In 8-dimensional geometry, the 2<sub>41</sub> is a uniform 8-polytope, constructed within the symmetry of the E<sub>8</sub> group.
Its Coxeter symbol is 2<sub>41</sub>, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.
The rectified 2<sub>41</sub> is constructed by points at the mid-edges of the 2<sub>41</sub>. The birectified 2<sub>41</sub> is constructed by points at the triangle face centers of the 2<sub>41</sub>, and is the same as the rectified 1<sub>42</sub>.
These polytopes are part of a family of 255 (2<sup>8</sup> − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
The 2<sub>41</sub> is composed of 17,520 facets (240 2<sub>31</sub> polytopes and 17,280 7-simplices), 144,960 6-faces (6,720 2<sub>21</sub> polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 2<sub>11</sub> and 483,840 5-simplices), 1,209,600 4-faces (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.
This polytope is a facet in the uniform tessellation, 2<sub>51</sub> with Coxeter-Dynkin diagram:
The 2160 vertices can be defined as follows:
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 short branch leaves the 7-simplex: . There are 17280 of these facets
Removing the node on the end of the 4-length branch leaves the 2<sub>31</sub>, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4<sub>21</sub> polytope.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 1<sub>41</sub>, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.
The rectified 2<sub>41</sub> is a rectification of the 2<sub>41</sub> polytope, with vertices positioned at the mid-edges of the 2<sub>41</sub>.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E<sub>8</sub> Coxeter group.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the short branch leaves the rectified 7-simplex: .
Removing the node on the end of the 4-length branch leaves the rectified 2<sub>31</sub>, .
Removing the node on the end of the 2-length branch leaves the 7-demicube, 1<sub>41</sub>.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .
Petrie polygon projections are 12, 18, or 30-sided based on the E6, E7, and E8 symmetries (respectively). The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.