Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.
The rectified 231 is constructed by points at the mid-edges of the 231.
These polytopes are part of a family of 127 (or 27-1) 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: .
| bgcolor=#e7dcc3 colspan=2 | Gosset 231 polytope | |
|---|---|---|
| Type | Uniform 7-polytope | |
| Family | 2k1 polytope | |
| Schläfli symbol | ||
| Coxeter symbol | 231 | |
| Coxeter diagram | ||
| 6-faces | 632: 56 221 576 | |
| 5-faces | 4788: 756 211 4032 | |
| 4-faces | 16128: 4032 201 12096 | |
| Cells | 20160 | |
| Faces | 10080 | |
| Edges | 2016 | |
| Vertices | 126 | |
| Vertex figure | 131 | |
| Petrie polygon | Octadecagon | |
| Coxeter group | E7, [3<sup>3,2,1</sup>] | |
| Properties | convex |
This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 331.
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 321 polytope, .
Removing the node on the end of the 3-length branch leaves the 221. There are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 131, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]
| E7 | width=70 | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | k-figures | notes | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| D6 | f0 | 126 | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | E7/D6 = 72x8!/32/6 | = 126 | |||
| A5A1 | f1 | 2 | 2016 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | E7/A5A1 = 72x8!/6 | /2 = 2016 | |||
| A3A2A1 | f2 | 3 | 3 | 10080 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | E7/A3A2A1 = 72x8!/4 | /3!/2 = 10080 | |||
| A3A2 | f3 | 4 | 6 | 4 | 20160 | 1 | 3 | 3 | 3 | 3 | 1 | E7/A3A2 = 72x8!/4 | /3! = 20160 | |||
| A4A2 | f4 | 5 | 10 | 10 | 5 | 4032 | 3 | 0 | 3 | 0 | E7/A4A2 = 72x8!/5 | /3! = 4032 | ||||
| A4A1 | 5 | 10 | 10 | 5 | 12096 | 1 | 2 | 2 | 1 | E7/A4A1 = 72x8!/5 | /2 = 12096 | |||||
| D5A1 | f5 | 10 | 40 | 80 | 80 | 16 | 16 | 756 | 2 | 0 | E7/D5A1 = 72x8!/32/5 | = 756 | ||||
| A5 | 6 | 15 | 20 | 15 | 0 | 6 | 4032 | 1 | 1 | E7/A5 = 72x8!/6 | = 72*8*7 = 4032 | |||||
| E6 | f6 | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 56 | E7/E6 = 72x8!/72x6 | = 8*7 = 56 | ||||
| A6 | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | 576 | E7/A6 = 72x8!/7 | = 72×8 = 576 | |||||
| bgcolor=#e7dcc3 colspan=2 | Rectified 231 polytope | |
|---|---|---|
| Type | Uniform 7-polytope | |
| Family | 2k1 polytope | |
| Schläfli symbol | ||
| Coxeter symbol | t1(231) | |
| Coxeter diagram | ||
| 6-faces | 758 | |
| 5-faces | 10332 | |
| 4-faces | 47880 | |
| Cells | 100800 | |
| Faces | 90720 | |
| Edges | 30240 | |
| Vertices | 2016 | |
| Vertex figure | 6-demicube | |
| Petrie polygon | Octadecagon | |
| Coxeter group | E7, [3<sup>3,2,1</sup>] | |
| Properties | convex |
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 221, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node.