,0,0,0) : 72 vertices

(-2,-2,-2,1,1,1,1,1,1) : 84 vertices

(2,2,2,-1,-1,-1,-1,-1,-1) : 84 vertices

This arises similarly to the relation of the A8 lattice and E8 lattice, sharing 8 mirrors of A8: .

A7 Coxeter plane projections

Name	421	expanded 8-simplex	birectified 8-simplex	birectified 8-simplex
Vertices	240	72	84	84
Image

This polytope is the vertex figure for a uniform tessellation of 8-dimensional space, represented by symbol 5₂₁ and Coxeter-Dynkin diagram:

The facet information of this polytope can be extracted from its Coxeter-Dynkin diagram:

Removing the node on the short branch leaves the 7-simplex:

Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form (4₁₁):

Every 7-simplex facet touches only 7-orthoplex facets, while alternate facets of an orthoplex facet touch either a simplex or another orthoplex. There are 17,280 simplex facets and 2160 orthoplex facets.

Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, the 4₂₁ polytope has 120,960 (7×17,280) 6-simplex faces that are facets of 7-simplexes. Since every 7-orthoplex has 128 (2⁷) 6-simplex facets, half of which are not incident to 7-simplexes, the 4₂₁ polytope has 138,240 (2⁶×2160) 6-simplex faces that are not facets of 7-simplexes. The 4₂₁ polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope. The total number of 6-simplex faces is 259200 (120,960+138,240).

The vertex figure of a single-ring polytope is obtained by removing the ringed node and ringing its neighbor(s). This makes the 3₂₁ polytope.

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

		Configuration matrix
E8		k-face	fk	f0	f1	f2	f3	f4	f5	f6		f7		k-figure	notes
E7		( )	f0	240	56	756	4032	10080	12096	4032	2016	576	126	3_21 polytope	E8/E7 = 192×10!/(72×8!) = 240
A1E6		{ }	f1	2	6720	27	216	720	1080	432	216	72	27	2_21 polytope	E8/A1E6 = 192×10!/(2×72×6!) = 6720
A2D5		{3}	f2	3	3	60480	16	80	160	80	40	16	10	5-demicube	E8/A2D5 = 192×10!/(6×24×5!) = 60480
A3A4		{3,3}	f3	4	6	4	241920	10	30	20	10	5	5	Rectified 5-cell	E8/A3A4 = 192×10!/(4!×5!) = 241920
A4A2A1		{3,3,3}	f4	5	10	10	5	483840	6	6	3	2	3	Triangular prism	E8/A4A2A1 = 192×10!/(5!×3!×2) = 483840
A5A1		{3,3,3,3}	f5	6	15	20	15	6	483840	2	1	1	2	Isosceles triangle	E8/A5A1 = 192×10!/(6!×2) = 483840
A6		{3,3,3,3,3}	f6	7	21	35	35	21	7	138240	*	1	1	{ }	E8/A6 = 192×(10!×7!) = 138240
A6A1				7	21	35	35	21	7	*	69120	0	2		E8/A6A1 = 192×10!/(7!×2) = 69120
A7		{3,3,3,3,3,3}	f7	8	28	56	70	56	28	8	0	17280	*	( )	E8/A7 = 192×10!/8! = 17280
D7		{3,3,3,3,3,4}		14	84	280	560	672	448	64	64	*	2160		E8/D7 = 192×10!/(26×7!) = 2160

These graphs represent orthographic projections in the E₈, E₇, E₆, and B₈, D₈, D₇, D₆, D₅, D₄, D₃, A₇, A₅ Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

Orthogonal projections
E8 / H4 [30]	[20]	[24]
(Colors: 1)	(Colors: 1)	(Colors: 1)
E7 [18]	E6 / F4 [12]	[6]
(Colors: 1,3,6)	(Colors: 1,8,24)	(Colors: 1,2,3)
D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
(Colors: 1,12,32,60)	(Colors: 1,27,72)	(Colors: 1,8,24)
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
(Colors: 1,5,10,20)	(Colors: 1,3,9,12)	(Colors: 1,2,3)
B8 [16/2]	A5 [6]	A7 [8]
(Colors: 1)	(Colors: 3,8,24,30)	(Colors: 1,2,4,8)

The 4₂₁ polytope is last in a family called the k₂₁ polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).

The 4₂₁ polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric 600-cells (at the golden ratio) using Zome tools.[6] (Not all of the 3360 edges of length √2(√5-1) are represented.)

The 4₂₁ is related to the 600-cell by a geometric folding of the Coxeter-Dynkin diagrams. This can be seen in the E8/H4 Coxeter plane projections. The 240 vertices of the 4₂₁ polytope are projected into 4-space as two copies of the 120 vertices of the 600-cell, one copy smaller (scaled by the golden ratio) than the other with the same orientation. Seen as a 2D orthographic projection in the E8/H4 Coxeter plane, the 120 vertices of the 600-cell are projected in the same four rings as seen in the 4₂₁. The other 4 rings of the 4₂₁ graph also match a smaller copy of the four rings of the 600-cell.

E8/H4 Coxeter plane foldings
E8	H4
421	600-cell
[20] symmetry planes
421	600-cell

In 4-dimensional complex geometry, the regular complex polytope ₃{3}₃{3}₃{3}₃, and Coxeter diagram exists with the same vertex arrangement as the 4₂₁ polytope. It is self-dual. Coxeter called it the Witting polytope, after Alexander Witting. Coxeter expresses its Shephard group symmetry by ₃[3]₃[3]₃[3]₃.[7]

The 4₂₁ is sixth 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

• Rectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for rectified 2160-17280 polyzetton (Acronym riffy) (Jonathan Bowers)[8]

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a rectification of the 4₂₁. Vertices are positioned at the midpoint of all the edges of 4₂₁, and new edges connecting them.

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

Removing the node on the short branch leaves the rectified 7-simplex:

Removing the node on the end of the 2-length branch leaves the rectified 7-orthoplex in its alternated form:

Removing the node on the end of the 4-length branch leaves the 3₂₁:

The vertex figure is determined by removing the ringed node and adding a ring to the neighboring node. This makes a 2₂₁ prism.

The Cartesian coordinates of the 6720 vertices of the rectified 4₂₁ is given by all permutations of coordinates from three other uniform polytope:

• hexic 8-cube - odd negatives: ½(±1,±1,±1,±1,±1,±1,±3,±3) - 3584 vertices[9]
• birectified 8-cube - (0,0,±1,±1,±1,±1,±1,±1) - 1792 vertices[10]
• cantellated 8-orthoplex - (0,0,0,0,0,0,±1,±1,±2) - 1344 vertices[11]

D8 Coxeter plane projections

Name	Rectified 421	birectified 8-cube =	hexic 8-cube =	cantellated 8-orthoplex =
Vertices	6720	1792	3584	1344
Image

These graphs represent orthographic projections in the E₈, E₇, E₆, and B₈, D₈, D₇, D₆, D₅, D₄, D₃, A₇, A₅ Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

Orthogonal projections
E8 / H4 [30]	[20]	[24]
E7 [18]	E6 / F4 [12]	[6]
D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
B8 [16/2]	A5 [6]	A7 [8]

• Birectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for birectified 2160-17280 polyzetton (acronym borfy) (Jonathan Bowers)[12]

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a birectification of the 4₂₁. Vertices are positioned at the center of all the triangle faces of 4₂₁.

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

Removing the node on the short branch leaves the birectified 7-simplex. There are 17280 of these facets.

Removing the node on the end of the 2-length branch leaves the birectified 7-orthoplex in its alternated form. There are 2160 of these facets.

Removing the node on the end of the 4-length branch leaves the rectified 3₂₁. There are 240 of these facets.

The vertex figure is determined by removing the ringed node and adding rings to the neighboring nodes. This makes a 5-demicube-triangular duoprism.

These graphs represent orthographic projections in the E₈, E₇, E₆, and B₈, D₈, D₇, D₆, D₅, D₄, D₃, A₇, A₅ Coxeter planes. Edges are not drawn. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green, etc.

Orthogonal projections
E8 / H4 [30]	[20]	[24]
E7 [18]	E6 / F4 [12]	[6]
D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
B8 [16/2]	A5 [6]	A7 [8]

• Trirectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for trirectified 2160-17280 polyzetton (acronym torfy) (Jonathan Bowers)[13]

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a birectification of the 4₂₁. Vertices are positioned at the center of all the triangle faces of 4₂₁.

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

Removing the node on the short branch leaves the trirectified 7-simplex:

Removing the node on the end of the 2-length branch leaves the trirectified 7-orthoplex in its alternated form:

Removing the node on the end of the 4-length branch leaves the birectified 3₂₁:

The vertex figure is determined by removing the ringed node and ring the neighbor nodes. This makes a tetrahedron-rectified 5-cell duoprism.

These graphs represent orthographic projections in the E₇, E₆, B₈, D₈, D₇, D₆, D₅, D₄, D₃, A₇, and A₅ Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

(E₈ and B₈ were too large to display)

Orthogonal projections
E7 [18]	E6 / F4 [12]	D4 - E6 [6]
D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
A5 [6]	A7 [8]