In 7-dimensional geometry, 2<sub>31</sub> is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 2<sub>31</sub>, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.
The rectified 2<sub>31</sub> is constructed by points at the mid-edges of the 2<sub>31</sub>.
These polytopes are part of a family of 127 (or 2<sup>7</sup>−1) convex uniform polytopes in seven dimensions, made of uniform polytope facets and vertex figures, defined by all combinations of rings in this Coxeter-Dynkin diagram: .
The 2<sub>31</sub> is composed of 126 vertices, 2016 edges, 10080 faces (triangles), 20160 cells (tetrahedra), 16128 4-faces (4-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 2<sub>21</sub>). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E<sub>7</sub>.
This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 3<sub>31</sub>.
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 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 3<sub>21</sub> polytope, .
Removing the node on the end of the 3-length branch leaves the 2<sub>21</sub>. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1<sub>32</sub> polytope, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1<sub>31</sub>, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
The rectified 2<sub>31</sub> is a rectification of the 2<sub>31</sub> polytope, creating new vertices on the center of edge of the 2<sub>31</sub>.
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<sub>21</sub>, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node.