2 41 polytope
Rectified 2_41 polytope
Rectified 2\_41 polytope
------------------------

|Rectified 241 polytope

|  |
|Type Uniform [8\-polytope](/wiki/8-polytope "8-polytope") |
|[Schläfli symbol](/wiki/Schl%C3%A4fli_symbol "Schläfli symbol") t1{3,3,34,1} |
|[Coxeter symbol](/wiki/Coxeter_symbol "Coxeter symbol") t1(241) |
|[Coxeter diagram](/wiki/Coxeter_diagram "Coxeter diagram")  |
|7\-faces 19680 total: 240 [t1(221)](/wiki/Rectified_2_21_polytope "Rectified 2 21 polytope") 17280 [t1{36}](/wiki/Rectified_7-simplex "Rectified 7-simplex") 2160 [141](/wiki/7-demicube "7-demicube") |
|6\-faces 313440 |
|5\-faces 1693440 |
|4\-faces 4717440 |
|Cells 7257600 |
|Faces 5322240 |
|Edges 19680 |
|Vertices 69120 |
|[Vertex figure](/wiki/Vertex_figure "Vertex figure") [rectified 6\-simplex](/wiki/Rectified_6-simplex "Rectified 6-simplex") prism |
|[Petrie polygon](/wiki/Petrie_polygon "Petrie polygon") [30\-gon](/wiki/Regular_polygon "Regular polygon") |
|[Coxeter group](/wiki/Coxeter_group "Coxeter group") [E8](/wiki/E8_%28mathematics%29 "E8 (mathematics)"), \[34,2,1] |
|Properties [convex](/wiki/Convex_polytope "Convex polytope") |

The **rectified 241** is a [rectification](/wiki/Rectification_%28geometry%29 "Rectification (geometry)") of the 241 polytope, with vertices positioned at the mid\-edges of the 241.

### Alternate names

* Rectified Diacositetracont\-myriaheptachiliadiacosioctaconta\-zetton for rectified 240\-17280 facetted polyzetton (known as robay for short)Jonathan BowersKlitzing, (o3x3o3o \*c3o3o3o3o \- robay)

### Construction

It is created by a [Wythoff construction](/wiki/Wythoff_construction "Wythoff construction") upon a set of 8 [hyperplane](/wiki/Hyperplane "Hyperplane") mirrors in 8\-dimensional space, defined by root vectors of the [E8](/wiki/E8_%28mathematics%29 "E8 (mathematics)") [Coxeter group](/wiki/Coxeter_group "Coxeter group").

The facet information can be extracted from its [Coxeter\-Dynkin diagram](/wiki/Coxeter-Dynkin_diagram "Coxeter-Dynkin diagram"): .

Removing the node on the short branch leaves the [rectified 7\-simplex](/wiki/Rectified_7-simplex "Rectified 7-simplex"): .

Removing the node on the end of the 4\-length branch leaves the [rectified 231](/wiki/Rectified_2_31_polytope "Rectified 2 31 polytope"), .

Removing the node on the end of the 2\-length branch leaves the [7\-demicube](/wiki/7-demicube "7-demicube"), 141.

The [vertex figure](/wiki/Vertex_figure "Vertex figure") is determined by removing the ringed node and ringing the neighboring node. This makes the [rectified 6\-simplex](/wiki/Rectified_6-simplex "Rectified 6-simplex") prism, .

### Visualizations

[Petrie polygon](/wiki/Petrie_polygon "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.

| E8\[30] | \[20] | \[24] |
| --- | --- | --- |
| [200px](/wiki/File:2_41_t1_E8.svg "2 41 t1 E8.svg")(1\) | [200px](/wiki/File:2_41_t1_p20.svg "2 41 t1 p20.svg") | [200px](/wiki/File:2_41_t1_p24.svg "2 41 t1 p24.svg") |
| E7\[18] | E6\[12] | \[6] |
| [200px](/wiki/File:2_41_t1_E7.svg "2 41 t1 E7.svg") | [200px](/wiki/File:2_41_t1_E6.svg "2 41 t1 E6.svg")(1,8,24,32\) | [200px](/wiki/File:2_41_t1_mox.svg "2 41 t1 mox.svg") |

| D3 / B2 / A3\[4] | D4 / B3 / A2\[6] | D5 / B4\[8] |
| --- | --- | --- |
| [200px](/wiki/File:2_41_t1_B2.svg "2 41 t1 B2.svg") | [200px](/wiki/File:2_41_t1_B3.svg "2 41 t1 B3.svg") | [200px](/wiki/File:2_41_t1_B4.svg "2 41 t1 B4.svg") |
| D6 / B5 / A4\[10] | D7 / B6\[12] | D8 / B7 / A6\[14] |
| [200px](/wiki/File:2_41_t1_B5.svg "2 41 t1 B5.svg") | [200px](/wiki/File:2_41_t1_B6.svg "2 41 t1 B6.svg")(1,3,9,12,18,21,36\) | [200px](/wiki/File:2_41_t1_B7.svg "2 41 t1 B7.svg") |
| B8\[16/2] | A5\[6] | A7\[8] |
| [200px](/wiki/File:2_41_t1_B8.svg "2 41 t1 B8.svg") | [200px](/wiki/File:2_41_t1_A5.svg "2 41 t1 A5.svg") | [200px](/wiki/File:2_41_t1_A7.svg "2 41 t1 A7.svg") |


What is the difference between the rectified 2_41 polytope and its non-rectified counterpart?