# 2 31 polytope

In 7-dimensional geometry, **2 _{31}** is a uniform polytope, constructed from the E7 group.

Its Coxeter symbol is **2 _{31}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

The **rectified 2 _{31}** is constructed by points at the mid-edges of the

**2**.

_{31}These polytopes are part of a family of 127 (or 2^{7}−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

The **2 _{31}** is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56

**2**). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E

_{21}_{7}.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, **3 _{31}**.

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

Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 3_{21} polytope, .

Removing the node on the end of the 3-length branch leaves the 2_{21}. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1_{32} polytope, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1_{31}, .

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

The **rectified 2 _{31}** is a rectification of the 2

_{31}polytope, creating new vertices on the center of edge of the 2

_{31}.

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.

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

Removing the node on the short branch leaves the rectified 6-simplex, .

Removing the node on the end of the 2-length branch leaves the, 6-demicube, .

Removing the node on the end of the 3-length branch leaves the rectified 2_{21}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node.