In 8-dimensional geometry, the **1 _{42}** is a uniform 8-polytope, constructed within the symmetry of the E

_{8}group.

## Contents

Its Coxeter symbol is **1 _{42}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.

The **rectified 1 _{42}** is constructed by points at the mid-edges of the

**1**and is the same as the birectified 2

_{42}_{41}, and the quadrirectified 4

_{21}.

These polytopes are part of a family of 255 (2^{8} − 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

The **1 _{42}** is composed of 2400 facets: 240

**1**polytopes, and 2160 7-demicubes (

_{32}**1**). Its vertex figure is a birectified 7-simplex.

_{41}This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol **1 _{52}**, and Coxeter-Dynkin diagram: .

## Alternate names

_{17280}for its 17280 vertices.

**1**for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.

_{42}**Diacositetracont-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√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, 1_{41}, .

Removing the node on the end of the 4-length branch leaves the 1_{32}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 0_{42}, .

## Projections

Orthographic projections are shown for the sub-symmetries of E_{8}: E_{7}, E_{6}, B_{8}, B_{7}, B_{6}, B_{5}, B_{4}, B_{3}, B_{2}, A_{7}, and A_{5} 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.

## Rectified 142 polytope

The **rectified 1 _{42}** is named from being a rectification of the 1

_{42}polytope, with vertices positioned at the mid-edges of the 1

_{42}.

## Alternate names

_{41}polytope

_{21}polytope

## 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 1-length branch leaves the birectified 7-simplex,

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1_{41}, .

Removing the node on the end of the 3-length branch leaves the 1_{32}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, .

## Projections

Orthographic projections are shown for the sub-symmetries of B_{6}, B_{5}, B_{4}, B_{3}, B_{2}, A_{7}, and A_{5} Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane.

(Planes for E_{8}: E_{7}, E_{6}, B_{8}, B_{7}, [20], [24] are not shown for being too large to display.)