4₂₁ polytope

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

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]

Coxeter named it 4₂₁ by 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: .

4_21 polytope

4₂₁
Type	Uniform 8-polytope
Family	k₂₁ polytope
Schläfli symbol	{3,3,3,3,3^2,1}
Coxeter symbol	4₂₁
Coxeter-Dynkin diagram
7-faces	19440 total: 2160 4₁₁ 17280 {3⁶}
6-faces	207360: 138240 {3⁵} 68120 {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]
Properties	convex

The 4₂₁ is composed of 17,280 7-simplex and 2,160 7-orthoplex facets. Its vertex figure is the 3₂₁ polytope.

For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a 30-gonal regular polygon (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.

As its 240 vertices represent the root vectors of the simple Lie group E₈, the polytope is sometimes referred to as the E₈ polytope.

Alternate names

It 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]

Using a complex number coordinate system, it can also be constructed as a 4-dimensional regular complex polytope, named as: 3{3}3{3}3{3}3. Coxeter called it the Witting polytope, after Alexander Witting.^[4]

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 even).

Tessellations

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

Construction

The facet information 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 simplex facet touches an 7-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the red node and node_1 the neighbor node. This makes the 3₂₁ polytope.

Projections

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

3D

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.^[5] (Not all of the 3360 edges of length √2(√5-1) are represented.)	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^[6]

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.

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 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 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 planes
E₈	H₄
4₂₁	600-cell
[20] symmetry planes
4₂₁	600-cell

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-Dynkin 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)^[7]

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 red node and node the neighbor nodea. This makes a 2₂₁ prism.

Projections

2D

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 4_21 polytope

Birectified 4₂₁ polytope
Type	Uniform 8-polytope
Schläfli symbol	t₂{3,3,3,3,3^2,1}
Coxeter symbol	t₂(4₂₁)
Coxeter-Dynkin diagram
7-faces	19680 total: 17280 t₂{3⁶} 2160 t₂{3⁵,4} 240 t₁(3₂₁)
6-faces	382560
5-faces	2600640
4-faces	7741440
Cells	9918720
Faces	5806080
Edges	1451520
Vertices	60480
Vertex figure	5-demicube-triangular duoprism
Coxeter group	E₈, [3^4,2,1]
Properties	convex

The birectified 4₂₁can be seen as a second rectification of the uniform 4₂₁ polytope. Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 4₂₁.

Alternative names

Birectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for birectified 2160-17280 polyzetton (acronym borfy) (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 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 red node and node the neighbor nodea. This makes a 5-demicube-triangular duoprism.

Projections

2D

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.

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 4_21 polytope

Trirectified 4₂₁ polytope
Type	Uniform 8-polytope
Schläfli symbol	t₃{3,3,3,3,3^2,1}
Coxeter symbol	t₃(4₂₁)
Coxeter-Dynkin diagram
7-faces	19680
6-faces	382560
5-faces	2661120
4-faces	9313920
Cells	16934400
Faces	14515200
Edges	4838400
Vertices	241920
Vertex figure	tetrahedron-rectified 5-cell duoprism
Coxeter group	E₈, [3^4,2,1]
Properties	convex

Alternative names

Trirectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for trirectified 2160-17280 polyzetton (acronym torfy) (Jonathan Bowers)^[9]

Construction

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.

Projections

2D

These graphs represent orthographic projections in the 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.

(E₈ and B₈ were too large to display)

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]

Notes

^ ^a ^b Gosset, 1900
^ Elte, 1912
^ Klitzing, (o3o3o3o *c3o3o3o3x - fy)
^ Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
^ David Richter: Gosset's Figure in 8 Dimensions, A Zome Model
^ e8Flyer.nb
^ Klitzing, (o3o3o3o *c3o3o3x3o - riffy)
^ Klitzing, (o3o3o3o *c3o3x3o3o - borfy)
^ Klitzing, (o3o3o3o *c3x3o3o3o - torfy)

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
Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, (1974).
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, ISBN 978-0-471-01003-6 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p347 (figure 3.8c) by Peter mcMullen: (30-gonal node-edge graph of 4₂₁)
Richard Klitzing, 8D, uniform polytopes (polyzetta) o3o3o3o *c3o3o3o3x - fy, o3o3o3o *c3o3o3x3o - riffy, o3o3o3o *c3o3x3o3o - borfy, o3o3o3o *c3x3o3o3o - torfy


Family	A_n	BC_n	D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes

4₂₁ polytope

Contents

4_21 polytope

Alternate names

Coordinates

Tessellations

Construction

Projections

3D

2D

k₂₁ family

Geometric folding

Rectified 4_21 polytope

Alternative names

Construction

Projections

2D

Birectified 4_21 polytope

Alternative names

Construction

Projections

2D

Trirectified 4_21 polytope

Alternative names

Construction

Projections

2D

See also

Notes

References

4 21 polytope

Contents

4_21 polytope

Alternate names

Coordinates

Tessellations

Construction

Projections

3D

2D

k21 family

Geometric folding

Rectified 4_21 polytope

Alternative names

Construction

Projections

2D

Birectified 4_21 polytope

Alternative names

Construction

Projections

2D

Trirectified 4_21 polytope

Alternative names

Construction

Projections

2D

See also

Notes

References

4₂₁ polytope

k₂₁ family