2 41 polytope
Construction
### 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.

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 [7\-simplex](/wiki/7-simplex "7-simplex"): . There are 17280 of these facets

Removing the node on the end of the 4\-length branch leaves the [231](/wiki/Gosset_2_31_polytope "Gosset 2 31 polytope"), . There are 240 of these facets. They are centered at the positions of the 240 vertices in the [421](/wiki/Gosset_4_21_polytope "Gosset 4 21 polytope") polytope.

The [vertex figure](/wiki/Vertex_figure "Vertex figure") is determined by removing the ringed node and ringing the neighboring node. This makes the [7\-demicube](/wiki/7-demicube "7-demicube"), 141, .

Seen in a [configuration matrix](/wiki/Configuration_%28polytope%29 "Configuration (polytope)"), the element counts can be derived by mirror removal and ratios of [Coxeter group](/wiki/Coxeter_group%23Properties "Coxeter group#Properties") orders.Coxeter, Regular Polytopes, 11\.8 Gossett figures in six, seven, and eight dimensions, p. 202\-203

|

 [Configuration matrix](/wiki/Configuration_%28polytope%29 "Configuration (polytope)") |
| --- |
| E8 |  | [*k*\-face](/wiki/Face_%28geometry%29%23k-face "Face (geometry)#k-face") | fk | f0 | f1 | f2 | f3 |f4f5f6f7 [*k*\-figure](/wiki/Vertex_figure "Vertex figure") | notes |
| D7 |  | ( ) | f0 |**2160** 64 | 672 | 2240 | 560 | 2240 | 280 | 1344 | 84 | 448 | 14 | 64 | [h{4,3,3,3,3,3}](/wiki/7-demicube "7-demicube") | E8/D7 \= 192\*10!/64/7! \= 2160 |
| A6A1 |  | { } | f1 | 2 |**69120** 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | [r{3,3,3,3,3}](/wiki/Rectified_6-simplex "Rectified 6-simplex") | E8/A6A1 \= 192\*10!/7!/2 \= 69120 |
| A4A2A1 |  | [{3}](/wiki/Triangle "Triangle") | f2 | 3 | 3 |**483840** 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | [{}x{3,3,3}](/wiki/5-cell_prism "5-cell prism") | E8/A4A2A1 \= 192\*10!/5!/3!/2 \= 483840 |
| A3A3 |  | [{3,3}](/wiki/Tetrahedron "Tetrahedron") | f3 | 4 | 6 | 4 |**1209600** 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | [{3,3}V( )](/wiki/Tetrahedral_pyramid "Tetrahedral pyramid") | E8/A3A3 \= 192\*10!/4!/4! \= 1209600 |
| A4A3 |  | [{3,3,3}](/wiki/5-cell "5-cell") |f4
 5 | 10 | 10 | 5 |**241920**\* 4 | 0 | 6 | 0 | 4 | 0 | [{3,3}](/wiki/Tetrahedron "Tetrahedron") | E8/A4A3 \= 192\*10!/5!/4! \= 241920 |
| A4A2 |  | 5 | 10 | 10 | 5 |\***967680** 1 | 3 | 3 | 3 | 3 | 1 | [{3}V( )](/wiki/Triangular_pyramid "Triangular pyramid") | E8/A4A2 \= 192\*10!/5!/3! \= 967680 |
| D5A2 |  | [{3,3,31,1}](/wiki/5-orthoplex "5-orthoplex") |f5
 10 | 40 | 80 | 80 | 16 | 16 |**60480**\* 3 | 0 | 3 | 0 | [{3}](/wiki/Equilateral_triangle "Equilateral triangle") | E8/D5A2 \= 192\*10!/16/5!/2 \= 40480 |
| A5A1 |  | [{3,3,3,3}](/wiki/5-simplex "5-simplex") | 6 | 15 | 20 | 15 | 0 | 6 |\***483840** 1 | 2 | 2 | 1 | [{ }V( )](/wiki/Isosceles_triangle "Isosceles triangle") | E8/A5A1 \= 192\*10!/6!/2 \= 483840 |
| E6A1 |  | [{3,3,32,1}](/wiki/2_21_polytope "2 21 polytope") |f6
 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 |**6720**\* 2 | 0 |{ }  E8/E6A1 \= 192\*10!/72/6! \= 6720 |
| A6 |  | [{3,3,3,3,3}](/wiki/6-simplex "6-simplex") | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 |\***138240** 1 | 1 | E8/A6 \= 192\*10!/7! \= 138240 |
| E7 |  | [{3,3,33,1}](/wiki/2_31_polytope "2 31 polytope") |f7
 126 | 2016 | 10080 | 20160 | 4032 | 12096 | 756 | 4032 | 56 | 576 |**240**\*( )  E8/E7 \= 192\*10!/72!/8! \= 240 |
| A7 |  | [{3,3,3,3,3,3}](/wiki/7-simplex "7-simplex") | 8 | 28 | 56 | 70 | 0 | 56 | 0 | 28 | 0 | 8 |\***17280** E8/A7 \= 192\*10!/8! \= 17280 |


How many elements does a 6-simplex have?