4₂₁ polytope

In 8-dimensional geometry, the 4₂₁ is a semiregular uniform 8-polytope, constructed within the symmetry of the E₈ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.[1]

Orthogonal projections in E₆ Coxeter plane
4₂₁	1₄₂	2₄₁
Rectified 4₂₁	Rectified 1₄₂	Rectified 2₄₁
Birectified 4₂₁	Trirectified 4₂₁

Its Coxeter symbol is 4₂₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .

The rectified 4₂₁ is constructed by points at the mid-edges of the 4₂₁. The birectified 4₂₁ is constructed by points at the triangle face centers of the 4₂₁. The trirectified 4₂₁ is constructed by points at the tetrahedral centers of the 4₂₁, and is the same as the rectified 1₄₂.

These polytopes are part of a family of 255 = 2⁸ − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: .

4₂₁ polytope

4₂₁
Type	Uniform 8-polytope
Family	k₂₁ polytope
Schläfli symbol	{3,3,3,3,3^2,1}
Coxeter symbol	4₂₁
Coxeter diagrams	=
7-faces	19440 total: 2160 4₁₁ 17280 {3⁶}
6-faces	207360: 138240 {3⁵} 69120 {3⁵}
5-faces	483840 {3⁴}
4-faces	483840 {3³}
Cells	241920 {3,3}
Faces	60480 {3}
Edges	6720
Vertices	240
Vertex figure	3₂₁ polytope
Petrie polygon	30-gon
Coxeter group	E₈, [3^4,2,1], order 696729600
Properties	convex

The 4₂₁ polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 3₂₁ polytope. As its vertices represent the root vectors of the simple Lie group E₈, this polytope is sometimes referred to as the E₈ root polytope.

The vertices of this polytope can also be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop.

For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

Alternate names

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]

Coordinates

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 $(\pm 2,\pm 2,0,0,0,0,0,0)\,$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2⁷) with coordinates obtained from $(\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,$ 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 $(1,1,1,1,1,1,1,1)$ are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of $(2,2,0,0,0,0,0,0)\,$ and the 28 obtained by permuting the coordinates of $(1,1,1,1,1,1,-1,-1)$ . 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 $(1,1,1,1,1,1,1,1)$ are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of $(2,-2,0,0,0,0,0,0)\,$ and the 70 obtained by permuting the coordinates of $(1,1,1,1,-1,-1,-1,-1)$ . 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 $1+56+126+56+1=240$ 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	Rectified 4₂₁	expanded 8-simplex	birectified 8-simplexes	birectified 8-simplexes
Vertices	240	72	84	84
Image

Tessellations

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

Construction and faces

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]

E₈	k-face	f_k	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	{3,3,3,3,3}	f₆	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}	f₇	14	84	280	560	672	448	64	64	*	2160	( )	E₈/D₇ = 192×10!/(2⁶×7!) = 2160

Projections

The 4₂₁ graph created as string art.	E₈ Coxeter plane projection

3D

Mathematical representation of the physical Zome model isomorphic (?) to E8. This is constructed from VisibLie_E8 pictured with all 3360 edges of length √2(√5-1) from two concentric 600-cells (at the golden ratio) with orthogonal projections to perspective 3-space	The actual split real even E8 4₂₁ polytope projected into perspective 3-space pictured with all 6720 edges of length √2 [5]	E8 rotated to H4+H4φ, projected to 3D, converted to STL, and printed in nylon plastic. Projection basis used: x = {1, φ, 0, -1, φ, 0,0,0} y = {φ, 0, 1, φ, 0, -1,0,0} z = {0, 1, φ, 0, -1, φ,0,0}

2D

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)

k₂₁ family

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).

Geometric folding

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

Related polytopes

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 dimensional
Space	Finite						Euclidean	Hyperbolic
E_n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
Symmetry	[3^−1,2,1]	[3^0,2,1]	[3^1,2,1]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−1₂₁	0₂₁	1₂₁	2₂₁	3₂₁	4₂₁	5₂₁	6₂₁

Rectified 4_21 polytope

Rectified 4₂₁
Type	Uniform 8-polytope
Schläfli symbol	t₁{3,3,3,3,3^2,1}
Coxeter symbol	t₁(4₂₁)
Coxeter diagram
7-faces	19680 total: 240 3₂₁ 17280 t₁{3⁶} 2160 t₁{3⁵,4}
6-faces	375840
5-faces	1935360
4-faces	3386880
Cells	2661120
Faces	1028160
Edges	181440
Vertices	6720
Vertex figure	2₂₁ prism
Coxeter group	E₈, [3^4,2,1]
Properties	convex

The rectified 4₂₁ can be seen as a rectification of the 4₂₁ polytope, creating new vertices on the center of edges of the 4₂₁.

Alternative names

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

Construction

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.