In 7-dimensional geometry, 1<sub>32</sub> is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 1<sub>32</sub>, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.
The rectified 1<sub>32</sub> is constructed by points at the mid-edges of the 1<sub>32</sub>.
These polytopes are part of a family of 127 (2<sup>7</sup>âÂÂ1) 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: .
This polytope can tessellate 7-dimensional space, with symbol 1<sub>33</sub>, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E<sub>7</sub><sup>*</sup> lattice.
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 end of the 2-length branch leaves the 6-demicube, 1<sub>31</sub>,
Removing the node on the end of the 3-length branch leaves the 1<sub>22</sub>,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 0<sub>32</sub>,
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
The 1<sub>32</sub> is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1<sub>3k</sub> series. The next figure is the Euclidean honeycomb 1<sub>33</sub> and the final is a noncompact hyperbolic honeycomb, 1<sub>34</sub>.
The rectified 1<sub>32</sub> (also called 0<sub>321</sub>) is a rectification of the 1<sub>32</sub> polytope, creating new vertices on the center of edge of the 1<sub>32</sub>. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}ÃÂ{3}ÃÂ{}.
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).
Removing the node on the end of the 3-length branch leaves the rectified 1<sub>22</sub> polytope,
Removing the node on the end of the 2-length branch leaves the demihexeract, 1<sub>31</sub>,
Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}ÃÂ{3}ÃÂ{},
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.