Papyrus 90

Orthogonal projections in E6 Coxeter plane
421 Template:CDD	File:1 42 polytope E6 Coxeter plane.svg 142 Template:CDD	File:2 41 t0 E6.svg 241 Template:CDD
File:4 21 t1 E6.svg Rectified 421 Template:CDD	File:4 21 t4 E6.svg Rectified 142 Template:CDD	File:2 41 t1 E6.svg Rectified 241 Template:CDD
File:4 21 t2 E6.svg Birectified 421 Template:CDD	File:4 21 t3 E6.svg Trirectified 421 Template:CDD

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.

Coxeter named it 4₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, Template:CDD.

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: Template:CDD.

4₂₁ polytope

421
Type	Uniform 8-polytope
Family	k21 polytope
Schläfli symbol	{3,3,3,3,32,1}
Coxeter symbol	421
Coxeter-Dynkin diagram	Template:CDD
7-faces	19440 total: 2160 411File:7-orthoplex.svg 17280 {36}File:7-simplex t0.svg
6-faces	207360: 138240 {35}File:6-simplex t0.svg 69120 {35}File:6-simplex t0.svg
5-faces	483840 {34}File:5-simplex t0.svg
4-faces	483840 {33}File:4-simplex t0.svg
Cells	241920 {3,3}File:3-simplex t0.svg
Faces	60480 {3}File:2-simplex t0.svg
Edges	6720
Vertices	240
Vertex figure	321 polytope
Petrie polygon	30-gon
Coxeter group	E8, [34,2,1]
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. 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.
• 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)

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:

Template:CDD

Construction

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

Template:CDD

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

Template:CDD

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

Template:CDD

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

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.

Template:CDD

Projections

File:E8-with-thread.jpg The 421 graph created as string art.

File:E8Petrie.svg E8 Coxeter plane projection

3D

File:Zome-like.png 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

File:E8 3D.png The actual split real even E8 421 polytope projected into perspective 3-space pictured with all 6720 edges of length √2

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]
File:4 21 t0 E8.svg (Colors: 1)	File:4 21 t0 p20.svg (Colors: 1)	File:4 21 t0 p24.svg (Colors: 1)
E7 [18]	E6 / F4 [12]	[6]
File:4 21 t0 E7.svg (Colors: 1,3,6)	Error creating thumbnail: (Colors: 1,8,24)	File:4 21 t0 mox.svg (Colors: 1,2,3)

D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
File:4 21 t0 B2.svg (Colors: 1,12,32,60)	File:4 21 t0 B3.svg (Colors: 1,27,72)	File:4 21 t0 B4.svg (Colors: 1,8,24)
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
File:4 21 t0 B5.svg (Colors: 1,5,10,20)	File:4 21 t0 B6.svg (Colors: 1,3,9,12)	File:4 21 t0 B7.svg (Colors: 1,2,3)
B8 [16/2]	A5 [6]	A7 [8]
File:4 21 t0 B8.svg (Colors: 1)	File:4 21 t0 A5.svg (Colors: 3,8,24,30)	File:4 21 t0 A7.svg (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

File:E8 roots zome.jpg

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. (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 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 planes
E8 Template:CDD	H4 Template:CDD
File:4 21 t0 E8.svg 421	File:600-cell graph H4.svg 600-cell
[20] symmetry planes
File:4 21 t0 p20.svg 421	File:600-cell t0 p20.svg 600-cell

Related polytopes

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.

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. Template:Gosset semiregular polytopes

Rectified 4_21 polytope

Rectified 421
Type	Uniform 8-polytope
Schläfli symbol	t1{3,3,3,3,32,1}
Coxeter symbol	t1(421)
Coxeter-Dynkin diagram	Template:CDD
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, [34,2,1]
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)

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.

Template:CDD

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

Template:CDD

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

Template:CDD

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

Template:CDD

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

Template:CDD

Projections

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]
File:4 21 t1 E8.svg	File:4 21 t1 p20.svg	File:4 21 t1 p24.svg
E7 [18]	E6 / F4 [12]	[6]
File:4 21 t1 E7.svg	File:4 21 t1 E6.svg	File:4 21 t1 mox.svg

D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
File:4 21 t1 B2.svg	File:4 21 t1 B3.svg	File:4 21 t1 B4.svg
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
File:4 21 t1 B5.svg	File:4 21 t1 B6.svg	File:4 21 t1 B7.svg
B8 [16/2]	A5 [6]	A7 [8]
File:4 21 t1 B8.svg	File:4 21 t1 A5.svg	File:4 21 t1 A7.svg

Birectified 4_21 polytope

Birectified 421 polytope
Type	Uniform 8-polytope
Schläfli symbol	t2{3,3,3,3,32,1}
Coxeter symbol	t2(421)
Coxeter-Dynkin diagram	Template:CDD
7-faces	19680 total: 17280 t2{36} File:7-simplex t2.svg 2160 t2{35,4} File:7-cube t4.svg 240 t1(321) File:Up2 3 21 t1 E7.svg
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, [34,2,1]
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)

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.

Template:CDD

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

Template:CDD

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.

Template:CDD

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

Template:CDD

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

Template:CDD

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]
File:4 21 t2 E8.svg	File:4 21 t2 p20.svg	File:4 21 t2 p24.svg
E7 [18]	E6 / F4 [12]	[6]
File:4 21 t2 E7.svg	File:4 21 t2 E6.svg	File:4 21 t2 mox.svg

D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
File:4 21 t2 B2.svg	File:4 21 t2 B3.svg	File:4 21 t2 B4.svg
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
File:4 21 t2 B5.svg	File:4 21 t2 B6.svg	File:4 21 t2 B7.svg
B8 [16/2]	A5 [6]	A7 [8]
File:4 21 t2 B8.svg	File:4 21 t2 A5.svg	File:4 21 t2 A7.svg

Trirectified 4_21 polytope

Trirectified 421 polytope
Type	Uniform 8-polytope
Schläfli symbol	t3{3,3,3,3,32,1}
Coxeter symbol	t3(421)
Coxeter-Dynkin diagram	Template:CDD
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, [34,2,1]
Properties	convex

Alternative names

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

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.

Template:CDD

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

Template:CDD

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

Template:CDD

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

Template:CDD

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

Template:CDD

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]
File:4 21 t3 E7.svg	File:4 21 t3 E6.svg	File:4 21 t3 mox.svg

D3 / B2 / A3 [4]	D4 / B3 / A2 / G2 [6]	D5 / B4 [8]
File:4 21 t3 B2.svg	File:4 21 t3 B3.svg	File:4 21 t3 B4.svg
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
File:4 21 t3 B6.svg	File:4 21 t3 B7.svg
A5 [6]	A7 [8]
File:4 21 t3 A5.svg	File:4 21 t3 A7.svg

See also

• List of E8 polytopes

Notes

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References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
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• 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)
• Template:KlitzingPolytopes o3o3o3o *c3o3o3o3x - fy, o3o3o3o *c3o3o3x3o - riffy, o3o3o3o *c3o3x3o3o - borfy, o3o3o3o *c3x3o3o3o - torfy