1 ₄₂ polytope

In 8-dimensional geometry, the 1₄₂ is a uniform 8-polytope, constructed within the symmetry of the E₈ group.

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

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: .

1₄₂ polytope

1₄₂
Type	Uniform 8-polytope
Family	1_k2 polytope
Schläfli symbol	{3,3^4,2}
Coxeter symbol	1₄₂
Coxeter diagrams
7-faces	2400: 240 1₃₂ 2160 1₄₁
6-faces	106080: 6720 1₂₂ 30240 1₃₁ 69120 {3⁵}
5-faces	725760: 60480 1₁₂ 181440 1₂₁ 483840 {3⁴}
4-faces	2298240: 241920 1₀₂ 604800 1₁₁ 1451520 {3³}
Cells	3628800: 1209600 1₀₁ 2419200 {3²}
Faces	2419200 {3}
Edges	483840
Vertices	17280
Vertex figure	t₂{3⁶}
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: .

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

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

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

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1₄₁, .

Removing the node on the end of the 4-length branch leaves the 1₃₂, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 0₄₂, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]

	Configuration matrix
E₈	k-face	f_k	f₀	f₁	f₂	f₃		f₄			f₅			f₆			f₇		k-figure	notes
A₇	( )	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	{ }	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	{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	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	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	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	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	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	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	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	1₃₀		6	15	20	0	15	0	0	6	*	*	483840	0	2	1	1	2	{ }v( )	E₈/A₅A₁ = 192*10!/6!/2 = 483840
E₆A₁	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	1₃₁		32	240	640	160	480	0	60	192	0	12	32	*	30240	*	1	1		E₈/D₆ = 192*10!/32/6! = 30240
A₆A₁	1₄₀		7	21	35	0	35	0	0	21	0	0	7	*	*	69120	0	2		E₈/A₆A₁ = 192*10!/7!/2 = 69120
E₇	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	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

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:

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]
(1)	(1,3,6)	(8,16,24,32,48,64,96)
[20]	[24]	[6]
		(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]
(32,160,192,240,480,512,832,960)	(72,216,432,720,864,1080)	(8,16,24,32,48,64,96)
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]

B8 [16/2]	A5 [6]	A7 [8]

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
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							-	-
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
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.