1 42 polytope

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: .

1₄₂ polytope

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.
• Coxeter named it 1₄₂ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
• Diacositetraconta-dischiliahectohexaconta-zetton (acronym: bif) - 240-2160 facetted polyzetton (Jonathan Bowers)

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 in this coordinate set, and the polytope radius is 4.

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.

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.

[[File:E8_142-3D_Concentric_Hulls.png|thumb|230px|Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry: