1 42 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 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 permutations 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, .

Projections

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.

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)

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<sup>2,2,1</sup>]]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	192	103,680	2,903,040	696,729,600	∞
Graph							-	-
Name	1_-1,2	1₀₂	1₁₂	1₂₂	1₃₂	1₄₂	1₅₂	1₆₂