# 1 22 polytope

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

_{6}group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V

_{72}(for its 72 vertices).

^{[1]}

Its Coxeter symbol is **1 _{22}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1

_{22}, constructed by positions points on the elements of 1

_{22}. The

**rectified 1**is constructed by points at the mid-edges of the

_{22}**1**. The

_{22}**birectified 1**is constructed by points at the triangle face centers of the

_{22}**1**.

_{22}These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

The 1_22 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E_{6}.

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

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

Removing the node on either of 2-length branches leaves the 5-demicube, 1_{31}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0_{22}, .

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

Along with the semiregular polytope, **2 _{21}**, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

The **1 _{22}** is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to

**1**in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1

_{22}_{22}.

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, **2 _{22}**, .

The **rectified 1 _{22}** polytope (also called

**0**) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).

_{221}^{[5]}

Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the birectified 5-simplex, .

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: **t _{2}(2_{11})**, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[7]}^{[8]}

Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.