1 ₄₂ polytope

Orthogonal projections in E₆ Coxeter plane
File:4 21 t0 E6.svg 4₂₁ File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:1 42 polytope E6 Coxeter plane.svg 1₄₂ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:2 41 t0 E6.svg 2₄₁ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.png
File:4 21 t1 E6.svg Rectified 4₂₁ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:4 21 t4 E6.svg Rectified 1₄₂ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:2 41 t1 E6.svg Rectified 2₄₁ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.png
File:4 21 t2 E6.svg Birectified 4₂₁ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:4 21 t3 E6.svg Trirectified 4₂₁ File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

In 8-dimensional geometry, the 1₄₂ is a uniform 8-polytope, constructed within the symmetry of the E₈ group. Its Coxeter symbol is 1₄₂, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 1₄₂ is constructed by points at the mid-edges of the 1₄₂ and is the same as the birectified 2₄₁, and the quadrirectified 4₂₁. These polytopes are part of a family of 255 (2⁸ − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex figures, defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram: File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png.

1₄₂ polytope

1₄₂
Type	Uniform 8-polytope
Family	1_k2 polytope
Schläfli symbol	{3,3^4,2}
Coxeter symbol	1₄₂
Coxeter diagrams	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
7-faces	2400: 240 1₃₂File:Gosset 1 32 petrie.svg 2160 1₄₁File:Demihepteract ortho petrie.svg
6-faces	106080: 6720 1₂₂File:Gosset 1 22 polytope.svg 30240 1₃₁File:Demihexeract ortho petrie.svg 69120 {3⁵}File:6-simplex t0.svg
5-faces	725760: 60480 1₁₂File:Demipenteract graph ortho.svg 181440 1₂₁File:Demipenteract graph ortho.svg 483840 {3⁴}File:5-simplex t0.svg
4-faces	2298240: 241920 1₀₂File:4-simplex t0.svg 604800 1₁₁File:4-cube t3.svg 1451520 {3³}File:4-simplex t0.svg
Cells	3628800: 1209600 1₀₁File:3-simplex t0.svg 2419200 {3²}File:3-simplex t0.svg
Faces	2419200 {3}File:2-simplex t0.svg
Edges	483840
Vertices	17280
Vertex figure	t₂{3⁶} File:7-simplex t2.svg
Petrie polygon	30-gon
Coxeter group	E₈, [3^4,2,1]
Properties	convex

The 1₄₂ is composed of 2400 facets: 240 1₃₂ polytopes, and 2160 7-demicubes (1₄₁). Its vertex figure is a birectified 7-simplex. This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 1₅₂, and Coxeter-Dynkin diagram: File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png.

Alternate names

E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V₁₇₂₈₀ for its 17280 vertices.^[1]
Coxeter named it 1₄₂ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
Diacositetracont-dischiliahectohexaconta-zetton (acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers)^[2]

Coordinates

The 17280 vertices can be defined as sign and location permutations of: All sign combinations (32): (280×32=8960 vertices)

(4, 2, 2, 2, 2, 0, 0, 0)

Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)

(2, 2, 2, 2, 2, 2, 2, 2)

(5, 1, 1, 1, 1, 1, 1, 1)

(3, 3, 3, 1, 1, 1, 1, 1)

The edge length is 2√2 in this coordinate set, and the polytope radius is 4√2.

Construction

		Configuration matrix
E₈	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	k-face	f_k	f₀	f₁	f₂	f₃		f₄			f₅			f₆			f₇		k-figure	notes
A₇	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 0x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	( )	f₀	17280	56	420	280	560	70	280	420	56	168	168	28	56	28	8	8	2r{3⁶}	E₈/A₇ = 192*10!/8! = 17280
A₄A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodes x1.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	{ }	f₁	2	483840	15	15	30	5	30	30	10	30	15	10	15	3	5	3	{3}x{3,3,3}	E₈/A₄A₂A₁ = 192*10!/5!/2/2 = 483840
A₃A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch 01.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	{3}	f₂	3	3	2419200	2	4	1	8	6	4	12	4	6	8	1	4	2	{3.3}v{ }	E₈/A₃A₂A₁ = 192*10!/4!/3!/2 = 2419200
A₃A₃	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch 01r.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	1₁₀	f₃	4	6	4	1209600	*	1	4	0	4	6	0	6	4	0	4	1	{3,3}v( )	E₈/A₃A₃ = 192*10!/4!/4! = 1209600
A₃A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01l.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	1₁₀	f₃	4	6	4	*	2419200	0	2	3	1	6	3	3	6	1	3	2	{3}v{ }	E₈/A₃A₂A₁ = 192*10!/4!/3!/2 = 2419200
A₄A₃	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch 01r.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	1₂₀	f₄	5	10	10	5	0	241920	*	*	4	0	0	6	0	0	4	0	{3,3}	E₈/A₄A₃ = 192*10!/4!/4! = 241920
D₄A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	1₁₁		8	24	32	8	8	*	604800	*	1	3	0	3	3	0	3	1	{3}v( )	E₈/D₄A₂ = 192*10!/8/4!/3! = 604800
A₄A₁A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01l.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	1₂₀		5	10	10	0	5	*	*	1451520	0	2	2	1	4	1	2	2	{ }v{ }	E₈/A₄A₁A₁ = 192*10!/5!/2/2 = 1451520
D₅A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	1₂₁	f₅	16	80	160	80	40	16	10	0	60480	*	*	3	0	0	3	0	{3}	E₈/D₅A₂ = 192*10!/16/5!/3! = 40480
D₅A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	1₂₁		16	80	160	40	80	0	10	16	*	181440	*	1	2	0	2	1	{ }v( )	E₈/D₅A₁ = 192*10!/16/5!/2 = 181440
A₅A₁	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01l.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	1₃₀		6	15	20	0	15	0	0	6	*	*	483840	0	2	1	1	2	{ }v( )	E₈/A₅A₁ = 192*10!/6!/2 = 483840
E₆A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	1₂₂	f₆	72	720	2160	1080	1080	216	270	216	27	27	0	6720	*	*	2	0	{ }	E₈/E₆A₁ = 192*10!/72/6!/2 = 6720
D₆	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	1₃₁		32	240	640	160	480	0	60	192	0	12	32	*	30240	*	1	1		E₈/D₆ = 192*10!/32/6! = 30240
A₆A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01l.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	1₄₀		7	21	35	0	35	0	0	21	0	0	7	*	*	69120	0	2		E₈/A₆A₁ = 192*10!/7!/2 = 69120
E₇	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	1₃₂	f₇	576	10080	40320	20160	30240	4032	7560	12096	756	1512	2016	56	126	0	240	*	( )	E₈/E₇ = 192*10!/72/8! = 240
D₇	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	1₄₁	f₇	64	672	2240	560	2240	0	280	1344	0	84	448	0	14	64	*	2160	( )	E₈/D₇ = 192*10!/64/7! = 2160

Projections

File:E8 142 Petrie Projection.png

The projection of 1₄₂ to the E₈ Coxeter plane (aka. the Petrie projection) with polytope radius

4 \sqrt{2}

is shown below with 483,840 edges of length

2 \sqrt{2}

culled 53% on the interior to only 226,444:

File:E8 142-3D Concentric Hulls.png

Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:

u = (1, φ, 0, −1, φ, 0,0,0)
v = (φ, 0, 1, φ, 0, −1,0,0)
w = (0, 1, φ, 0, −1, φ,0,0)

The 17280 projected 1₄₂ polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls for each set of tallied norms. Notice the last two outer hulls are a combination of two overlapped Dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).

E8 [30]	E7 [18]	E6 [12]
File:Gosset 1 42 polytope petrie.svg (1)	File:1 42 t0 e7.svg (1,3,6)	File:1 42 polytope E6 Coxeter plane.svg (8,16,24,32,48,64,96)
[20]	[24]	[6]
File:1 42 t0 p20.svg	File:1 42 t0 p24.svg	File:1 42 t0 mox.svg (1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20)

Orthographic projections are shown for the sub-symmetries of E₈: E₇, E₆, B₈, B₇, B₆, B₅, B₄, B₃, B₂, A₇, and A₅ Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
File:1 42 t0 B2.svg (32,160,192,240,480,512,832,960)	File:1 42 t0 B3.svg (72,216,432,720,864,1080)	File:1 42 t0 B4.svg (8,16,24,32,48,64,96)
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
File:1 42 t0 B5.svg	File:1 42 t0 B6.svg	File:1 42 t0 B7.svg
B8 [16/2]	A5 [6]	A7 [8]
File:1 42 t0 B8.svg	File:1 42 t0 A5.svg	File:1 42 t0 A7.svg

Related polytopes and honeycombs

1_k2 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
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	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01l.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
Symmetry (order)	[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	103,680	2,903,040	696,729,600	∞
Graph	File:Trigonal hosohedron.png	File:4-simplex t0.svg	File:Demipenteract graph ortho.svg	File:Up 1 22 t0 E6.svg	File:Up2 1 32 t0 E7.svg	File:Gosset 1 42 polytope petrie.svg	-	-
Name	1_−1,2	1₀₂	1₁₂	1₂₂	1₃₂	1₄₂	1₅₂	1₆₂

Rectified 1₄₂ polytope

Rectified 1₄₂
Type	Uniform 8-polytope
Schläfli symbol	t₁{3,3^4,2}
Coxeter symbol	0₄₂₁
Coxeter diagrams	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
7-faces	19680
6-faces	382560
5-faces	2661120
4-faces	9072000
Cells	16934400
Faces	16934400
Edges	7257600
Vertices	483840
Vertex figure	{3,3,3}×{3}×{}
Coxeter group	E₈, [3^4,2,1]
Properties	convex

The rectified 1₄₂ is named from being a rectification of the 1₄₂ polytope, with vertices positioned at the mid-edges of the 1₄₂. It can also be called a 0₄₂₁ polytope with the ring at the center of 3 branches of length 4, 2, and 1.

Alternate names

0₄₂₁ polytope
Birectified 2₄₁ polytope
Quadrirectified 4₂₁ polytope
Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym buffy) (Jonathan Bowers)^[4]

Construction

		Configuration matrix
E₈	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	k-face	f_k	f₀	f₁	f₂			f₃					f₄						f₅						f₆					f₇			k-figure
A₄A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodes x0.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	( )	f₀	483840	30	30	15	60	10	15	60	30	60	5	20	30	60	30	30	10	20	30	30	15	6	10	10	15	6	3	5	2	3	{3,3,3}x{3,3}x{}
A₃A₁A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodes 1x.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	{ }	f₁	2	7257600	2	1	4	1	2	8	4	6	1	4	8	12	6	4	4	6	12	8	4	1	6	4	8	2	1	4	1	2
A₃A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	{3}	f₂	3	3	4838400	*	*	1	1	4	0	0	1	4	4	6	0	0	4	6	6	4	0	0	6	4	4	1	0	4	1	1
A₃A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch 10.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png			3	3	*	2419200	*	0	2	0	4	0	1	0	8	0	6	0	4	0	12	0	4	0	6	0	8	0	1	4	0	2
A₂A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png			3	3	*	*	9676800	0	0	2	1	3	0	1	2	6	3	3	1	3	6	6	3	1	3	3	6	2	1	3	1	2
A₃A₃	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₂₀₀	f₃	4	6	4	0	0	1209600	*	*	*	*	1	4	0	0	0	0	4	6	0	0	0	0	6	4	0	0	0	4	1	0
A₃A₃	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₁₁₀		6	12	4	4	0	*	1209600	*	*	*	1	0	4	0	0	0	4	0	6	0	0	0	6	0	4	0	0	4	0	1
A₃A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png			6	12	4	0	4	*	*	4838400	*	*	0	1	1	3	0	0	1	3	3	3	0	0	3	3	3	1	0	3	1	1
A₃A₂A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png			6	12	0	4	4	*	*	*	2419200	*	0	0	2	0	3	0	1	0	6	0	3	0	3	0	6	0	1	3	0	2
A₃A₁A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	0₂₀₀		4	6	0	0	4	*	*	*	*	7257600	0	0	0	2	1	2	0	1	2	4	2	1	1	2	4	2	1	2	1	2
A₄A₃	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₂₁₀	f₄	10	30	20	10	0	5	5	0	0	0	241920	*	*	*	*	*	4	0	0	0	0	0	6	0	0	0	0	4	0	0
A₄A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₂₁₀		10	30	20	0	10	5	0	5	0	0	*	967680	*	*	*	*	1	3	0	0	0	0	3	3	0	0	0	3	1	0
D₄A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₁₁₁		24	96	32	32	32	0	8	8	8	0	*	*	604800	*	*	*	1	0	3	0	0	0	3	0	3	0	0	3	0	1
A₄A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₂₁₀		10	30	10	0	20	0	0	5	0	5	*	*	*	2903040	*	*	0	1	1	2	0	0	1	2	2	1	0	2	1	1
A₄A₁A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	0₂₁₀		10	30	0	10	20	0	0	0	5	5	*	*	*	*	1451520	*	0	0	2	0	2	0	1	0	4	0	1	2	0	2
A₄A₁	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	0₃₀₀		5	10	0	0	10	0	0	0	0	5	*	*	*	*	*	2903040	0	0	0	2	1	1	0	1	2	2	1	1	1	2
D₅A₂	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₂₁₁	f₅	80	480	320	160	160	80	80	80	40	0	16	16	10	0	0	0	60480	*	*	*	*	*	3	0	0	0	0	3	0	0	{3}
A₅A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₂₂₀		20	90	60	0	60	15	0	30	0	15	0	6	0	6	0	0	*	483840	*	*	*	*	1	2	0	0	0	2	1	0	{ }v()
D₅A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₂₁₁		80	480	160	160	320	0	40	80	80	80	0	0	10	16	16	0	*	*	181440	*	*	*	1	0	2	0	0	2	0	1	{ }v()
A₅	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₃₁₀		15	60	20	0	60	0	0	15	0	30	0	0	0	6	0	6	*	*	*	967680	*	*	0	1	1	1	0	1	1	1	( )v( )v()
A₅A₁	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	0₃₁₀		15	60	0	20	60	0	0	0	15	30	0	0	0	0	6	6	*	*	*	*	483840	*	0	0	2	0	1	1	0	2	{ }v()
A₅A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	0₄₀₀		6	15	0	0	20	0	0	0	0	15	0	0	0	0	0	6	*	*	*	*	*	483840	0	0	0	2	1	0	1	2	{ }v()
E₆A₁	File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₂₂₁	f₆	720	6480	4320	2160	4320	1080	1080	2160	1080	1080	216	432	270	432	216	0	27	72	27	0	0	0	6720	*	*	*	*	2	0	0	{ }
A₆	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₃₂₀		35	210	140	0	210	35	0	105	0	105	0	21	0	42	0	21	0	7	0	7	0	0	*	138240	*	*	*	1	1	0
D₆	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₃₁₁		240	1920	640	640	1920	0	160	480	480	960	0	0	60	192	192	192	0	0	12	32	32	0	*	*	30240	*	*	1	0	1
A₆	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₄₁₀		21	105	35	0	140	0	0	35	0	105	0	0	0	21	0	42	0	0	0	7	0	7	*	*	*	138240	*	0	1	1
A₆A₁	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 2.pngFile:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.png	0₄₁₀		21	105	0	35	140	0	0	0	35	105	0	0	0	0	21	42	0	0	0	0	7	7	*	*	*	*	69120	0	0	2
E₇	File:CDel nodea x.pngFile:CDel 2.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₃₂₁	f₇	10080	120960	80640	40320	120960	20160	20160	60480	30240	60480	4032	12096	7560	24192	12096	12096	756	4032	1512	4032	2016	0	56	576	126	0	0	240	*	*	( )
A₇	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodes 1x.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	0₄₂₀		56	420	280	0	560	70	0	280	0	420	0	56	0	168	0	168	0	28	0	56	0	28	0	8	0	8	0	*	17280	*
D₇	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 10.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 2.pngFile:CDel nodea x.png	0₄₁₁		672	6720	2240	2240	8960	0	560	2240	2240	6720	0	0	280	1344	1344	2688	0	0	84	448	448	448	0	0	14	64	64	*	*	2160

Projections

Orthographic projections are shown for the sub-symmetries of B₆, B₅, B₄, B₃, B₂, A₇, and A₅ Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane. (Planes for E₈: E₇, E₆, B₈, B₇, [24] are not shown for being too large to display.)

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
File:4 21 t4 B2.svg	File:4 21 t4 B3.svg	File:4 21 t4 B4.svg
D6 / B5 / A4 [10]	D7 / B6 [12]	[6]
File:4 21 t4 B5.svg	File:4 21 t4 B6.svg	File:4 21 t4 mox.svg
A5 [6]	A7 [8]	[20]
File:4 21 t4 A5.svg	File:4 21 t4 A7.svg	File:4 21 t4 p20.svg

Notes

↑ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
↑ Klitzing, (o3o3o3x *c3o3o3o3o - bif)
↑ ^3.0 ^3.1 Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
↑ Klitzing, (o3o3o3x *c3o3o3o3o - buffy)

References

H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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 Kaleidoscopes: Selected Writings of H.S.M. Coxeter | Wiley
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Klitzing, Richard. "8D Uniform polyzetta". o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen

[2] Klitzing, (o3o3o3x *c3o3o3o3o - bif)

[auto-3] 3.0 ^3.1 Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

[4] Klitzing, (o3o3o3x *c3o3o3o3o - buffy)

[1]

[2]

[3]

[4]

1 ₄₂ polytope

Contents