User:Tomruen/Configuration
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The configuration matrix shows the number of k-face elements along the diagonal, while the nondiagonal element show the incidence counts between all elements. The number of elements of its facets can be seen on the bottom row, left of the diagonal, and k-face elements above that. The top row, right of the diagonal represent the number of elements of the vertex figure. The second row contains the edge-figures, and so on. These figures are the duals of the k-faces of the dual polytope, which can be seen by rotating the matrix 180 degrees.
For regular n-polytopes, the there are only one type of element, so the matrix is n×n. For irregular polytopes, the matrix is expanded with one row per element type, which in the limit contains one row for every element. Like a general polyhedron with v vertices, e edges and f faces would have v+e+f total rows and columns.
Polygons
| regular polygon | Triangle |
Square |
Pentagon |
Hexagon | |
|---|---|---|---|---|---|
xno . . | n | 2 ----+---+-- x . | 2 | n |
x-2n-o . . | 2n | 2 -------+----+--- x . | 2 | 2n |
x3o . . | 3 | 2 ----+---+-- x . | 2 | 3 |
x4o . . | 4 | 2 ----+---+-- x . | 2 | 4 |
x5o . . | 5 | 2 ----+---+-- x . | 2 | 5 |
x6o . . | 6 | 2 ----+---+-- x . | 2 | 6 |
Triangle
| 1,1 | 2,2 | 3,3 |
|---|---|---|
| Equilateral {3}     |
Isosceles { }∨( )   |
Scalene ( )∨( )∨( )  |
| (v:3; e:3) | (v:2+1; e:2+1) | (v:1+1+1; e:1+1+1) |
| A | a --+---+--- A | 3 | 2 --+---+--- a | 2 | 3 |
| A B | a b --+-----+----- A | 2 * | 1 1 B | * 1 | 2 0 --+-----+----- a | 1 1 | 2 * b | 2 0 | * 1 |
| A B C | a b c --+-------+------- A | 1 * * | 0 1 1 B | * 1 * | 1 0 1 C | * * 1 | 1 1 0 --+-------+------- a | 0 1 1 | 1 * * b | 1 0 1 | * 1 * c | 1 1 0 | * * 1 |
Quadrilateral
| 1,1 | 1,2 | 2,1 | 2,2 | 2,3 | 3,2 | 4,4 |
|---|---|---|---|---|---|---|
| Square {4}   |
Rectangle { }×{ }   |
Rhombus { }+{ }   |
Parallelogram |
Isosceles trapezoid { }||{ }  |
Kite |
General |
| (v:4; e:4) | (v:4; e:2+2) | (v:2+2; e:4) | (v:2+2; e:2+2) | (v:2+2; e:1+1+2) | (v:1+1+2; e:2+2) | (v:1+1+1+1; e:1+1+1+1) |
| A | a --+---+-- A | 4 | 2 --+---+-- a | 2 | 4 |
| A | a b --+---+---- A | 4 | 1 1 --+---+---- a | 2 | 2 * b | 2 | * 2 |
| A B | a --+-----+-- A | 2 * | 2 B | * 2 | 2 --+-----+-- a | 1 1 | 4 |
| A B | a b --+-----+---- A | 2 * | 1 1 B | * 2 | 1 1 --+-----+---- a | 1 1 | 2 * b | 1 1 | * 2 |
| A B | a b c --+-----+------ A | 2 * | 1 0 1 B | * 2 | 0 1 1 --+-----+------ a | 2 0 | 1 * * b | 0 2 | * 1 * c | 1 1 | * * 2 |
| A B C | a b --+-------+---- A | 1 * * | 2 0 B | * 1 * | 0 2 C | * * 2 | 1 1 --+-------+---- a | 1 0 1 | 2 * b | 0 1 1 | * 2 |
| A B C D | a b c d --+---------+-------- A | 1 * * * | 1 0 0 1 B | * 1 * * | 1 1 0 0 C | * * 1 * | 0 1 1 0 D | * * * 1 | 0 0 1 1 --+---------+-------- a | 1 1 0 0 | 1 * * * b | 0 1 1 0 | * 1 * * c | 0 0 1 1 | * * 1 * d | 1 0 0 1 | * * * 1 |
Polyhedra
| Platonic solid {p,q} |
Tetrahedron [1] {3,3} (v:4; e:6; f:4) |
Icosahedron[2] {3,5} (v:12; e:30; f:20) |
Dodecahedron [3] {5,3} (v:20; e:30; f:12) |
Stellated dodecahedron [4] {5/2,5} (v:12; e:30; f:12) | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
x3o3o . . . | 4 | 3 | 3 ------+---+---+-- x . . | 2 | 6 | 2 ------+---+---+-- x3o . | 3 | 3 | 4 |
x3o5o . . . | 12 | 5 | 5 ------+----+----+--- x . . | 2 | 30 | 2 ------+----+----+--- x3o . | 3 | 3 | 20 |
o3o5x . . . | 20 | 3 | 3 ------+----+----+--- . . x | 2 | 30 | 2 ------+----+----+--- . o5x | 5 | 5 | 12 |
x5/2o5o . . . | 12 | 5 | 5 --------+----+----+--- x . . | 2 | 30 | 2 --------+----+----+--- x5/2o . | 5 | 5 | 12 | ||||||||||||||||
| Octahedron [5] {3,4} (v:6; e:12; f:8) |
Cube [6] {4,3} (v:8; e:12; f:6) |
Great icosahedron [7] {3,5/2} (v:12; e:30; f:20) |
Great stellated dodecahedron [8] {5/2,3} (v:20; e:30; f:12) |
Great dodecahedron [9] {5,5/2} (v:12; e:30; f:12) | ||||||||||||||||
x3o4o . . . | 6 | 4 | 4 ------+---+----+-- x . . | 2 | 12 | 2 ------+---+----+-- x3o . | 3 | 3 | 8 |
o3o4x . . . | 8 | 3 | 3 ------+---+----+-- . . x | 2 | 12 | 2 ------+---+----+-- . o4x | 4 | 4 | 6 |
o5/2o3x . . . | 12 | 5 | 5 --------+----+----+--- . . x | 2 | 30 | 2 --------+----+----+--- . o3x | 3 | 3 | 20 |
x5/2o3o . . . | 20 | 3 | 3 --------+----+----+--- x . . | 2 | 30 | 2 --------+----+----+--- x5/2o . | 5 | 5 | 12 |
o5/2o5x . . . | 12 | 5 | 5 --------+----+----+--- . . x | 2 | 30 | 2 --------+----+----+--- . o5x | 5 | 5 | 12 |
Tetrahedra
| 1,1,1 | 1,2,1 | 1,3,1 | 2,3,2 | 2,4,2 |
|---|---|---|---|---|
| Regular (v:4; e:6; f:4)  |
tetragonal disphenoid (v:4; e:2+4; f:4)  |
Rhombic disphenoid (v:4; e:2+2+2; f:4)  |
Digonal disphenoid (v:2+2; e:4+1+1; f:2+2)  |
Phyllic disphenoid (v:2+2; e:2+2+1+1; f:2+2)  |
A| 4 | 3 | 3 ---+---+---+-- a| 2 | 6 | 2 ---+---+---+-- aaa| 3 | 3 | 4 |
A| 4 | 2 1 | 3 ---+---+-----+-- a| 2 | 4 * | 2 b| 2 | * 2 | 2 ---+---+-----+-- aab| 3 | 2 1 | 4 |
A| 4 | 1 1 1 | 3 ----+---+-------+-- a| 2 | 2 * * | 2 b| 2 | * 2 * | 2 c| 2 | * * 2 | 2 ----+---+-------+-- abc| 3 | 1 1 1 | 4 |
A| 2 * | 2 1 0 | 2 1 B| * 2 | 2 0 1 | 1 2 ---+-----+-------+---- a| 1 1 | 4 * * | 1 1 b| 2 0 | * 1 * | 2 0 c| 0 2 | * * 1 | 0 2 ---+-----+-------+---- aab| 2 1 | 2 1 0 | 2 * aac| 1 2 | 2 0 1 | * 2 |
A| 2 * | 1 0 1 1 | 1 2 B| * 2 | 1 1 1 0 | 2 1 ---+-----+---------+---- a| 1 1 | 2 * * * | 1 1 b| 1 1 | * 2 * * | 1 1 c| 0 2 | * * 1 * | 2 0 d| 2 0 | * * * 1 | 0 2 ---+-----+---------+---- abc| 1 2 | 1 1 1 0 | 2 * bcd| 2 1 | 1 1 0 1 | * 2 |
| 2,2,2 | 3,4,3 | 4,6,4 | ||
| Triangular pyramid (v:3+1; e:3+3; f:3+1)  |
Mirrored spheroid (v:2+1+1; e:2+2+1+1; f:2+1+1)  |
No symmetry (v:1+1+1+1; e:1+1+1+1+1+1; f:1+1+1+1)  | ||
A| 3 * | 2 1 | 2 1 B| * 1 | 0 3 | 3 0 ---+-----+-----+---- a| 2 0 | 3 * | 1 1 b| 1 1 | * 3 | 2 0 ---+-----+-----+---- abb| 2 1 | 1 2 | 3 * aaa| 3 0 | 3 0 | * 1 |
A| 1 * * | 2 0 1 0 | 2 1 0 B| * 1 * | 0 2 1 0 | 2 0 1 C| * * 2 | 1 1 0 1 | 1 1 1 ---+-------+---------+------ a| 1 0 1 | 2 * * * | 1 1 0 b| 0 1 1 | * 2 * * | 1 0 1 c| 1 1 0 | * * 1 * | 2 0 0 d| 0 0 2 | * * * 1 | 0 1 1 ---+-------+---------+------ ABC| 1 1 1 | 1 1 1 0 | 2 * * ACC| 1 0 2 | 2 0 0 1 | * 1 * BCC| 0 1 2 | 0 2 0 1 | * * 1 |
A | 1 0 0 0 | 1 1 1 0 0 0 | 1 1 1 0 B | 0 1 0 0 | 1 0 0 1 1 0 | 1 1 0 1 C | 0 0 1 0 | 0 1 0 1 0 1 | 1 0 1 1 D | 0 0 0 1 | 0 0 1 0 1 1 | 0 1 1 1 ----+---------+-------------+-------- a | 1 1 0 0 | 1 0 0 0 0 0 | 1 1 0 0 b | 1 0 1 0 | 0 1 0 0 0 0 | 1 0 1 0 c | 1 0 0 1 | 0 0 1 0 0 0 | 0 1 1 0 d | 0 1 1 0 | 0 0 0 1 0 0 | 1 0 0 1 e | 0 1 0 1 | 0 0 0 0 1 0 | 0 1 0 1 f | 0 0 1 1 | 0 0 0 0 0 1 | 0 0 1 1 ----+---------+-------------+-------- ABC | 1 1 1 0 | 1 1 0 1 0 0 | 1 0 0 0 ABD | 1 1 0 1 | 1 0 1 0 1 0 | 0 1 0 0 ACD | 1 0 1 1 | 0 1 1 0 0 1 | 0 0 1 0 BCD | 0 1 1 1 | 0 0 0 1 1 1 | 0 0 0 1 | ||
Uniform polyhedra
| Tetratetrahedron [10] (v:6; e:12; f:4+4) |
Truncated tetrahedron [11] (v:12; e:6+12; f:4+4) | |
|---|---|---|
o3x3o . . . | 6 | 4 | 2 2 ------+---+----+---- . x . | 2 | 12 | 1 1 ------+---+----+---- o3x . | 3 | 3 | 4 * . x3o | 3 | 3 | * 4 |
x3x3o . . . | 12 | 1 2 | 2 1 ------+----+------+---- x . . | 2 | 6 * | 2 0 . x . | 2 | * 12 | 1 1 ------+----+------+---- x3x . | 6 | 3 3 | 4 * . x3o | 3 | 0 3 | * 4 |
o3x3x . . . | 12 | 2 1 | 1 2 ------+----+------+---- . x . | 2 | 12 * | 1 1 . . x | 2 | * 6 | 0 2 ------+----+------+---- o3x . | 3 | 3 0 | 4 * . x3x | 6 | 3 3 | * 4 |
| Rhombitetratetrahedron [12] (v:12; e:12+12; f:4+6+4) |
Truncated tetratetrahedron [13] (v:12; e:12+12; f:4+6+4) |
Snub tetrahedron [14]  (v:12; e:6+12+12; f:4+4+12) |
x3o3x . . . | 12 | 2 2 | 1 2 1 ------+----+-------+------ x . . | 2 | 12 * | 1 1 0 . . x | 2 | * 12 | 0 1 1 ------+----+-------+------ x3o . | 3 | 3 0 | 4 * * x . x | 4 | 2 2 | * 6 * . o3x | 3 | 0 3 | * * 4 |
x3x3x . . . | 24 | 1 1 1 | 1 1 1 ------+----+----------+------ x . . | 2 | 12 * * | 1 1 0 . x . | 2 | * 12 * | 1 0 1 . . x | 2 | * * 12 | 0 1 1 ------+----+----------+------ x3x . | 6 | 3 3 0 | 4 * * x . x | 4 | 2 0 2 | * 6 * . x3x | 6 | 0 3 3 | * * 4 |
s3s3s
demi( . . . ) | 12 | 1 2 2 | 1 1 3
--------------+----+---------+-------
s 2 s | 2 | 6 * * | 0 0 2
sefa( s3s . ) | 2 | * 12 * | 1 0 1
sefa( . s3s ) | 2 | * * 12 | 0 1 1
--------------+----+---------+-------
s3s . ♦ 3 | 0 3 0 | 4 * *
. s3s ♦ 3 | 0 0 3 | * 4 *
sefa( s3s3s ) | 3 | 1 1 1 | * * 12
|
| Cuboctahedron [15] (v:12; e:24; f:8+6) |
Truncated cube [16] (v:24; e:12+24; f:8+6) |
Truncated octahedron [17] (v:24; e:24+12; f:8+6) |
|---|---|---|
o3x4o . . . | 12 | 4 | 2 2 ------+----+----+---- . x . | 2 | 24 | 1 1 ------+----+----+---- o3x . | 3 | 3 | 8 * . x4o | 4 | 4 | * 6 |
x3x4o . . . | 24 | 1 2 | 2 1 ------+----+-------+---- x . . | 2 | 12 * | 2 0 . x . | 2 | * 24 | 1 1 ------+----+-------+---- x3x . | 6 | 3 3 | 8 * . x4o | 4 | 0 4 | * 6 |
o3x4x . . . | 24 | 2 1 | 1 2 ------+----+-------+---- . x . | 2 | 24 * | 1 1 . . x | 2 | * 12 | 0 2 ------+----+-------+---- o3x . | 3 | 3 0 | 8 * . x4x | 8 | 4 4 | * 6 |
| Rhombicuboctahedron [18] (v:24; e:24+24; f:8+12+6) |
Truncated cuboctahedron [19] (v:48; e:24+24+24; f:8+12+6) |
Snub cube [20] (v:24; e:12+24+24; f:8+6+24) |
x3o4x . . . | 24 | 2 2 | 1 2 1 ------+----+-------+------- x . . | 2 | 24 * | 1 1 0 . . x | 2 | * 24 | 0 1 1 ------+----+-------+------- x3o . | 3 | 3 0 | 8 * * x . x | 4 | 2 2 | * 12 * . o4x | 4 | 0 4 | * * 6 |
x3x4x . . . | 48 | 1 1 1 | 1 1 1 ------+----+----------+------- x . . | 2 | 24 * * | 1 1 0 . x . | 2 | * 24 * | 1 0 1 . . x | 2 | * * 24 | 0 1 1 ------+----+----------+------- x3x . | 6 | 3 3 0 | 8 * * x . x | 4 | 2 0 2 | * 12 * . x4x | 8 | 0 4 4 | * * 6 |
s3s4s
demi( . . . ) | 24 | 1 2 2 | 1 1 3
--------------+----+----------+-------
s 2 s ♦ 2 | 12 * * | 0 0 2
sefa( s3s . ) | 2 | * 24 * | 1 0 1
sefa( . s4s ) | 2 | * * 24 | 0 1 1
--------------+----+----------+-------
s3s . ♦ 3 | 0 3 0 | 8 * *
. s4s ♦ 4 | 0 0 4 | * 6 *
sefa( s3s4s ) | 3 | 1 1 1 | * * 24
|
Icosidodecahedron [21]  (v:30; e:60; f:20+12) |
Truncated dodecahedron [22] (v:60; e:30+60; f:20+12) |
Truncated icosahedron [23] (v:60; e:60+30; f:20+12) |
|---|---|---|
o3x5o . . . | 30 | 4 | 2 2 ------+----+----+------ . x . | 2 | 60 | 1 1 ------+----+----+------ o3x . | 3 | 3 | 20 * . x5o | 5 | 5 | * 12 |
x3x5o . . . | 60 | 1 2 | 2 1 ------+----+-------+------ x . . | 2 | 30 * | 2 0 . x . | 2 | * 60 | 1 1 ------+----+-------+------ x3x . | 6 | 3 3 | 20 * . x5o | 5 | 0 5 | * 12 |
o3x5x . . . | 60 | 2 1 | 1 2 ------+----+-------+------ . x . | 2 | 60 * | 1 1 . . x | 2 | * 30 | 0 2 ------+----+-------+------ o3x . | 3 | 3 0 | 20 * . x5x | 10 | 5 5 | * 12 |
| Rhombicosidodecahedron [24] (v:60; e:60+60; f:20+30+12) |
Truncated icosidodecahedron [25] (v:120; e:60+60+60; f:20+30+12) |
Snub dodecahedron [26] (v:60; e:30+60+60; f:20+12+60) |
x3o5x . . . | 60 | 2 2 | 1 2 1 ------+----+-------+--------- x . . | 2 | 60 * | 1 1 0 . . x | 2 | * 60 | 0 1 1 ------+----+-------+--------- x3o . | 3 | 3 0 | 20 * * x . x | 4 | 2 2 | * 30 * . o5x | 5 | 0 5 | * * 12 |
x3x5x . . . | 120 | 1 1 1 | 1 1 1 ------+-----+----------+--------- x . . | 2 | 60 * * | 1 1 0 . x . | 2 | * 60 * | 1 0 1 . . x | 2 | * * 60 | 0 1 1 ------+-----+----------+--------- x3x . | 6 | 3 3 0 | 20 * * x . x | 4 | 2 0 2 | * 30 * . x5x | 10 | 0 5 5 | * * 12 |
s3s5s
demi( . . . ) | 60 | 1 2 2 | 1 1 3
--------------+----+----------+---------
s 2 s ♦ 2 | 30 * * | 0 0 2
sefa( s3s . ) | 2 | * 60 * | 1 0 1
sefa( . s5s ) | 2 | * * 60 | 0 1 1
--------------+----+----------+---------
s3s . ♦ 3 | 0 3 0 | 20 * *
. s5s ♦ 5 | 0 0 5 | * 12 *
sefa( s3s5s ) | 3 | 1 1 1 | * * 60
|
Higher polytopes
4D
Regular 4-polytopes
| {p,q,r} | 5-cell {3,3,3} [27] | 16-cell {3,3,4} [28] | 600-cell {3,3,5} [29] | 120-cell {5,3,3} [30] | |||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
x3o3o3o . . . . | 5 ♦ 4 | 6 | 4 --------+---+----+----+-- x . . . | 2 | 10 | 3 | 3 --------+---+----+----+-- x3o . . | 3 | 3 | 10 | 2 --------+---+----+----+-- x3o3o . ♦ 4 | 6 | 4 | 5 |
x3o3o4o . . . . | 8 ♦ 6 | 12 | 8 --------+---+----+----+--- x . . . | 2 | 24 | 4 | 4 --------+---+----+----+--- x3o . . | 3 | 3 | 32 | 2 --------+---+----+----+--- x3o3o . ♦ 4 | 6 | 4 | 16 |
x3o3o5o . . . . | 120 ♦ 12 | 30 | 20 --------+-----+-----+------+---- x . . . | 2 | 720 | 5 | 5 --------+-----+-----+------+---- x3o . . | 3 | 3 | 1200 | 2 --------+-----+-----+------+---- x3o3o . ♦ 4 | 6 | 4 | 600 |
o3o3o5x . . . . | 600 ♦ 4 | 6 | 4 --------+-----+------+-----+---- . . . x | 2 | 1200 | 3 | 3 --------+-----+------+-----+---- . . o5x | 5 | 5 | 720 | 2 --------+-----+------+-----+---- . o3o5x ♦ 20 | 30 | 12 | 120 | |||||||||||||||||||||||||
| 24-cell {3,4,3} [31] | tesseract {4,3,3} [32] | grand 600-cell {3,3,5/2} [33] | great grand stellated 120-cell {5/2,3,3} [34] | ||||||||||||||||||||||||||
x3o4o3o . . . . | 24 ♦ 8 | 12 | 6 --------+----+----+----+--- x . . . | 2 | 96 | 3 | 3 --------+----+----+----+--- x3o . . | 3 | 3 | 96 | 2 --------+----+----+----+--- x3o4o . ♦ 6 | 12 | 8 | 24 |
o3o3o4x . . . . | 16 ♦ 4 | 6 | 4 --------+----+----+----+-- . . . x | 2 | 32 | 3 | 3 --------+----+----+----+-- . . o4x | 4 | 4 | 24 | 2 --------+----+----+----+-- . o3o4x ♦ 8 | 12 | 6 | 8 |
x3o3o5/2o . . . . | 120 ♦ 12 | 30 | 20 ----------+-----+-----+------+---- x . . . | 2 | 720 | 5 | 5 ----------+-----+-----+------+---- x3o . . | 3 | 3 | 1200 | 2 ----------+-----+-----+------+---- x3o3o . ♦ 4 | 6 | 4 | 600 |
o3o3o5/2x . . . . | 600 ♦ 4 | 6 | 4 ----------+-----+------+-----+---- . . . x | 2 | 1200 | 3 | 3 ----------+-----+------+-----+---- . . o5/2x | 5 | 5 | 720 | 2 ----------+-----+------+-----+---- . o3o5/2x ♦ 20 | 30 | 12 | 120 |
| great stellated 120-cell {5/2,3,5} [35] | icosahedral 120-cell {3,5,5/2} [36] | small stellated 120-cell {5/2,5,3} [37] | great 120-cell {5,5/2,5} [38] |
|---|---|---|---|
o5o3o5/2x . . . . | 120 ♦ 12 | 30 | 20 ----------+-----+-----+-----+---- . . . x | 2 | 720 | 5 | 5 ----------+-----+-----+-----+---- . . o5/2x | 5 | 5 | 720 | 2 ----------+-----+-----+-----+---- . o3o5/2x ♦ 20 | 30 | 12 | 120 |
x3o5o5/2o . . . . | 120 ♦ 12 | 30 | 12 ----------+-----+-----+------+---- x . . . | 2 | 720 | 5 | 5 ----------+-----+-----+------+---- x3o . . | 3 | 3 | 1200 | 2 ----------+-----+-----+------+---- x3o5o . ♦ 12 | 30 | 20 | 120 |
x5/2o5o3o . . . . | 120 ♦ 20 | 30 | 12 ----------+-----+------+-----+---- x . . . | 2 | 1200 | 3 | 3 ----------+-----+------+-----+---- x5/2o . . | 5 | 5 | 720 | 2 ----------+-----+------+-----+---- x5/2o5o . ♦ 12 | 30 | 12 | 120 |
x5o5/2o5o . . . . | 120 ♦ 12 | 30 | 12 ----------+-----+-----+-----+---- x . . . | 2 | 720 | 5 | 5 ----------+-----+-----+-----+---- x5o . . | 5 | 5 | 720 | 2 ----------+-----+-----+-----+---- x5o5/2o . ♦ 12 | 30 | 12 | 120 |
| grand 120-cell {5,3,5/2} [39] | great icosahedral 120-cell {3,5/2,5} [40] | great grand 120-cell {5,5/2,3} [41] | grand stellated 120-cell {5/2,5,5/2} [42] |
x5o3o5/2o . . . . | 120 ♦ 12 | 30 | 20 ----------+-----+-----+-----+---- x . . . | 2 | 720 | 5 | 5 ----------+-----+-----+-----+---- x5o . . | 5 | 5 | 720 | 2 ----------+-----+-----+-----+---- x5o3o . ♦ 20 | 30 | 12 | 120 |
x3o5/2o5o . . . . | 120 ♦ 12 | 30 | 12 ----------+-----+-----+------+---- x . . . | 2 | 720 | 5 | 5 ----------+-----+-----+------+---- x3o . . | 3 | 3 | 1200 | 2 ----------+-----+-----+------+---- x3o5/2o . ♦ 12 | 30 | 20 | 120 |
x5o5/2o3o . . . . | 120 ♦ 20 | 30 | 12 ----------+-----+------+-----+---- x . . . | 2 | 1200 | 3 | 3 ----------+-----+------+-----+---- x5o . . | 5 | 5 | 720 | 2 ----------+-----+------+-----+---- x5o5/2o . ♦ 12 | 30 | 12 | 120 |
x5/2o5o5/2o . . . . | 120 ♦ 12 | 30 | 12 ----------+-----+-----+-----+---- x . . . | 2 | 720 | 5 | 5 ----------+-----+-----+-----+---- x5/2o . . | 5 | 5 | 720 | 2 ----------+-----+-----+-----+---- x5/2o5o . ♦ 12 | 30 | 12 | 120 |
5-cells
| 5-cell {3,3,3} [43] | Tetrahedral pyramid {3,3}∨( ) | {3}∨{ } |
|---|---|---|
x3o3o3o . . . . | 5 ♦ 4 | 6 | 4 --------+---+----+----+-- x . . . | 2 | 10 | 3 | 3 --------+---+----+----+-- x3o . . | 3 | 3 | 10 | 2 --------+---+----+----+-- x3o3o . ♦ 4 | 6 | 4 | 5 |
(pt || tet) o.3o.3o. | 1 * ♦ 4 0 | 6 0 | 4 0 .o3.o3.o | * 4 ♦ 1 3 | 3 3 | 3 1 ------------+-----+-----+-----+---- oo3oo3oo&#x | 1 1 | 4 * | 3 0 | 3 0 .x .. .. | 0 2 | * 6 | 1 2 | 2 1 ------------+-----+-----+-----+---- ox .. ..&#x | 1 2 | 2 1 | 6 * | 2 0 .x3.o .. | 0 3 | 0 3 | * 4 | 1 1 ------------+-----+-----+-----+---- ox3oo ..&#x ♦ 1 3 | 3 3 | 3 1 | 4 * .x3.o3.o ♦ 0 4 | 0 6 | 0 4 | * 1 |
(line || perp {3})
o. o.3o. | 2 * ♦ 1 3 0 | 3 3 0 | 3 1
.o .o3.o | * 3 ♦ 0 2 2 | 1 4 1 | 2 2
------------+-----+-------+-------+----
x. .. .. | 2 0 | 1 * * | 3 0 0 | 3 0
oo oo3oo&#x | 1 1 | * 6 * | 1 2 0 | 2 1
.. .x .. | 0 2 | * * 3 | 0 2 1 | 1 2
------------+-----+-------+-------+----
xo .. ..&#x | 2 1 | 1 2 0 | 3 * * | 2 0
.. ox ..&#x | 1 2 | 0 2 1 | * 6 * | 1 1
.. .x3.o | 0 3 | 0 0 3 | * * 1 | 0 2
------------+-----+-------+-------+----
xo ox ..&#x ♦ 2 2 | 1 4 1 | 2 2 0 | 3 *
.. ox3oo&#x ♦ 1 3 | 0 3 3 | 0 3 1 | * 2
|
| {3}∨( )∨( ) | Digonal disphenoid pyramid, { }∨{ }∨( ) | |
( (pt || {3}) || pt)
o..3o.. | 1 * * ♦ 3 1 0 0 | 3 3 0 0 | 1 3 0
.o.3.o. | * 3 * ♦ 1 0 2 1 | 2 1 1 2 | 1 2 1
..o3..o | * * 1 ♦ 0 1 0 3 | 0 3 0 3 | 0 3 1
-----------+-------+---------+---------+------
oo.3oo.&#x | 1 1 0 | 3 * * * | 2 1 0 0 | 1 2 0
o.o3o.o&#x | 1 0 1 | * 1 * * | 0 3 0 0 | 0 3 0
.x. ... | 0 2 0 | * * 3 * | 1 0 1 1 | 1 1 1
.oo3.oo&#x | 0 1 1 | * * * 3 | 0 1 0 2 | 0 2 1
-----------+-------+---------+---------+------
ox. ...&#x | 1 2 0 | 2 0 1 0 | 3 * * * | 1 1 0
ooo ...&#x | 1 1 1 | 1 1 0 1 | * 3 * * | 0 2 0
.x.3.o. | 0 3 0 | 0 0 3 0 | * * 1 * | 1 0 1
.xo ...&#x | 0 2 1 | 0 0 1 2 | * * * 3 | 0 1 1
-----------+-------+---------+---------+------
ox.3oo.&#x ♦ 1 3 0 | 3 0 3 0 | 3 0 1 0 | 1 * *
oxo ...&#x ♦ 1 2 1 | 2 1 1 2 | 1 2 0 1 | * 3 *
.xo3.oo&#x ♦ 0 3 1 | 0 0 3 3 | 0 0 1 3 | * * 1
|
( (pt || line) || perp line) o.. o.. | 1 * * ♦ 2 2 0 0 0 | 1 4 1 0 0 | 2 2 0 .o. .o. | * 2 * ♦ 1 0 1 2 0 | 1 2 0 2 1 | 2 1 1 ..o ..o | * * 2 ♦ 0 1 0 2 1 | 0 2 1 1 2 | 1 2 1 -----------+-------+-----------+-----------+------ oo. oo.&#x | 1 1 0 | 2 * * * * | 1 2 0 0 0 | 2 1 0 o.o o.o&#x | 1 0 1 | * 2 * * * | 0 2 1 0 0 | 1 2 0 .x. ... | 0 2 0 | * * 1 * * | 1 0 0 2 0 | 2 0 1 .oo .oo&#x | 0 1 1 | * * * 4 * | 0 1 0 1 1 | 1 1 1 ... ..x | 0 0 2 | * * * * 1 | 0 0 1 0 2 | 0 2 1 -----------+-------+-----------+-----------+------ ox. ...&#x | 1 2 0 | 2 0 1 0 0 | 1 * * * * | 2 0 0 ooo ooo&#x | 1 1 1 | 1 1 0 1 0 | * 4 * * * | 1 1 0 ... o.x&#x | 1 0 2 | 0 2 0 0 1 | * * 1 * * | 0 2 0 .xo ...&#x | 0 2 1 | 0 0 1 2 0 | * * * 2 * | 1 0 1 ... .ox&#x | 0 1 2 | 0 0 0 2 1 | * * * * 2 | 0 1 1 -----------+-------+-----------+-----------+------ oxo ...&#x ♦ 1 2 1 | 2 1 1 2 0 | 1 2 0 1 0 | 2 * * ... oox&#x ♦ 1 1 2 | 1 2 0 2 1 | 0 2 1 0 1 | * 2 * .xo .ox&#x ♦ 0 2 2 | 0 0 1 4 1 | 0 0 0 2 2 | * * 1 | |
Uniform 4D
o3x3o3o - rap . . . . | 10 ♦ 6 | 3 6 | 3 2 --------+----+----+-------+---- . x . . | 2 | 30 | 1 2 | 2 1 --------+----+----+-------+---- o3x . . | 3 | 3 | 10 * | 2 0 . x3o . | 3 | 3 | * 20 | 1 1 --------+----+----+-------+---- o3x3o . ♦ 6 | 12 | 4 4 | 5 * . x3o3o ♦ 4 | 6 | 0 4 | * 5 |
x3x3o3o - tip . . . . | 20 | 1 3 | 3 3 | 3 1 --------+----+-------+-------+---- x . . . | 2 | 10 * | 3 0 | 3 0 . x . . | 2 | * 30 | 1 2 | 2 1 --------+----+-------+-------+---- x3x . . | 6 | 3 3 | 10 * | 2 0 . x3o . | 3 | 0 3 | * 20 | 1 1 --------+----+-------+-------+---- x3x3o . ♦ 12 | 6 12 | 4 4 | 5 * . x3o3o ♦ 4 | 0 6 | 0 4 | * 5 |
x3o3x3o - srip . . . . | 30 ♦ 2 4 | 1 4 2 2 | 2 2 1 --------+----+-------+-------------+------- x . . . | 2 | 30 * | 1 2 0 0 | 2 1 0 . . x . | 2 | * 60 | 0 1 1 1 | 1 1 1 --------+----+-------+-------------+------- x3o . . | 3 | 3 0 | 10 * * * | 2 0 0 x . x . | 4 | 2 2 | * 30 * * | 1 1 0 . o3x . | 3 | 0 3 | * * 20 * | 1 0 1 . . x3o | 3 | 0 3 | * * * 20 | 0 1 1 --------+----+-------+-------------+------- x3o3x . ♦ 12 | 12 12 | 4 6 4 0 | 5 * * x . x3o ♦ 6 | 3 6 | 0 3 0 2 | * 10 * . o3x3o ♦ 6 | 0 12 | 0 0 4 4 | * * 5 |
x3o3o3x - spid . . . . | 20 ♦ 3 3 | 3 6 3 | 1 3 3 1 --------+----+-------+----------+---------- x . . . | 2 | 30 * | 2 2 0 | 1 2 1 0 . . . x | 2 | * 30 | 0 2 2 | 0 1 2 1 --------+----+-------+----------+---------- x3o . . | 3 | 3 0 | 20 * * | 1 1 0 0 x . . x | 4 | 2 2 | * 30 * | 0 1 1 0 . . o3x | 3 | 0 3 | * * 20 | 0 0 1 1 --------+----+-------+----------+---------- x3o3o . ♦ 4 | 6 0 | 4 0 0 | 5 * * * x3o . x ♦ 6 | 6 3 | 2 3 0 | * 10 * * x . o3x ♦ 6 | 3 6 | 0 3 2 | * * 10 * . o3o3x ♦ 4 | 0 6 | 0 0 4 | * * * 5 | |
o3x3x3o - deca . . . . | 30 | 2 2 | 1 4 1 | 2 2 --------+----+-------+----------+---- . x . . | 2 | 30 * | 1 2 0 | 2 1 . . x . | 2 | * 30 | 0 2 1 | 1 2 --------+----+-------+----------+---- o3x . . | 3 | 3 0 | 10 * * | 2 0 . x3x . | 6 | 3 3 | * 20 * | 1 1 . . x3o | 3 | 0 3 | * * 10 | 0 2 --------+----+-------+----------+---- o3x3x . ♦ 12 | 12 6 | 4 4 0 | 5 * . x3x3o ♦ 12 | 6 12 | 0 4 4 | * 5 |
x3x3x3o - grip . . . . | 60 | 1 1 2 | 1 2 2 1 | 2 1 1 --------+----+----------+-------------+------- x . . . | 2 | 30 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 30 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 60 | 0 1 1 1 | 1 1 1 --------+----+----------+-------------+------- x3x . . | 6 | 3 3 0 | 10 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 30 * * | 1 1 0 . x3x . | 6 | 0 3 3 | * * 20 * | 1 0 1 . . x3o | 3 | 0 0 3 | * * * 20 | 0 1 1 --------+----+----------+-------------+------- x3x3x . ♦ 24 | 12 12 12 | 4 6 4 0 | 5 * * x . x3o ♦ 6 | 3 0 6 | 0 3 0 2 | * 10 * . x3x3o ♦ 12 | 0 6 12 | 0 0 4 4 | * * 5 |
x3x3o3x - prip . . . . | 60 | 1 2 2 | 2 2 1 2 1 | 1 2 1 1 --------+----+----------+----------------+---------- x . . . | 2 | 30 * * | 2 2 0 0 0 | 1 2 1 0 . x . . | 2 | * 60 * | 1 0 1 1 0 | 1 1 0 1 . . . x | 2 | * * 60 | 0 1 0 1 1 | 0 1 1 1 --------+----+----------+----------------+---------- x3x . . | 6 | 3 3 0 | 20 * * * * | 1 1 0 0 x . . x | 4 | 2 0 2 | * 30 * * * | 0 1 1 0 . x3o . | 3 | 0 3 0 | * * 20 * * | 1 0 0 1 . x . x | 4 | 0 2 2 | * * * 30 * | 0 1 0 1 . . o3x | 3 | 0 0 3 | * * * * 20 | 0 0 1 1 --------+----+----------+----------------+---------- x3x3o . ♦ 12 | 6 12 0 | 4 0 4 0 0 | 5 * * * x3x . x ♦ 12 | 6 6 6 | 2 3 0 3 0 | * 10 * * x . o3x ♦ 6 | 3 0 6 | 0 3 0 0 2 | * * 10 * . x3o3x ♦ 12 | 0 12 12 | 0 0 4 6 4 | * * * 5 |
x3x3x3x - gippid . . . . | 120 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 --------+-----+-------------+-------------------+---------- x . . . | 2 | 60 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 60 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 60 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 2 | * * * 60 | 0 0 1 0 1 1 | 0 1 1 1 --------+-----+-------------+-------------------+---------- x3x . . | 6 | 3 3 0 0 | 20 * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 30 * * * * | 1 0 1 0 x . . x | 4 | 2 0 0 2 | * * 30 * * * | 0 1 1 0 . x3x . | 6 | 0 3 3 0 | * * * 20 * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 30 * | 0 1 0 1 . . x3x | 6 | 0 0 3 3 | * * * * * 20 | 0 0 1 1 --------+-----+-------------+-------------------+---------- x3x3x . ♦ 24 | 12 12 12 0 | 4 6 0 4 0 0 | 5 * * * x3x . x ♦ 12 | 6 6 0 6 | 2 0 3 0 3 0 | * 10 * * x . x3x ♦ 12 | 6 0 6 6 | 0 3 3 0 0 2 | * * 10 * . x3x3x ♦ 24 | 0 12 12 12 | 0 0 0 4 6 4 | * * * 5 | |
| 4-demicube {3,31,1} [44] | 24-cell {31,1,1} [45] |
|---|---|
x3o3o *b3o - hex . . . . | 8 ♦ 6 | 12 | 4 4 -----------+---+----+----+---- x . . . | 2 | 24 | 4 | 2 2 -----------+---+----+----+---- x3o . . | 3 | 3 | 32 | 1 1 -----------+---+----+----+---- x3o3o . ♦ 4 | 6 | 4 | 8 * x3o . *b3o ♦ 4 | 6 | 4 | * 8 |
o3x3o *b3o - ico . . . . | 24 ♦ 8 | 4 4 4 | 2 2 2 -----------+----+----+----------+------ . x . . | 2 | 96 | 1 1 1 | 1 1 1 -----------+----+----+----------+------ o3x . . | 3 | 3 | 32 * * | 1 1 0 . x3o . | 3 | 3 | * 32 * | 1 0 1 . x . *b3o | 3 | 3 | * * 32 | 0 1 1 -----------+----+----+----------+------ o3x3o . ♦ 6 | 12 | 4 4 0 | 8 * * o3x . *b3o ♦ 6 | 12 | 4 0 4 | * 8 * . x3o *b3o ♦ 6 | 12 | 0 4 4 | * * 8 |
o3x3o4o - ico . . . . | 24 ♦ 8 | 4 8 | 4 2 --------+----+----+-------+----- . x . . | 2 | 96 | 1 2 | 2 1 --------+----+----+-------+----- o3x . . | 3 | 3 | 32 * | 2 0 . x3o . | 3 | 3 | * 64 | 1 1 --------+----+----+-------+----- o3x3o . ♦ 6 | 12 | 4 4 | 16 * . x3o4o ♦ 6 | 12 | 0 8 | * 8 | |
o3o3x4o - rit . . . . | 32 ♦ 6 | 6 3 | 2 3 --------+----+----+-------+----- . . x . | 2 | 96 | 1 2 | 1 2 --------+----+----+-------+----- . o3x . | 3 | 3 | 64 * | 1 1 . . x4o | 4 | 4 | * 24 | 0 2 --------+----+----+-------+----- o3o3x . ♦ 4 | 6 | 4 0 | 16 * . o3x4o ♦ 12 | 24 | 8 6 | * 8 |
x3x3o4o - thex . . . . | 48 | 1 4 | 4 4 | 4 1 --------+----+-------+-------+----- x . . . | 2 | 24 * | 4 0 | 4 0 . x . . | 2 | * 96 | 1 2 | 2 1 --------+----+-------+-------+----- x3x . . | 6 | 3 3 | 32 * | 2 0 . x3o . | 3 | 0 3 | * 64 | 1 1 --------+----+-------+-------+----- x3x3o . ♦ 12 | 6 12 | 4 4 | 16 * . x3o4o ♦ 6 | 0 12 | 0 8 | * 8 |
x3o3x4o - rico . . . . | 96 ♦ 2 4 | 1 4 2 2 | 2 2 1 --------+----+--------+-------------+-------- x . . . | 2 | 96 * | 1 2 0 0 | 2 1 0 . . x . | 2 | * 192 | 0 1 1 1 | 1 1 1 --------+----+--------+-------------+-------- x3o . . | 3 | 3 0 | 32 * * * | 2 0 0 x . x . | 4 | 2 2 | * 96 * * | 1 1 0 . o3x . | 3 | 0 3 | * * 64 * | 1 0 1 . . x4o | 4 | 0 4 | * * * 48 | 0 1 1 --------+----+--------+-------------+-------- x3o3x . ♦ 12 | 12 12 | 4 6 4 0 | 16 * * x . x4o ♦ 8 | 4 8 | 0 4 0 2 | * 24 * . o3x4o ♦ 12 | 0 24 | 0 0 8 6 | * * 8 |
x3o3o4x - sidpith . . . . | 64 | 3 3 | 3 6 3 | 1 3 3 1 --------+----+-------+----------+----------- x . . . | 2 | 96 * | 2 2 0 | 1 2 1 0 . . . x | 2 | * 96 | 0 2 2 | 0 1 2 1 --------+----+-------+----------+----------- x3o . . | 3 | 3 0 | 64 * * | 1 1 0 0 x . . x | 4 | 2 2 | * 96 * | 0 1 1 0 . . o4x | 4 | 0 4 | * * 48 | 0 0 1 1 --------+----+-------+----------+----------- x3o3o . ♦ 4 | 6 0 | 4 0 0 | 16 * * * x3o . x ♦ 6 | 6 3 | 2 3 0 | * 32 * * x . o4x ♦ 8 | 4 8 | 0 4 2 | * * 24 * . o3o4x ♦ 8 | 0 12 | 0 0 6 | * * * 8 |
o3x3x4o - tah . . . . | 96 | 2 2 | 1 4 1 | 2 2 --------+----+-------+----------+----- . x . . | 2 | 96 * | 1 2 0 | 2 1 . . x . | 2 | * 96 | 0 2 1 | 1 2 --------+----+-------+----------+----- o3x . . | 3 | 3 0 | 32 * * | 2 0 . x3x . | 6 | 3 3 | * 64 * | 1 1 . . x4o | 4 | 0 4 | * * 24 | 0 2 --------+----+-------+----------+----- o3x3x . ♦ 12 | 12 6 | 4 4 0 | 16 * . x3x4o ♦ 24 | 12 24 | 0 8 6 | * 8 |
o3x3o4x - srit . . . . | 96 | 4 2 | 2 2 4 1 | 1 2 2 (A),(B) --------+----+--------+-------------+-------- . x . . | 2 | 192 * | 1 1 1 0 | 1 1 1 (1),(/),(\) . . . x | 2 | * 96 | 0 0 2 1 | 0 1 2 (2),(3) --------+----+--------+-------------+-------- o3x . . | 3 | 3 0 | 64 * * * | 1 1 0 . x3o . | 3 | 3 0 | * 64 * * | 1 0 1 . x . x | 4 | 2 2 | * * 96 * | 0 1 1 . . o4x | 4 | 0 4 | * * * 24 | 0 0 2 --------+----+--------+-------------+-------- o3x3o . ♦ 6 | 12 0 | 4 4 0 0 | 16 * * o3x . x ♦ 6 | 6 3 | 2 0 3 0 | * 32 * . x3o4x ♦ 24 | 24 24 | 0 8 12 6 | * * 8 |
o3o3x4x - tat . . . . | 64 | 3 1 | 3 3 | 1 3 --------+----+-------+-------+----- . . x . | 2 | 96 * | 2 1 | 1 2 . . . x | 2 | * 32 | 0 3 | 0 3 --------+----+-------+-------+----- . o3x . | 3 | 3 0 | 64 * | 1 1 . . x4x | 8 | 4 4 | * 24 | 0 2 --------+----+-------+-------+----- o3o3x . ♦ 4 | 6 0 | 4 0 | 16 * . o3x4x ♦ 24 | 24 12 | 8 6 | * 8 |
x3x3x4o - tico . . . . | 192 | 1 1 2 | 1 2 2 1 | 2 1 1 --------+-----+-----------+-------------+-------- x . . . | 2 | 96 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 96 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 192 | 0 1 1 1 | 1 1 1 --------+-----+-----------+-------------+-------- x3x . . | 6 | 3 3 0 | 32 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 96 * * | 1 1 0 . x3x . | 6 | 0 3 3 | * * 64 * | 1 0 1 . . x4o | 4 | 0 0 4 | * * * 48 | 0 1 1 --------+-----+-----------+-------------+-------- x3x3x . ♦ 24 | 12 12 12 | 4 6 4 0 | 16 * * x . x4o ♦ 8 | 4 0 8 | 0 4 0 2 | * 24 * . x3x4o ♦ 24 | 0 12 24 | 0 0 8 6 | * * 8 |
x3x3o4x - prit . . . . | 192 | 1 2 2 | 2 2 1 2 1 | 1 2 1 1 --------+-----+------------+----------------+----------- x . . . | 2 | 96 * * | 2 2 0 0 0 | 1 2 1 0 . x . . | 2 | * 192 * | 1 0 1 1 0 | 1 1 0 1 . . . x | 2 | * * 192 | 0 1 0 1 1 | 0 1 1 1 --------+-----+------------+----------------+----------- x3x . . | 6 | 3 3 0 | 64 * * * * | 1 1 0 0 x . . x | 4 | 2 0 2 | * 96 * * * | 0 1 1 0 . x3o . | 3 | 0 3 0 | * * 64 * * | 1 0 0 1 . x . x | 4 | 0 2 2 | * * * 96 * | 0 1 0 1 . . o4x | 4 | 0 0 4 | * * * * 48 | 0 0 1 1 --------+-----+------------+----------------+----------- x3x3o . ♦ 12 | 6 12 0 | 4 0 4 0 0 | 16 * * * x3x . x ♦ 12 | 6 6 6 | 2 3 0 3 0 | * 32 * * x . o4x ♦ 8 | 4 0 8 | 0 4 0 0 2 | * * 24 * . x3o4x ♦ 24 | 0 24 24 | 0 0 8 12 6 | * * * 8 |
x3o3x4x - proh . . . . | 192 | 2 2 1 | 1 2 2 1 2 | 1 1 2 1 --------+-----+------------+----------------+----------- x . . . | 2 | 192 * * | 1 1 1 0 0 | 1 1 1 0 . . x . | 2 | * 192 * | 0 1 0 1 1 | 1 0 1 1 . . . x | 2 | * * 96 | 0 0 2 0 2 | 0 1 2 1 --------+-----+------------+----------------+----------- x3o . . | 3 | 3 0 0 | 64 * * * * | 1 1 0 0 x . x . | 4 | 2 2 0 | * 96 * * * | 1 0 1 0 x . . x | 4 | 2 0 2 | * * 96 * * | 0 1 1 0 . o3x . | 3 | 0 3 0 | * * * 64 * | 1 0 0 1 . . x4x | 8 | 0 4 4 | * * * * 48 | 0 0 1 1 --------+-----+------------+----------------+----------- x3o3x . ♦ 12 | 12 12 0 | 4 6 0 4 0 | 16 * * * x3o . x ♦ 6 | 6 0 3 | 2 0 3 0 0 | * 32 * * x . x4x ♦ 16 | 8 8 8 | 0 4 4 0 2 | * * 24 * . o3x4x ♦ 24 | 0 24 12 | 0 0 0 8 6 | * * * 8 |
o3x3x4x - grit . . . . | 192 | 2 1 1 | 1 2 2 1 | 1 1 2 --------+-----+-----------+-------------+-------- . x . . | 2 | 192 * * | 1 1 1 0 | 1 1 1 . . x . | 2 | * 96 * | 0 2 0 1 | 1 0 2 . . . x | 2 | * * 96 | 0 0 2 1 | 0 1 2 --------+-----+-----------+-------------+-------- o3x . . | 3 | 3 0 0 | 64 * * * | 1 1 0 . x3x . | 6 | 3 3 0 | * 64 * * | 1 0 1 . x . x | 4 | 2 0 2 | * * 96 * | 0 1 1 . . x4x | 8 | 0 4 4 | * * * 24 | 0 0 2 --------+-----+-----------+-------------+-------- o3x3x . ♦ 12 | 12 6 0 | 4 4 0 0 | 16 * * o3x . x ♦ 6 | 6 0 3 | 2 0 3 0 | * 32 * . x3x4x ♦ 48 | 24 24 24 | 0 8 12 6 | * * 8 |
x3x3x4x - gidpith . . . . | 384 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 --------+-----+-----------------+-------------------+----------- x . . . | 2 | 192 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 192 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 192 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 2 | * * * 192 | 0 0 1 0 1 1 | 0 1 1 1 --------+-----+-----------------+-------------------+----------- x3x . . | 6 | 3 3 0 0 | 64 * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 96 * * * * | 1 0 1 0 x . . x | 4 | 2 0 0 2 | * * 96 * * * | 0 1 1 0 . x3x . | 6 | 0 3 3 0 | * * * 64 * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 96 * | 0 1 0 1 . . x4x | 8 | 0 0 4 4 | * * * * * 48 | 0 0 1 1 --------+-----+-----------------+-------------------+----------- x3x3x . ♦ 24 | 12 12 12 0 | 4 6 0 4 0 0 | 16 * * * x3x . x ♦ 12 | 6 6 0 6 | 2 0 3 0 3 0 | * 32 * * x . x4x ♦ 16 | 8 0 8 8 | 0 4 4 0 0 2 | * * 24 * . x3x4x ♦ 48 | 0 24 24 24 | 0 0 0 8 12 6 | * * * 8 | |||
o3x4o3o - rico . . . . | 96 ♦ 6 | 3 6 | 3 2 --------+----+-----+--------+------ . x . . | 2 | 288 | 1 2 | 2 1 --------+----+-----+--------+------ o3x . . | 3 | 3 | 96 * | 2 0 . x4o . | 4 | 4 | * 144 | 1 1 --------+----+-----+--------+------ o3x4o . ♦ 12 | 24 | 8 6 | 24 * . x4o3o ♦ 8 | 12 | 0 6 | * 24 |
x3x4o3o - tico . . . . | 192 | 1 3 | 3 3 | 3 1 --------+-----+--------+--------+------ x . . . | 2 | 96 * | 3 0 | 3 0 . x . . | 2 | * 288 | 1 2 | 2 1 --------+-----+--------+--------+------ x3x . . | 6 | 3 3 | 96 * | 2 0 . x4o . | 4 | 0 4 | * 144 | 1 1 --------+-----+--------+--------+------ x3x4o . ♦ 24 | 12 24 | 8 6 | 24 * . x4o3o ♦ 8 | 0 12 | 0 6 | * 24 |
x3o4x3o - srico . . . . | 288 | 2 4 | 1 4 2 2 | 2 2 1 --------+-----+---------+----------------+--------- x . . . | 2 | 288 * | 1 2 0 0 | 2 1 0 . . x . | 2 | * 576 | 0 1 1 1 | 1 1 1 --------+-----+---------+----------------+--------- x3o . . | 3 | 3 0 | 96 * * * | 2 0 0 x . x . | 4 | 2 2 | * 288 * * | 1 1 0 . o4x . | 4 | 0 4 | * * 144 * | 1 0 1 . . x3o | 3 | 0 3 | * * * 192 | 0 1 1 --------+-----+---------+----------------+--------- x3o4x . ♦ 24 | 24 24 | 8 12 6 0 | 24 * * x . x3o ♦ 6 | 3 6 | 0 3 0 2 | * 96 * . o4x3o ♦ 12 | 0 24 | 0 0 6 8 | * * 24 |
x3o4o3x - spic . . . . | 144 ♦ 4 4 | 4 8 4 | 1 4 4 1 --------+-----+---------+-------------+------------ x . . . | 2 | 288 * | 2 2 0 | 1 2 1 0 . . . x | 2 | * 288 | 0 2 2 | 0 1 2 1 --------+-----+---------+-------------+------------ x3o . . | 3 | 3 0 | 192 * * | 1 1 0 0 x . . x | 4 | 2 2 | * 288 * | 0 1 1 0 . . o3x | 3 | 0 3 | * * 192 | 0 0 1 1 --------+-----+---------+-------------+------------ x3o4o . ♦ 6 | 12 0 | 8 0 0 | 24 * * * x3o . x ♦ 6 | 6 3 | 2 3 0 | * 96 * * x . o3x ♦ 6 | 3 6 | 0 3 2 | * * 96 * . o4o3x ♦ 6 | 0 12 | 0 0 8 | * * * 24 |
o3x4x3o - cont . . . . | 288 | 2 2 | 1 4 1 | 2 2 --------+-----+---------+-----------+------ . x . . | 2 | 288 * | 1 2 0 | 2 1 . . x . | 2 | * 288 | 0 2 1 | 1 2 --------+-----+---------+-----------+------ o3x . . | 3 | 3 0 | 96 * * | 2 0 . x4x . | 8 | 4 4 | * 144 * | 1 1 . . x3o | 3 | 0 3 | * * 96 | 0 2 --------+-----+---------+-----------+------ o3x4x . ♦ 24 | 24 12 | 8 6 0 | 24 * . x4x3o ♦ 24 | 12 24 | 0 6 8 | * 24 |
x3x4x3o - grico . . . . | 576 | 1 1 2 | 1 2 2 1 | 2 1 1 --------+-----+-------------+----------------+--------- x . . . | 2 | 288 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 288 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 576 | 0 1 1 1 | 1 1 1 --------+-----+-------------+----------------+--------- x3x . . | 6 | 3 3 0 | 96 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 288 * * | 1 1 0 . x4x . | 8 | 0 4 4 | * * 144 * | 1 0 1 . . x3o | 3 | 0 0 3 | * * * 192 | 0 1 1 --------+-----+-------------+----------------+--------- x3x4x . ♦ 48 | 24 24 24 | 8 12 6 0 | 24 * * x . x3o ♦ 6 | 3 0 6 | 0 3 0 2 | * 96 * . x4x3o ♦ 24 | 0 12 24 | 0 0 6 8 | * * 24 |
x3x4o3x - prico . . . . | 576 | 1 2 2 | 2 2 1 2 1 | 1 2 1 1 --------+-----+-------------+---------------------+------------ x . . . | 2 | 288 * * | 2 2 0 0 0 | 1 2 1 0 . x . . | 2 | * 576 * | 1 0 1 1 0 | 1 1 0 1 . . . x | 2 | * * 576 | 0 1 0 1 1 | 0 1 1 1 --------+-----+-------------+---------------------+------------ x3x . . | 6 | 3 3 0 | 192 * * * * | 1 1 0 0 x . . x | 4 | 2 0 2 | * 288 * * * | 0 1 1 0 . x4o . | 4 | 0 4 0 | * * 144 * * | 1 0 0 1 . x . x | 4 | 0 2 2 | * * * 288 * | 0 1 0 1 . . o3x | 3 | 0 0 3 | * * * * 192 | 0 0 1 1 --------+-----+-------------+---------------------+------------ x3x4o . ♦ 24 | 12 24 0 | 8 0 6 0 0 | 24 * * * x3x . x ♦ 12 | 6 6 6 | 2 3 0 3 0 | * 96 * * x . o3x ♦ 6 | 3 0 6 | 0 3 0 0 2 | * * 96 * . x4o3x ♦ 24 | 0 24 24 | 0 0 6 12 8 | * * * 24 |
x3x4x3x - gippic . . . . | 1152 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 --------+------+-----------------+-------------------------+------------ x . . . | 2 | 576 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 576 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 576 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 2 | * * * 576 | 0 0 1 0 1 1 | 0 1 1 1 --------+------+-----------------+-------------------------+------------ x3x . . | 6 | 3 3 0 0 | 192 * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 288 * * * * | 1 0 1 0 x . . x | 4 | 2 0 0 2 | * * 288 * * * | 0 1 1 0 . x4x . | 8 | 0 4 4 0 | * * * 144 * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 288 * | 0 1 0 1 . . x3x | 6 | 0 0 3 3 | * * * * * 192 | 0 0 1 1 --------+------+-----------------+-------------------------+------------ x3x4x . ♦ 48 | 24 24 24 0 | 8 12 0 6 0 0 | 24 * * * x3x . x ♦ 12 | 6 6 0 6 | 2 0 3 0 3 0 | * 96 * * x . x3x ♦ 12 | 6 0 6 6 | 0 3 3 0 0 2 | * * 96 * . x4x3x ♦ 48 | 0 24 24 24 | 0 0 0 6 12 8 | * * * 24 |
s3s4o3o - sadi
demi( . . . . ) | 96 ♦ 3 6 | 3 9 3 | 3 1 4
----------------+----+---------+-----------+---------
. s4o . | 2 | 144 * | 0 2 2 | 1 1 2
sefa( s3s . . ) | 2 | * 288 | 1 2 0 | 2 0 1
----------------+----+---------+-----------+---------
s3s . . ♦ 3 | 0 3 | 96 * * | 2 0 0
sefa( s3s4o . ) | 3 | 1 2 | * 288 * | 1 0 1
sefa( . s4o3o ) | 3 | 3 0 | * * 96 | 0 1 1
----------------+----+---------+-----------+---------
s3s4o . ♦ 12 | 6 24 | 8 12 0 | 24 * *
. s4o3o ♦ 4 | 6 0 | 0 0 4 | * 24 *
sefa( s3s4o3o ) ♦ 4 | 3 3 | 0 3 1 | * * 96
|
o3x3o5o - rox . . . . | 720 ♦ 10 | 5 10 | 5 2 --------+-----+------+-----------+-------- . x . . | 2 | 3600 | 1 2 | 2 1 --------+-----+------+-----------+-------- o3x . . | 3 | 3 | 1200 * | 2 0 . x3o . | 3 | 3 | * 2400 | 1 1 --------+-----+------+-----------+-------- o3x3o . ♦ 6 | 12 | 4 4 | 600 * . x3o5o ♦ 12 | 30 | 0 20 | * 120 |
o3o3x5o - rahi . . . . | 1200 ♦ 6 | 6 3 | 2 3 --------+------+------+----------+-------- . . x . | 2 | 3600 | 2 1 | 1 2 --------+------+------+----------+-------- . o3x . | 3 | 3 | 2400 * | 1 1 . . x5o | 5 | 5 | * 720 | 0 2 --------+------+------+----------+-------- o3o3x . ♦ 4 | 6 | 4 0 | 600 * . o3x5o ♦ 30 | 60 | 20 12 | * 120 |
x3x3o5o - tex . . . . | 1440 | 1 5 | 5 5 | 5 1 --------+------+----------+-----------+-------- x . . . | 2 | 720 * | 5 0 | 5 0 . x . . | 2 | * 3600 | 1 2 | 2 1 --------+------+----------+-----------+-------- x3x . . | 6 | 3 3 | 1200 * | 2 0 . x3o . | 3 | 0 3 | * 2400 | 1 1 --------+------+----------+-----------+-------- x3x3o . ♦ 12 | 6 12 | 4 4 | 600 * . x3o5o ♦ 12 | 0 30 | 0 20 | * 120 |
x3o3x5o - srix . . . . | 3600 | 2 4 | 1 4 2 2 | 2 2 1 --------+------+-----------+---------------------+------------ x . . . | 2 | 3600 * | 1 2 0 0 | 2 1 0 . . x . | 2 | * 7200 | 0 1 1 1 | 1 1 1 --------+------+-----------+---------------------+------------ x3o . . | 3 | 3 0 | 1200 * * * | 2 0 0 x . x . | 4 | 2 2 | * 3600 * * | 1 1 0 . o3x . | 3 | 0 3 | * * 2400 * | 1 0 1 . . x5o | 5 | 0 5 | * * * 1440 | 0 1 1 --------+------+-----------+---------------------+------------ x3o3x . ♦ 12 | 12 12 | 4 6 4 0 | 600 * * x . x5o ♦ 10 | 5 10 | 0 5 0 2 | * 720 * . o3x5o ♦ 30 | 0 60 | 0 0 20 12 | * * 120 |
x3o3o5x - sidpixhi . . . . | 2400 | 3 3 | 3 6 3 | 1 3 3 1 --------+------+-----------+----------------+----------------- x . . . | 2 | 3600 * | 2 2 0 | 1 2 1 0 . . . x | 2 | * 3600 | 0 2 2 | 0 1 2 1 --------+------+-----------+----------------+----------------- x3o . . | 3 | 3 0 | 2400 * * | 1 1 0 0 x . . x | 4 | 2 2 | * 3600 * | 0 1 1 0 . . o5x | 5 | 0 5 | * * 1440 | 0 0 1 1 --------+------+-----------+----------------+----------------- x3o3o . ♦ 4 | 6 0 | 4 0 0 | 600 * * * x3o . x ♦ 6 | 6 3 | 2 3 0 | * 1200 * * x . o5x ♦ 10 | 5 10 | 0 5 2 | * * 720 * . o3o5x ♦ 20 | 0 30 | 0 0 12 | * * * 120 |
o3x3x5o - xhi . . . . | 3600 | 2 2 | 1 4 1 | 2 2 --------+------+-----------+---------------+-------- . x . . | 2 | 3600 * | 1 2 0 | 2 1 . . x . | 2 | * 3600 | 0 2 1 | 1 2 --------+------+-----------+---------------+-------- o3x . . | 3 | 3 0 | 1200 * * | 2 0 . x3x . | 6 | 3 3 | * 2400 * | 1 1 . . x5o | 5 | 0 5 | * * 720 | 0 2 --------+------+-----------+---------------+-------- o3x3x . ♦ 12 | 12 6 | 4 4 0 | 600 * . x3x5o ♦ 60 | 30 60 | 0 20 12 | * 120 |
o3x3o5x - srahi . . . . | 3600 | 4 2 | 2 2 4 1 | 1 2 2 --------+------+-----------+--------------------+------------- . x . . | 2 | 7200 * | 1 1 1 0 | 1 1 1 . . . x | 2 | * 3600 | 0 0 2 1 | 0 1 2 --------+------+-----------+--------------------+------------- o3x . . | 3 | 3 0 | 2400 * * * | 1 1 0 . x3o . | 3 | 3 0 | * 2400 * * | 1 0 1 . x . x | 4 | 2 2 | * * 3600 * | 0 1 1 . . o5x | 5 | 0 5 | * * * 720 | 0 0 2 --------+------+-----------+--------------------+------------- o3x3o . ♦ 6 | 12 0 | 4 4 0 0 | 600 * * o3x . x ♦ 6 | 6 3 | 2 0 3 0 | * 1200 * . x3o5x ♦ 60 | 60 60 | 0 20 30 12 | * * 120 |
o3o3x5x - thi . . . . | 2400 | 3 1 | 3 3 | 1 3 --------+------+-----------+----------+-------- . . x . | 2 | 3600 * | 2 1 | 1 2 . . . x | 2 | * 1200 | 0 3 | 0 3 --------+------+-----------+----------+-------- . o3x . | 3 | 3 0 | 2400 * | 1 1 . . x5x | 10 | 5 5 | * 720 | 0 2 --------+------+-----------+----------+-------- o3o3x . ♦ 4 | 6 0 | 4 0 | 600 * . o3x5x ♦ 60 | 60 30 | 20 12 | * 120 |
x3x3x5o - grix . . . . | 7200 | 1 1 2 | 1 2 2 1 | 2 1 1 --------+------+----------------+---------------------+------------ x . . . | 2 | 3600 * * | 1 2 0 0 | 2 1 0 . x . . | 2 | * 3600 * | 1 0 2 0 | 2 0 1 . . x . | 2 | * * 7200 | 0 1 1 1 | 1 1 1 --------+------+----------------+---------------------+------------ x3x . . | 6 | 3 3 0 | 1200 * * * | 2 0 0 x . x . | 4 | 2 0 2 | * 3600 * * | 1 1 0 . x3x . | 6 | 0 3 3 | * * 2400 * | 1 0 1 . . x5o | 5 | 0 0 5 | * * * 1440 | 0 1 1 --------+------+----------------+---------------------+------------ x3x3x . ♦ 24 | 12 12 12 | 4 6 4 0 | 600 * * x . x5o ♦ 10 | 5 0 10 | 0 5 0 2 | * 720 * . x3x5o ♦ 60 | 0 30 60 | 0 0 20 12 | * * 120 |
x3x3o5x - prahi . . . . | 7200 | 1 2 2 | 2 2 1 2 1 | 1 2 1 1 --------+------+----------------+--------------------------+----------------- x . . . | 2 | 3600 * * | 2 2 0 0 0 | 1 2 1 0 . x . . | 2 | * 7200 * | 1 0 1 1 0 | 1 1 0 1 . . . x | 2 | * * 7200 | 0 1 0 1 1 | 0 1 1 1 --------+------+----------------+--------------------------+----------------- x3x . . | 6 | 3 3 0 | 2400 * * * * | 1 1 0 0 x . . x | 4 | 2 0 2 | * 3600 * * * | 0 1 1 0 . x3o . | 3 | 0 3 0 | * * 2400 * * | 1 0 0 1 . x . x | 4 | 0 2 2 | * * * 3600 * | 0 1 0 1 . . o5x | 5 | 0 0 5 | * * * * 1440 | 0 0 1 1 --------+------+----------------+--------------------------+----------------- x3x3o . ♦ 12 | 6 12 0 | 4 0 4 0 0 | 600 * * * x3x . x ♦ 12 | 6 6 6 | 2 3 0 3 0 | * 1200 * * x . o5x ♦ 10 | 5 0 10 | 0 5 0 0 2 | * * 720 * . x3o5x ♦ 60 | 0 60 60 | 0 0 20 30 12 | * * * 120 |
x3o3x5x - prix . . . . | 7200 | 2 2 1 | 1 2 2 1 2 | 1 1 2 1 --------+------+----------------+--------------------------+----------------- x . . . | 2 | 7200 * * | 1 1 1 0 0 | 1 1 1 0 . . x . | 2 | * 7200 * | 0 1 0 1 1 | 1 0 1 1 . . . x | 2 | * * 3600 | 0 0 2 0 2 | 0 1 2 1 --------+------+----------------+--------------------------+----------------- x3o . . | 3 | 3 0 0 | 2400 * * * * | 1 1 0 0 x . x . | 4 | 2 2 0 | * 3600 * * * | 1 0 1 0 x . . x | 4 | 2 0 2 | * * 3600 * * | 0 1 1 0 . o3x . | 3 | 0 3 0 | * * * 2400 * | 1 0 0 1 . . x5x | 10 | 0 5 5 | * * * * 1440 | 0 0 1 1 --------+------+----------------+--------------------------+----------------- x3o3x . ♦ 12 | 12 12 0 | 4 6 0 4 0 | 600 * * * x3o . x ♦ 6 | 12 0 6 | 2 0 3 0 0 | * 1200 * * x . x5x ♦ 20 | 10 10 10 | 0 5 5 0 2 | * * 720 * . o3x5x ♦ 60 | 0 60 30 | 0 0 0 20 12 | * * * 120 |
o3x3x5x - grahi . . . . | 7200 | 2 1 1 | 1 2 2 1 | 1 1 2 --------+------+----------------+--------------------+------------- . x . . | 2 | 7200 * * | 1 1 1 0 | 1 1 1 . . x . | 2 | * 3600 * | 0 2 0 1 | 1 0 2 . . . x | 2 | * * 3600 | 0 0 2 1 | 0 1 2 --------+------+----------------+--------------------+------------- o3x . . | 3 | 3 0 0 | 2400 * * * | 1 1 0 . x3x . | 6 | 3 3 0 | * 2400 * * | 1 0 1 . x . x | 4 | 2 0 2 | * * 3600 * | 0 1 1 . . x5x | 10 | 0 5 5 | * * * 720 | 0 0 2 --------+------+----------------+--------------------+------------- o3x3x . ♦ 12 | 12 6 0 | 4 4 0 0 | 600 * * o3x . x ♦ 6 | 6 0 3 | 2 0 3 0 | * 1200 * . x3x5x ♦ 120 | 60 60 60 | 0 20 30 12 | * * 120 |
x3x3x5x - gidpixhi . . . . | 14400 | 1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 --------+-------+---------------------+-------------------------------+----------------- x . . . | 2 | 7200 * * * | 1 1 1 0 0 0 | 1 1 1 0 . x . . | 2 | * 7200 * * | 1 0 0 1 1 0 | 1 1 0 1 . . x . | 2 | * * 7200 * | 0 1 0 1 0 1 | 1 0 1 1 . . . x | 1 | * * * 7200 | 0 0 1 0 1 1 | 0 1 1 1 --------+-------+---------------------+-------------------------------+----------------- x3x . . | 6 | 3 3 0 0 | 2400 * * * * * | 1 1 0 0 x . x . | 4 | 2 0 2 0 | * 3600 * * * * | 1 0 1 0 x . . x | 4 | 2 0 0 2 | * * 3600 * * * | 0 1 1 0 . x3x . | 6 | 0 3 3 0 | * * * 2400 * * | 1 0 0 1 . x . x | 4 | 0 2 0 2 | * * * * 3600 * | 0 1 0 1 . . x5x | 10 | 0 0 5 5 | * * * * * 1440 | 0 0 1 1 --------+-------+---------------------+-------------------------------+----------------- x3x3x . ♦ 24 | 12 12 12 0 | 4 6 0 4 0 0 | 600 * * * x3x . x ♦ 12 | 6 6 0 6 | 2 0 3 0 3 0 | * 1200 * * x . x5x ♦ 20 | 10 0 10 10 | 0 5 5 0 0 2 | * * 720 * . x3x5x ♦ 120 | 0 60 60 60 | 0 0 0 20 30 12 | * * * 120 | ||
5D
| 5-simplex {3,3,3,3} [46] | 5-orthoplex {3,3,3,4} [47] | 5-cube {4,3,3,3} [48] |
|---|---|---|
x3o3o3o3o . . . . . | 6 ♦ 5 | 10 | 10 | 5 ----------+---+----+----+----+-- x . . . . | 2 | 15 ♦ 4 | 6 | 4 ----------+---+----+----+----+-- x3o . . . | 3 | 3 | 20 | 3 | 3 ----------+---+----+----+----+-- x3o3o . . ♦ 4 | 6 | 4 | 15 | 2 ----------+---+----+----+----+-- x3o3o3o . ♦ 5 | 10 | 10 | 5 | 6 |
x3o3o3o4o . . . . . | 10 ♦ 8 | 24 | 32 | 16 ----------+----+----+----+----+--- x . . . . | 2 | 40 ♦ 6 | 12 | 8 ----------+----+----+----+----+--- x3o . . . | 3 | 3 | 80 | 4 | 4 ----------+----+----+----+----+--- x3o3o . . ♦ 4 | 6 | 4 | 80 | 2 ----------+----+----+----+----+--- x3o3o3o . ♦ 5 | 10 | 10 | 5 | 32 |
o3o3o3o4x . . . . . | 32 ♦ 5 | 10 | 10 | 5 ----------+----+----+----+----+--- . . . . x | 2 | 80 ♦ 4 | 6 | 4 ----------+----+----+----+----+--- . . . o4x | 4 | 4 | 80 | 3 | 3 ----------+----+----+----+----+--- . . o3o4x ♦ 8 | 12 | 6 | 40 | 2 ----------+----+----+----+----+--- . o3o3o4x ♦ 16 | 32 | 24 | 8 | 10 |
5-simplexes
| 5-simplex {3,3,3,3} [49] | |||
|---|---|---|---|
x3o3o3o3o . . . . . | 6 ♦ 5 | 10 | 10 | 5 ----------+---+----+----+----+-- x . . . . | 2 | 15 ♦ 4 | 6 | 4 ----------+---+----+----+----+-- x3o . . . | 3 | 3 | 20 | 3 | 3 ----------+---+----+----+----+-- x3o3o . . ♦ 4 | 6 | 4 | 15 | 2 ----------+---+----+----+----+-- x3o3o3o . ♦ 5 | 10 | 10 | 5 | 6 |
(pt || pen) o.3o.3o.3o. | 1 * ♦ 5 0 | 10 0 | 10 0 | 5 0 .o3.o3.o3.o | * 5 ♦ 1 4 | 4 6 | 6 4 | 4 1 ---------------+-----+------+-------+------+---- oo3oo3oo3oo&#x | 1 1 | 5 * ♦ 4 0 | 6 0 | 4 0 .x .. .. .. | 0 2 | * 10 ♦ 1 3 | 3 3 | 3 1 ---------------+-----+------+-------+------+---- ox .. .. ..&#x | 1 2 | 2 1 | 10 * | 3 0 | 3 0 .x3.o .. .. | 0 3 | 0 3 | * 10 | 1 2 | 2 1 ---------------+-----+------+-------+------+---- ox3oo .. ..&#x ♦ 1 3 | 3 3 | 3 1 | 10 * | 2 0 .x3.o3.o .. ♦ 0 4 | 0 6 | 0 4 | * 5 | 1 1 ---------------+-----+------+-------+------+---- ox3oo3oo ..&#x ♦ 1 4 | 4 6 | 6 4 | 4 1 | 5 * .x3.o3.o3.o ♦ 0 5 | 0 10 | 0 10 | 0 5 | * 1 |
(line || perp tet) o. o.3o.3o. | 2 * ♦ 1 4 0 | 4 6 0 | 6 4 0 | 4 1 .o .o3.o3.o | * 4 ♦ 0 2 3 | 1 6 3 | 3 6 1 | 3 2 ---------------+-----+-------+--------+-------+---- x. .. .. .. | 2 0 | 1 * * ♦ 4 0 0 | 6 0 0 | 4 0 oo oo3oo3oo&#x | 1 1 | * 8 * ♦ 1 3 0 | 3 3 0 | 3 1 .. .x .. .. | 0 2 | * * 6 ♦ 0 2 2 | 1 4 1 | 2 2 ---------------+-----+-------+--------+-------+---- xo .. .. ..&#x | 2 1 | 1 2 0 | 4 * * | 3 0 0 | 3 0 .. ox .. ..&#x | 1 2 | 0 2 1 | * 12 * | 1 2 0 | 2 1 .. .x3.o .. | 0 3 | 0 0 3 | * * 4 | 0 2 1 | 1 2 ---------------+-----+-------+--------+-------+---- xo ox .. ..&#x ♦ 2 2 | 1 4 1 | 2 2 0 | 6 * * | 2 0 .. ox3oo ..&#x ♦ 1 3 | 0 3 3 | 0 3 1 | * 8 * | 1 1 .. .x3.o3.o ♦ 0 4 | 0 0 6 | 0 0 4 | * * 1 | 0 2 ---------------+-----+-------+--------+-------+---- xo ox3oo ..&#x ♦ 2 3 | 1 6 3 | 3 6 1 | 3 2 0 | 4 * .. ox3oo3oo&#x ♦ 1 4 | 0 4 6 | 0 6 4 | 0 4 1 | * 2 |
({3} || perp {3})
o.3o. o.3o. & | 6 ♦ 2 3 | 1 9 | 4 3 | 5
-----------------+---+-----+------+-----+--
x. .. .. .. & | 2 | 6 * ♦ 1 3 | 3 3 | 4
oo3oo oo3oo&#x | 2 | * 9 ♦ 0 4 | 2 4 | 4
-----------------+---+-----+------+-----+--
x.3o. .. .. & | 3 | 3 0 | 2 * | 3 0 | 3
xo .. .. ..&#x & | 3 | 1 2 | * 18 | 1 2 | 3
-----------------+---+-----+------+-----+--
xo3oo .. ..&#x & ♦ 4 | 3 3 | 1 3 | 6 * | 2
xo .. ox ..&#x ♦ 4 | 2 4 | 0 4 | * 9 | 2
-----------------+---+-----+------+-----+--
xo3oo ox ..&#x & ♦ 5 | 4 6 | 1 9 | 2 3 | 6
|
o..3o..3o.. | 1 * * ♦ 4 1 0 0 | 6 4 0 0 | 4 6 0 0 | 1 4 0 .o.3.o.3.o. | * 4 * ♦ 1 0 3 1 | 3 1 3 3 | 3 3 1 3 | 1 3 1 ..o3..o3..o | * * 1 ♦ 0 1 0 4 | 0 4 0 6 | 0 6 0 4 | 0 4 1 ---------------+-------+---------+---------+---------+------ oo.3oo.3oo.&#x | 1 1 0 | 4 * * * ♦ 3 1 0 0 | 3 3 0 0 | 1 3 0 o.o3o.o3o.o&#x | 1 0 1 | * 1 * * ♦ 0 4 0 0 | 0 6 0 0 | 0 4 0 .x. ... ... | 0 2 0 | * * 6 * ♦ 1 0 2 1 | 2 1 1 2 | 1 2 1 .oo3.oo3.oo&#x | 0 1 1 | * * * 4 ♦ 0 1 0 3 | 0 3 0 3 | 0 3 1 ---------------+-------+---------+---------+---------+------ ox. ... ...&#x | 1 2 0 | 2 0 1 0 | 6 * * * | 2 1 0 0 | 1 2 0 ooo3ooo3ooo&#x | 1 1 1 | 1 1 0 1 | * 4 * * | 0 3 0 0 | 0 3 0 .x.3.o. ... | 0 3 0 | 0 0 3 0 | * * 4 * | 1 0 1 1 | 1 1 1 .xo ... ...&#x | 0 2 1 | 0 0 1 2 | * * * 6 | 0 1 0 2 | 0 2 1 ---------------+-------+---------+---------+---------+------ ox.3oo. ...&#x ♦ 1 3 0 | 3 0 3 0 | 3 0 1 0 | 4 * * * | 1 1 0 oxo ... ...&#x ♦ 1 2 1 | 2 1 1 2 | 1 2 0 1 | * 6 * * | 0 2 0 .x.3.o.3.o. ♦ 0 4 0 | 0 0 6 0 | 0 0 4 0 | * * 1 * | 1 0 1 .xo3.oo ...&#x ♦ 0 3 1 | 0 0 3 3 | 0 0 1 3 | * * * 4 | 0 1 1 ---------------+-------+---------+---------+---------+------ ox.3oo.3oo.&#x ♦ 1 4 0 | 4 0 6 0 | 6 0 4 0 | 4 0 1 0 | 1 * * oxo3ooo ...&#x ♦ 1 3 1 | 3 1 3 3 | 3 3 1 3 | 1 3 0 1 | * 4 * .xo3.oo3.oo&#x ♦ 0 4 1 | 0 0 6 4 | 0 0 4 6 | 0 0 1 4 | * * 1 |
( (pt || {3}) || line )
o..3o.. o.. | 1 * * ♦ 3 2 0 0 0 | 3 1 6 0 0 0 | 1 6 3 0 0 | 2 3 0
.o.3.o. .o. | * 3 * ♦ 1 0 2 2 0 | 2 0 2 1 4 1 | 1 4 1 2 2 | 2 2 1
..o3..o ..o | * * 2 ♦ 0 1 0 3 1 | 0 1 3 0 3 3 | 0 3 3 1 3 | 1 3 1
---------------+-------+-----------+-------------+-----------+------
oo.3oo. oo.&#x | 1 1 0 | 3 * * * * ♦ 2 0 2 0 0 0 | 1 4 1 0 0 | 2 2 0
o.o3o.o o.o&#x | 1 0 1 | * 2 * * * ♦ 0 1 3 0 0 0 | 0 3 3 0 0 | 1 3 0
.x. ... ... | 0 2 0 | * * 3 * * ♦ 1 0 0 1 2 0 | 1 2 0 2 1 | 2 1 1
.oo3.oo .oo&#x | 0 1 1 | * * * 6 * ♦ 0 0 1 0 2 1 | 0 2 1 1 2 | 1 2 1
... ... ..x | 0 0 2 | * * * * 1 ♦ 0 1 0 0 0 3 | 0 0 3 0 3 | 0 3 1
---------------+-------+-----------+-------------+-----------+------
ox. ... ...&#x | 1 2 0 | 2 0 1 0 0 | 3 * * * * * | 1 2 0 0 0 | 2 1 0
... ... o.x&#x | 1 0 2 | 0 2 0 0 1 | * 1 * * * * | 0 0 3 0 0 | 0 3 0
ooo3ooo ooo&#x | 1 1 1 | 1 1 0 1 0 | * * 6 * * * | 0 2 1 0 0 | 1 2 0
.x.3.o. ... | 0 3 0 | 0 0 3 0 0 | * * * 1 * * | 1 0 0 2 0 | 2 0 1
.xo ... ...&#x | 0 2 1 | 0 0 1 2 0 | * * * * 6 * | 0 1 0 1 1 | 1 1 1
... ... .ox&#x | 0 1 2 | 0 0 0 2 1 | * * * * * 3 | 0 0 1 0 2 | 0 2 1
---------------+-------+-----------+-------------+-----------+------
ox.3oo. ...&#x ♦ 1 3 0 | 3 0 3 0 0 | 3 0 0 1 0 0 | 1 * * * * | 2 0 0
oxo ... ...&#x ♦ 1 2 1 | 2 1 1 2 0 | 1 0 2 0 1 0 | * 6 * * * | 1 1 0
... ... oox&#x ♦ 1 1 2 | 1 2 0 2 1 | 0 1 2 0 0 1 | * * 3 * * | 0 2 0
.xo3.oo ...&#x ♦ 0 3 1 | 0 0 3 3 0 | 0 0 0 1 3 0 | * * * 2 * | 1 0 1
.xo ... .ox&#x ♦ 0 2 2 | 0 0 1 4 1 | 0 0 0 0 2 2 | * * * * 3 | 0 1 1
---------------+-------+-----------+-------------+-----------+------
oxo3ooo ...&#x ♦ 1 3 1 | 3 1 3 3 0 | 3 0 3 1 3 0 | 1 3 0 1 0 | 2 * *
oxo ... oox&#x ♦ 1 2 2 | 2 2 1 4 1 | 1 1 4 0 2 2 | 0 2 2 0 1 | * 3 *
.xo3.oo .ox&#x ♦ 0 3 2 | 0 0 3 6 1 | 0 0 0 1 6 3 | 0 0 0 2 3 | * * 1
|
( (line || perp line) || perp line) o.. o.. o.. | 2 * * ♦ 1 2 2 0 0 0 | 2 2 1 1 4 0 0 | 4 1 1 2 2 0 | 2 2 1 .o. .o. .o. | * 2 * ♦ 0 2 0 1 2 0 | 1 0 2 0 4 2 1 | 2 1 0 4 2 1 | 2 1 2 ..o ..o ..o | * * 2 ♦ 0 0 2 0 2 1 | 0 1 0 2 4 1 2 | 2 0 1 2 4 1 | 1 2 2 ---------------+-------+-------------+---------------+-------------+------ x.. ... ... | 2 0 0 | 1 * * * * * ♦ 2 2 0 0 0 0 0 | 4 1 1 0 0 0 | 2 2 0 oo. oo. oo.&#x | 1 1 0 | * 4 * * * * ♦ 1 0 1 0 2 0 0 | 2 1 0 2 1 0 | 2 1 1 o.o o.o o.o&#x | 1 0 1 | * * 4 * * * ♦ 0 1 0 1 2 0 0 | 2 0 1 1 2 0 | 1 2 1 ... .x. ... | 0 2 0 | * * * 1 * * ♦ 0 0 2 0 0 2 0 | 0 1 0 4 0 1 | 2 0 2 .oo .oo .oo&#x | 0 1 1 | * * * * 4 * ♦ 0 0 0 0 2 1 1 | 1 0 0 2 2 1 | 1 1 2 ... ... ..x | 0 0 2 | * * * * * 1 ♦ 0 0 0 2 0 0 2 | 0 0 1 0 4 1 | 0 2 2 ---------------+-------+-------------+---------------+-------------+------ xo. ... ...&#x | 2 1 0 | 1 2 0 0 0 0 | 2 * * * * * * | 2 1 0 0 0 0 | 2 1 0 x.o ... ...&#x | 2 0 1 | 1 0 2 0 0 0 | * 2 * * * * * | 2 0 1 0 0 0 | 1 2 0 ... ox. ...&#x | 1 2 0 | 0 2 0 1 0 0 | * * 2 * * * * | 0 1 0 2 0 0 | 2 0 1 ... ... o.x&#x | 1 0 2 | 0 0 2 0 0 1 | * * * 2 * * * | 0 0 1 0 2 0 | 0 2 1 ooo ooo ooo&#x | 1 1 1 | 0 1 1 0 1 0 | * * * * 8 * * | 1 0 0 1 1 0 | 1 1 1 ... .xo ...&#x | 0 2 1 | 0 0 0 1 2 0 | * * * * * 2 * | 0 0 0 2 0 1 | 1 0 2 ... ... .ox&#x | 0 1 2 | 0 0 0 0 2 1 | * * * * * * 2 | 0 0 0 0 2 1 | 0 1 2 ---------------+-------+-------------+---------------+-------------+------ xoo ... ...&#x ♦ 2 1 1 | 1 2 2 0 1 0 | 1 1 0 0 2 0 0 | 4 * * * * * | 1 1 0 xo. ox. ...&#x ♦ 2 2 0 | 1 4 0 1 0 0 | 2 0 2 0 0 0 0 | * 1 * * * * | 2 0 0 x.o ... o.x&#x ♦ 2 0 2 | 1 0 4 0 0 1 | 0 2 0 2 0 0 0 | * * 1 * * * | 0 2 0 ... oxo ...&#x ♦ 1 2 1 | 0 2 1 1 2 0 | 0 0 1 0 2 1 0 | * * * 4 * * | 1 0 1 ... ... oox&#x ♦ 1 1 2 | 0 1 2 0 2 1 | 0 0 0 1 2 0 1 | * * * * 4 * | 0 1 1 ... .xo .ox&#x ♦ 0 2 2 | 0 0 0 1 4 1 | 0 0 0 0 0 2 2 | * * * * * 1 | 0 0 2 ---------------+-------+-------------+---------------+-------------+------ xoo oxo ...&#x ♦ 2 2 1 | 1 4 2 1 2 0 | 2 1 2 0 4 1 0 | 2 1 0 2 0 0 | 2 * * xoo ... oox&#x ♦ 2 1 2 | 1 2 4 0 2 1 | 1 2 0 2 4 0 1 | 2 0 1 0 2 0 | * 2 * ... oxo oox&#x ♦ 1 2 2 | 0 2 2 1 4 1 | 0 0 1 1 4 2 2 | 0 0 0 2 2 1 | * * 2 | |
Uniform 5D
| 5-demicube h{4,3,3,3}[50] |
r{3,3,3,3}[51] | 2r{3,3,3,3}[52] | 2r{4,3,3,3}[53] |
|---|---|---|---|
x3o3o *b3o3o - hin . . . . . | 16 ♦ 10 | 30 | 10 20 | 5 5 -------------+----+----+-----+-------+------ x . . . . | 2 | 80 ♦ 6 | 3 6 | 3 2 -------------+----+----+-----+-------+------ x3o . . . | 3 | 3 | 160 | 1 2 | 2 1 -------------+----+----+-----+-------+------ x3o3o . . ♦ 4 | 6 | 4 | 40 * | 2 0 x3o . *b3o . ♦ 4 | 6 | 4 | * 80 | 1 1 -------------+----+----+-----+-------+------ x3o3o *b3o . ♦ 8 | 24 | 32 | 8 8 | 10 * x3o . *b3o3o ♦ 5 | 10 | 10 | 0 5 | * 16 |
o3x3o3o3o - rix . . . . . | 15 ♦ 8 | 4 12 | 6 8 | 4 2 ----------+----+----+-------+-------+---- . x . . . | 2 | 60 | 1 3 | 3 3 | 3 1 ----------+----+----+-------+-------+---- o3x . . . | 3 | 3 | 20 * | 3 0 | 3 0 . x3o . . | 3 | 3 | * 60 | 1 2 | 2 1 ----------+----+----+-------+-------+---- o3x3o . . ♦ 6 | 12 | 4 4 | 15 * | 2 0 . x3o3o . ♦ 4 | 6 | 0 4 | * 30 | 1 1 ----------+----+----+-------+-------+---- o3x3o3o . ♦ 10 | 30 | 10 20 | 5 5 | 6 * . x3o3o3o ♦ 5 | 10 | 0 10 | 0 5 | * 6 |
o3o3x3o3o - dot . . . . . | 20 ♦ 9 | 9 9 | 3 9 3 | 3 3 ----------+----+----+-------+----------+---- . . x . . | 2 | 90 | 2 2 | 1 4 1 | 2 2 ----------+----+----+-------+----------+---- . o3x . . | 3 | 3 | 60 * | 1 2 0 | 2 1 . . x3o . | 3 | 3 | * 60 | 0 2 1 | 1 2 ----------+----+----+-------+----------+---- o3o3x . . ♦ 4 | 6 | 4 0 | 15 * * | 2 0 . o3x3o . ♦ 6 | 12 | 4 4 | * 30 * | 1 1 . . x3o3o ♦ 4 | 6 | 0 4 | * * 15 | 0 2 ----------+----+----+-------+----------+---- o3o3x3o . ♦ 10 | 30 | 20 10 | 5 5 0 | 6 * . o3x3o3o ♦ 10 | 30 | 10 20 | 0 5 5 | * 6 |
o3x3o *b3o3o - nit . . . . . | 80 ♦ 12 | 6 6 12 | 3 6 6 4 | 3 2 2 -------------+----+-----+-------------+-------------+--------- . x . . . | 2 | 480 | 1 1 2 | 1 2 2 1 | 2 1 1 -------------+----+-----+-------------+-------------+--------- o3x . . . | 3 | 3 | 160 * * | 1 2 0 0 | 2 1 0 . x3o . . | 3 | 3 | * 160 * | 1 0 2 0 | 2 0 1 . x . *b3o . | 3 | 3 | * * 320 | 0 1 1 1 | 1 1 1 -------------+----+-----+-------------+-------------+--------- o3x3o . . ♦ 6 | 12 | 4 4 0 | 40 * * * | 2 0 0 o3x . *b3o . ♦ 6 | 12 | 4 0 4 | * 80 * * | 1 1 0 . x3o *b3o . ♦ 6 | 12 | 0 4 4 | * * 80 * | 1 0 1 . x . *b3o3o ♦ 4 | 6 | 0 0 4 | * * * 80 | 0 1 1 -------------+----+-----+-------------+-------------+--------- o3x3o *b3o . ♦ 24 | 96 | 32 32 32 | 8 8 8 0 | 10 * * o3x . *b3o3o ♦ 10 | 30 | 10 0 20 | 0 5 0 5 | * 16 * . x3o *b3o3o ♦ 10 | 30 | 0 10 20 | 0 0 5 5 | * * 16 |
6D
| 6-simplex {3,3,3,3,3} [54] | 6-orthoplex {3,3,3,3,4} [55] | 6-cube {4,3,3,3,3} [56] |
|---|---|---|
x3o3o3o3o3o . . . . . . | 7 ♦ 6 | 15 | 20 | 15 | 6 ------------+---+----+----+----+----+-- x . . . . . | 2 | 21 ♦ 5 | 10 | 10 | 5 ------------+---+----+----+----+----+-- x3o . . . . | 3 | 3 | 35 ♦ 4 | 6 | 4 ------------+---+----+----+----+----+-- x3o3o . . . ♦ 4 | 6 | 4 | 35 | 3 | 3 ------------+---+----+----+----+----+-- x3o3o3o . . ♦ 5 | 10 | 10 | 5 | 21 | 2 ------------+---+----+----+----+----+-- x3o3o3o3o . ♦ 6 | 15 | 20 | 15 | 6 | 7 |
x3o3o3o3o4o . . . . . . | 12 ♦ 10 | 40 | 80 | 80 | 32 ------------+----+----+-----+-----+-----+--- x . . . . . | 2 | 60 ♦ 8 | 24 | 32 | 16 ------------+----+----+-----+-----+-----+--- x3o . . . . | 3 | 3 | 160 ♦ 6 | 12 | 8 ------------+----+----+-----+-----+-----+--- x3o3o . . . ♦ 4 | 6 | 4 | 240 | 4 | 4 ------------+----+----+-----+-----+-----+--- x3o3o3o . . ♦ 5 | 10 | 10 | 5 | 192 | 2 ------------+----+----+-----+-----+-----+--- x3o3o3o3o . ♦ 6 | 15 | 20 | 15 | 6 | 64 |
o3o3o3o3o4x . . . . . . | 64 ♦ 6 | 15 | 20 | 15 | 6 ------------+----+-----+-----+-----+----+--- . . . . . x | 2 | 192 ♦ 5 | 10 | 10 | 5 ------------+----+-----+-----+-----+----+--- . . . . o4x | 4 | 4 | 240 ♦ 4 | 6 | 4 ------------+----+-----+-----+-----+----+--- . . . o3o4x ♦ 8 | 12 | 6 | 160 | 3 | 3 ------------+----+-----+-----+-----+----+--- . . o3o3o4x ♦ 16 | 32 | 24 | 8 | 60 | 2 ------------+----+-----+-----+-----+----+--- . o3o3o3o4x ♦ 32 | 80 | 80 | 40 | 10 | 12 |
Uniform 6D
| 6-demicube h{4,3,3,3,3} [57] | 221 {3,3,32,1} [58] | 122 {3,32,2} [59] |
|---|---|---|
x3o3o *b3o3o3o . . . . . . | 32 ♦ 15 | 60 | 20 60 | 15 30 | 6 6 ---------------+----+-----+-----+---------+--------+------ x . . . . . | 2 | 240 ♦ 8 | 4 12 | 6 8 | 4 2 ---------------+----+-----+-----+---------+--------+------ x3o . . . . | 3 | 3 | 640 | 1 3 | 3 3 | 3 1 ---------------+----+-----+-----+---------+--------+------ x3o3o . . . ♦ 4 | 6 | 4 | 160 * | 3 0 | 3 0 x3o . *b3o . . ♦ 4 | 6 | 4 | * 480 | 1 2 | 2 1 ---------------+----+-----+-----+---------+--------+------ x3o3o *b3o . . ♦ 8 | 24 | 32 | 8 8 | 60 * | 2 0 x3o . *b3o3o . ♦ 5 | 10 | 10 | 0 5 | * 192 | 1 1 ---------------+----+-----+-----+---------+--------+------ x3o3o *b3o3o . ♦ 16 | 80 | 160 | 40 80 | 10 16 | 12 * x3o . *b3o3o3o ♦ 6 | 15 | 20 | 0 15 | 0 6 | * 32 |
x3o3o3o3o *c3o . . . . . . | 27 ♦ 16 | 80 | 160 | 80 40 | 16 10 ---------------+----+-----+-----+------+---------+------ x . . . . . | 2 | 216 ♦ 10 | 30 | 20 10 | 5 5 ---------------+----+-----+-----+------+---------+------ x3o . . . . | 3 | 3 | 720 ♦ 6 | 6 3 | 2 3 ---------------+----+-----+-----+------+---------+------ x3o3o . . . ♦ 4 | 6 | 4 | 1080 | 2 1 | 1 2 ---------------+----+-----+-----+------+---------+------ x3o3o3o . . ♦ 5 | 10 | 10 | 5 | 432 * | 1 1 x3o3o . . *c3o ♦ 5 | 10 | 10 | 5 | * 216 | 0 2 ---------------+----+-----+-----+------+---------+------ x3o3o3o3o . ♦ 6 | 15 | 20 | 15 | 6 0 | 72 * x3o3o3o . *c3o ♦ 10 | 40 | 80 | 80 | 16 16 | * 27 |
o3o3o3o3o *c3x . . . . . . | 72 ♦ 20 | 90 | 60 60 | 15 30 15 | 6 6 ---------------+----+-----+------+-----------+-------------+------ . . . . . x | 2 | 720 ♦ 9 | 9 9 | 3 9 3 | 3 3 ---------------+----+-----+------+-----------+-------------+------ . . o . . *c3x | 3 | 3 | 2160 | 2 2 | 1 4 1 | 2 2 ---------------+----+-----+------+-----------+-------------+------ . o3o . . *c3x ♦ 4 | 6 | 4 | 1080 * | 1 2 0 | 2 1 . . o3o . *c3x ♦ 4 | 6 | 4 | * 1080 | 0 2 1 | 1 2 ---------------+----+-----+------+-----------+-------------+------ o3o3o . . *c3x ♦ 5 | 10 | 10 | 5 0 | 216 * * | 2 0 . o3o3o . *c3x ♦ 8 | 24 | 32 | 8 8 | * 270 * | 1 1 . . o3o3o *c3x ♦ 5 | 10 | 10 | 0 5 | * * 216 | 0 2 ---------------+----+-----+------+-----------+-------------+------ o3o3o3o . *c3x ♦ 16 | 80 | 160 | 80 40 | 16 10 0 | 27 * . o3o3o3o *c3x ♦ 16 | 80 | 160 | 40 80 | 0 10 16 | * 27 |
| rectified 6-simplex r{3,3,3,3,3} [60] |
birectified 6-simplex r{3,3,3,3,3} [61] |
rectified 1_22 r{3,32,2} [62] |
o3x3o3o3o3o - ril . . . . . . | 21 ♦ 10 | 5 20 | 10 20 | 10 10 | 5 2 ------------+----+-----+--------+--------+-------+---- . x . . . . | 2 | 105 | 1 4 | 4 6 | 6 4 | 4 1 ------------+----+-----+--------+--------+-------+---- o3x . . . . | 3 | 3 | 35 * ♦ 4 0 | 6 0 | 4 0 . x3o . . . | 3 | 3 | * 140 | 1 3 | 3 3 | 3 1 ------------+----+-----+--------+--------+-------+---- o3x3o . . . ♦ 6 | 12 | 4 4 | 35 * | 3 0 | 3 0 . x3o3o . . ♦ 4 | 6 | 0 4 | * 105 | 1 2 | 2 1 ------------+----+-----+--------+--------+-------+---- o3x3o3o . . ♦ 10 | 30 | 10 20 | 5 5 | 21 * | 2 0 . x3o3o3o . ♦ 5 | 10 | 0 10 | 0 5 | * 42 | 1 1 ------------+----+-----+--------+--------+-------+---- o3x3o3o3o . ♦ 15 | 60 | 20 60 | 15 30 | 6 6 | 7 * . x3o3o3o3o ♦ 6 | 15 | 0 20 | 0 15 | 0 6 | * 7 |
o3o3x3o3o3o - bril . . . . . . | 35 ♦ 12 | 12 18 | 4 18 12 | 6 12 3 | 4 3 ------------+----+-----+---------+------------+----------+---- . . x . . . | 2 | 210 | 2 3 | 1 6 3 | 3 6 1 | 3 2 ------------+----+-----+---------+------------+----------+---- . o3x . . . | 3 | 3 | 140 * | 1 3 0 | 3 3 0 | 3 1 . . x3o . . | 3 | 3 | * 210 | 0 2 2 | 1 4 1 | 2 2 ------------+----+-----+---------+------------+----------+---- o3o3x . . . ♦ 4 | 6 | 4 0 | 35 * * | 3 0 0 | 3 0 . o3x3o . . ♦ 6 | 12 | 4 4 | * 105 * | 1 2 0 | 2 1 . . x3o3o . ♦ 4 | 6 | 0 4 | * * 105 | 0 2 1 | 1 2 ------------+----+-----+---------+------------+----------+---- o3o3x3o . . ♦ 10 | 30 | 20 10 | 5 5 0 | 21 * * | 2 0 . o3x3o3o . ♦ 10 | 30 | 10 20 | 0 5 5 | * 42 * | 1 1 . . x3o3o3o ♦ 5 | 10 | 0 10 | 0 0 5 | * * 21 | 0 2 ------------+----+-----+---------+------------+----------+---- o3o3x3o3o . ♦ 20 | 90 | 60 60 | 15 30 15 | 6 6 0 | 7 * . o3x3o3o3o ♦ 15 | 60 | 20 60 | 0 15 30 | 0 6 6 | * 7 |
o3o3x3o3o *c3o - ram . . . . . . | 720 ♦ 18 | 18 18 9 | 6 18 9 6 9 | 6 3 6 9 3 | 2 3 3 ---------------+-----+------+----------------+--------------------------+---------------------+--------- . . x . . . | 2 | 6480 | 2 2 1 | 1 4 2 1 2 | 2 1 2 4 1 | 1 2 2 ---------------+-----+------+----------------+--------------------------+---------------------+--------- . o3x . . . | 3 | 3 | 4320 * * | 1 2 1 0 0 | 2 1 1 2 0 | 1 2 1 . . x3o . . | 3 | 3 | * 4320 * | 0 2 0 1 1 | 1 0 2 2 1 | 1 1 2 . . x . . *c3o | 3 | 3 | * * 2160 | 0 0 2 0 2 | 0 1 0 4 1 | 0 2 2 ---------------+-----+------+----------------+--------------------------+---------------------+--------- o3o3x . . . ♦ 4 | 6 | 4 0 0 | 1080 * * * * | 2 1 0 0 0 | 1 2 0 . o3x3o . . ♦ 6 | 12 | 4 4 0 | * 2160 * * * | 1 0 1 1 0 | 1 1 1 . o3x . . *c3o ♦ 6 | 12 | 4 0 4 | * * 1080 * * | 0 1 0 2 0 | 0 2 1 . . x3o3o . ♦ 4 | 6 | 0 4 0 | * * * 1080 * | 0 0 2 0 1 | 1 0 2 . . x3o . *c3o ♦ 6 | 12 | 0 4 4 | * * * * 1080 | 0 0 0 2 1 | 0 1 2 ---------------+-----+------+----------------+--------------------------+---------------------+--------- o3o3x3o . . ♦ 10 | 30 | 20 10 0 | 5 5 0 0 0 | 432 * * * * | 1 1 0 o3o3x . . *c3o ♦ 10 | 30 | 20 0 10 | 5 0 5 0 0 | * 216 * * * | 0 2 0 . o3x3o3o . ♦ 10 | 30 | 10 20 0 | 0 5 0 5 0 | * * 432 * * | 1 0 1 . o3x3o . *c3o ♦ 24 | 96 | 32 32 32 | 0 8 8 0 8 | * * * 270 * | 0 1 1 . . x3o3o *c3o ♦ 10 | 30 | 0 20 10 | 0 0 0 5 5 | * * * * 216 | 0 0 2 ---------------+-----+------+----------------+--------------------------+---------------------+--------- o3o3x3o3o . ♦ 20 | 90 | 60 60 0 | 15 30 0 15 0 | 6 0 6 0 0 | 72 * * o3o3x3o . *c3o ♦ 80 | 480 | 320 160 160 | 80 80 80 0 40 | 16 16 0 10 0 | * 27 * . o3x3o3o *c3o ♦ 80 | 480 | 160 320 160 | 0 80 40 80 80 | 0 0 16 10 16 | * * 27 |
7D
| 7-simplex {3,3,3,3,3,3} [63] | 7-orthoplex {3,3,3,3,3,4} [64] | 7-cube {4,3,3,3,3,3} [65] |
|---|---|---|
x3o3o3o3o3o3o . . . . . . . | 8 ♦ 7 | 21 | 35 | 35 | 21 | 7 --------------+---+----+----+----+----+----+-- x . . . . . . | 2 | 28 ♦ 6 | 15 | 20 | 15 | 6 --------------+---+----+----+----+----+----+-- x3o . . . . . | 3 | 3 | 56 ♦ 5 | 10 | 10 | 5 --------------+---+----+----+----+----+----+-- x3o3o . . . . ♦ 4 | 6 | 4 | 70 ♦ 4 | 6 | 4 --------------+---+----+----+----+----+----+-- x3o3o3o . . . ♦ 5 | 10 | 10 | 5 | 56 | 3 | 3 --------------+---+----+----+----+----+----+-- x3o3o3o3o . . ♦ 6 | 15 | 20 | 15 | 6 | 28 | 2 --------------+---+----+----+----+----+----+-- x3o3o3o3o3o . ♦ 7 | 21 | 35 | 35 | 21 | 7 | 8 |
x3o3o3o3o3o4o . . . . . . . | 14 ♦ 12 | 60 | 160 | 240 | 192 | 64 --------------+----+----+-----+-----+-----+-----+---- x . . . . . . | 2 | 84 ♦ 10 | 40 | 80 | 80 | 32 --------------+----+----+-----+-----+-----+-----+---- x3o . . . . . | 3 | 3 | 280 ♦ 8 | 24 | 32 | 16 --------------+----+----+-----+-----+-----+-----+---- x3o3o . . . . ♦ 4 | 6 | 4 | 560 ♦ 6 | 12 | 8 --------------+----+----+-----+-----+-----+-----+---- x3o3o3o . . . ♦ 5 | 10 | 10 | 5 | 672 | 4 | 4 --------------+----+----+-----+-----+-----+-----+---- x3o3o3o3o . . ♦ 6 | 15 | 20 | 15 | 6 | 448 | 2 --------------+----+----+-----+-----+-----+-----+---- x3o3o3o3o3o . ♦ 7 | 21 | 35 | 35 | 21 | 7 | 128 |
o3o3o3o3o3o4x . . . . . . . | 128 ♦ 7 | 21 | 35 | 35 | 21 | 7 --------------+-----+-----+-----+-----+-----+----+--- . . . . . . x | 2 | 448 ♦ 6 | 15 | 20 | 15 | 6 --------------+-----+-----+-----+-----+-----+----+--- . . . . . o4x | 4 | 4 | 672 ♦ 5 | 10 | 10 | 5 --------------+-----+-----+-----+-----+-----+----+--- . . . . o3o4x ♦ 8 | 12 | 6 | 560 ♦ 4 | 6 | 4 --------------+-----+-----+-----+-----+-----+----+--- . . . o3o3o4x ♦ 16 | 32 | 24 | 8 | 280 | 3 | 3 --------------+-----+-----+-----+-----+-----+----+--- . . o3o3o3o4x ♦ 32 | 80 | 80 | 40 | 10 | 84 | 2 --------------+-----+-----+-----+-----+-----+----+--- . o3o3o3o3o4x ♦ 64 | 192 | 240 | 160 | 60 | 12 | 14 |
Uniform 7D
| 7-demicube h{4,3,3,3,3,3} [66] | 321 {3,3,3,32,1} [67] |
|---|---|
x3o3o *b3o3o3o3o . . . . . . . | 64 ♦ 21 | 105 | 35 140 | 35 105 | 21 42 | 7 7 -----------------+----+-----+------+----------+----------+--------+------ x . . . . . . | 2 | 672 ♦ 10 | 5 20 | 10 20 | 10 10 | 5 2 -----------------+----+-----+------+----------+----------+--------+------ x3o . . . . . | 3 | 3 | 2240 | 1 4 | 4 6 | 6 4 | 4 1 -----------------+----+-----+------+----------+----------+--------+------ x3o3o . . . . ♦ 4 | 6 | 4 | 560 * ♦ 4 0 | 6 0 | 4 0 x3o . *b3o . . . ♦ 4 | 6 | 4 | * 2240 | 1 3 | 3 3 | 3 1 -----------------+----+-----+------+----------+----------+--------+------ x3o3o *b3o . . . ♦ 8 | 24 | 32 | 8 8 | 280 * | 3 0 | 3 0 x3o . *b3o3o . . ♦ 5 | 10 | 10 | 0 5 | * 1344 | 1 2 | 2 1 -----------------+----+-----+------+----------+----------+--------+------ x3o3o *b3o3o . . ♦ 16 | 80 | 160 | 40 80 | 10 16 | 84 * | 2 0 x3o . *b3o3o3o . ♦ 6 | 15 | 20 | 0 15 | 0 6 | * 448 | 1 1 -----------------+----+-----+------+----------+----------+--------+------ x3o3o *b3o3o3o . ♦ 32 | 240 | 640 | 160 480 | 60 192 | 12 32 | 14 * x3o . *b3o3o3o3o ♦ 7 | 21 | 35 | 0 35 | 0 21 | 0 7 | * 64 |
o3o3o3o *c3o3o3x . . . . . . . | 56 ♦ 27 | 216 | 720 | 1080 | 432 216 | 72 27 -----------------+----+-----+------+-------+-------+-----------+-------- . . . . . . x | 2 | 756 ♦ 16 | 80 | 160 | 80 40 | 16 10 -----------------+----+-----+------+-------+-------+-----------+-------- . . . . . o3x | 3 | 3 | 4032 ♦ 10 | 30 | 20 10 | 5 5 -----------------+----+-----+------+-------+-------+-----------+-------- . . . . o3o3x ♦ 4 | 6 | 4 | 10080 ♦ 6 | 6 3 | 2 3 -----------------+----+-----+------+-------+-------+-----------+-------- . . o . *c3o3o3x ♦ 5 | 10 | 10 | 5 | 12096 | 2 1 | 1 2 -----------------+----+-----+------+-------+-------+-----------+-------- . o3o . *c3o3o3x ♦ 6 | 15 | 20 | 15 | 6 | 4032 * | 1 1 . . o3o *c3o3o3x ♦ 6 | 15 | 20 | 15 | 6 | * 2016 | 0 2 -----------------+----+-----+------+-------+-------+-----------+-------- o3o3o . *c3o3o3x ♦ 7 | 21 | 35 | 35 | 21 | 10 0 | 576 * . o3o3o *c3o3o3x ♦ 12 | 60 | 160 | 240 | 192 | 32 32 | * 126 |
| 231 {3,3,33,1} [68] | 132 {3,33,2} [69] |
x3o3o3o *c3o3o3o . . . . . . . | 126 ♦ 32 | 240 | 640 | 160 480 | 60 192 | 12 32 -----------------+-----+------+-------+-------+------------+----------+------- x . . . . . . | 2 | 2016 ♦ 15 | 60 | 20 60 | 15 30 | 6 6 -----------------+-----+------+-------+-------+------------+----------+------- x3o . . . . . | 3 | 3 | 10080 ♦ 8 | 4 12 | 6 8 | 4 2 -----------------+-----+------+-------+-------+------------+----------+------- x3o3o . . . . ♦ 4 | 6 | 4 | 20160 | 1 3 | 3 3 | 3 1 -----------------+-----+------+-------+-------+------------+----------+------- x3o3o3o . . . ♦ 5 | 10 | 10 | 5 | 4032 * | 3 0 | 3 0 x3o3o . *c3o . . ♦ 5 | 10 | 10 | 5 | * 12096 | 1 2 | 2 1 -----------------+-----+------+-------+-------+------------+----------+------- x3o3o3o *c3o . . ♦ 10 | 40 | 80 | 80 | 16 16 | 756 * | 2 0 x3o3o . *c3o3o . ♦ 6 | 15 | 20 | 15 | 0 6 | * 4032 | 1 1 -----------------+-----+------+-------+-------+------------+----------+------- x3o3o3o *c3o3o . ♦ 27 | 216 | 720 | 1080 | 216 432 | 27 72 | 56 * x3o3o . *c3o3o3o ♦ 7 | 21 | 35 | 35 | 0 21 | 0 7 | * 576 |
o3o3o3x *c3o3o3o . . . . . . . | 576 ♦ 35 | 210 | 140 210 | 35 105 105 | 21 42 21 | 7 7 -----------------+-----+-------+-------+-------------+-----------------+---------------+------- . . . x . . . | 2 | 10080 ♦ 12 | 12 18 | 4 12 12 | 6 12 3 | 4 3 -----------------+-----+-------+-------+-------------+-----------------+---------------+------- . . o3x . . . | 3 | 3 | 40320 | 2 3 | 1 6 3 | 3 6 1 | 3 2 -----------------+-----+-------+-------+-------------+-----------------+---------------+------- . o3o3x . . . ♦ 4 | 6 | 4 | 20160 * | 1 3 0 | 3 3 0 | 3 1 . . o3x *c3o . . ♦ 4 | 6 | 4 | * 30240 | 0 2 2 | 1 4 1 | 2 2 -----------------+-----+-------+-------+-------------+-----------------+---------------+------- o3o3o3x . . . ♦ 5 | 10 | 10 | 5 0 | 4032 * * | 3 0 0 | 3 0 . o3o3x *c3o . . ♦ 8 | 24 | 32 | 8 8 | * 7560 * | 1 2 0 | 2 1 . . o3x *c3o3o . ♦ 5 | 10 | 10 | 0 5 | * * 12096 | 0 2 1 | 1 2 -----------------+-----+-------+-------+-------------+-----------------+---------------+------- o3o3o3x *c3o . . ♦ 16 | 80 | 160 | 80 40 | 16 10 0 | 756 * * | 2 0 . o3o3x *c3o3o . ♦ 16 | 80 | 160 | 40 80 | 0 10 16 | * 1512 * | 1 1 . . o3x *c3o3o3o ♦ 6 | 15 | 20 | 0 15 | 0 0 6 | * * 2016 | 0 2 -----------------+-----+-------+-------+-------------+-----------------+---------------+------- o3o3o3x *c3o3o . ♦ 72 | 720 | 2160 | 1080 1080 | 216 270 216 | 27 27 0 | 56 * . o3o3x *c3o3o3o ♦ 32 | 240 | 640 | 160 480 | 0 60 192 | 0 12 32 | * 126 |
| 051 r{3,3,3,3,3,3} [70] | 042 2r{3,3,3,3,3,3} [71] |
o3x3o3o3o3o3o - roc . . . . . . . | 28 ♦ 12 | 6 30 | 15 40 | 20 30 | 15 12 | 6 2 --------------+----+-----+--------+--------+--------+-------+---- . x . . . . . | 2 | 168 | 1 5 | 5 10 | 10 10 | 10 5 | 5 1 --------------+----+-----+--------+--------+--------+-------+---- o3x . . . . . | 3 | 3 | 56 * ♦ 5 0 | 10 0 | 10 0 | 5 0 . x3o . . . . | 3 | 3 | * 280 | 1 4 | 4 6 | 6 4 | 4 1 --------------+----+-----+--------+--------+--------+-------+---- o3x3o . . . . ♦ 6 | 12 | 4 4 | 70 * ♦ 4 0 | 6 0 | 4 0 . x3o3o . . . ♦ 4 | 6 | 0 4 | * 280 | 1 3 | 3 3 | 3 1 --------------+----+-----+--------+--------+--------+-------+---- o3x3o3o . . . ♦ 10 | 30 | 10 20 | 5 5 | 56 * | 3 0 | 3 0 . x3o3o3o . . ♦ 5 | 10 | 0 10 | 0 5 | * 168 | 1 2 | 2 1 --------------+----+-----+--------+--------+--------+-------+---- o3x3o3o3o . . ♦ 15 | 60 | 20 60 | 15 30 | 6 6 | 28 * | 2 0 . x3o3o3o3o . ♦ 6 | 15 | 0 20 | 0 15 | 0 6 | * 56 | 1 1 --------------+----+-----+--------+--------+--------+-------+---- o3x3o3o3o3o . ♦ 21 | 105 | 35 140 | 35 105 | 21 42 | 7 7 | 8 * . x3o3o3o3o3o ♦ 7 | 21 | 0 35 | 0 35 | 0 21 | 0 7 | * 8 |
o3o3x3o3o3o3o - broc . . . . . . . | 56 ♦ 15 | 15 30 | 5 30 30 | 10 30 15 | 10 15 3 | 5 3 --------------+----+-----+---------+------------+------------+----------+---- . . x . . . . | 2 | 420 | 2 4 | 1 8 6 | 4 12 4 | 6 8 1 | 4 2 --------------+----+-----+---------+------------+------------+----------+---- . o3x . . . . | 3 | 3 | 280 * | 1 4 0 | 4 6 0 | 6 4 0 | 4 1 . . x3o . . . | 3 | 3 | * 560 | 0 2 3 | 1 6 3 | 3 6 1 | 3 2 --------------+----+-----+---------+------------+------------+----------+---- o3o3x . . . . ♦ 4 | 6 | 4 0 | 70 * * ♦ 4 0 0 | 6 0 0 | 4 0 . o3x3o . . . ♦ 6 | 12 | 4 4 | * 280 * | 1 3 0 | 3 3 0 | 3 1 . . x3o3o . . ♦ 4 | 6 | 0 4 | * * 420 | 0 2 2 | 1 4 1 | 2 2 --------------+----+-----+---------+------------+------------+----------+---- o3o3x3o . . . ♦ 10 | 30 | 20 10 | 5 5 0 | 56 * * | 3 0 0 | 3 0 . o3x3o3o . . ♦ 10 | 30 | 10 20 | 0 5 5 | * 168 * | 1 2 0 | 2 1 . . x3o3o3o . ♦ 5 | 10 | 0 10 | 0 0 5 | * * 168 | 0 2 1 | 1 2 --------------+----+-----+---------+------------+------------+----------+---- o3o3x3o3o . . ♦ 20 | 90 | 60 60 | 15 30 15 | 6 6 0 | 28 * * | 2 0 . o3x3o3o3o . ♦ 15 | 60 | 20 60 | 0 15 30 | 0 6 6 | * 56 * | 1 1 . . x3o3o3o3o ♦ 6 | 15 | 0 20 | 0 0 15 | 0 0 6 | * * 28 | 0 2 --------------+----+-----+---------+------------+------------+----------+---- o3o3x3o3o3o . ♦ 35 | 210 | 140 210 | 35 105 105 | 21 42 21 | 7 7 0 | 8 * . o3x3o3o3o3o ♦ 21 | 105 | 35 140 | 0 35 105 | 0 21 42 | 0 7 7 | * 8 |
| 033 3r{3,3,3,3,3,3} [72] | |
o3o3o3x3o3o3o - he . . . . . . . | 70 ♦ 16 | 24 24 | 16 36 16 | 4 24 24 4 | 6 16 6 | 4 4 --------------+----+-----+---------+-------------+---------------+----------+---- . . . x . . . | 2 | 560 | 3 3 | 3 9 3 | 1 9 9 1 | 3 9 3 | 3 3 --------------+----+-----+---------+-------------+---------------+----------+---- . . o3x . . . | 3 | 3 | 560 * | 2 3 0 | 1 6 3 0 | 3 6 1 | 3 2 . . . x3o . . | 3 | 3 | * 560 | 0 3 2 | 0 3 6 1 | 1 6 3 | 2 3 --------------+----+-----+---------+-------------+---------------+----------+---- . o3o3x . . . ♦ 4 | 6 | 4 0 | 280 * * | 1 3 0 0 | 3 3 0 | 3 1 . . o3x3o . . ♦ 6 | 12 | 4 4 | * 420 * | 0 2 2 0 | 1 4 1 | 2 2 . . . x3o3o . ♦ 4 | 6 | 0 4 | * * 280 | 0 0 3 1 | 0 3 3 | 1 3 --------------+----+-----+---------+-------------+---------------+----------+---- o3o3o3x . . . ♦ 5 | 10 | 10 0 | 5 0 0 | 56 * * * | 3 0 0 | 3 0 . o3o3x3o . . ♦ 10 | 30 | 20 10 | 5 5 0 | * 168 * * | 1 2 0 | 2 1 . . o3x3o3o . ♦ 10 | 30 | 10 20 | 0 5 5 | * * 168 * | 0 2 1 | 1 2 . . . x3o3o3o ♦ 5 | 10 | 0 10 | 0 0 5 | * * * 56 | 0 0 3 | 0 3 --------------+----+-----+---------+-------------+---------------+----------+---- o3o3o3x3o . . ♦ 15 | 60 | 60 20 | 30 15 0 | 6 6 0 0 | 28 * * | 2 0 . o3o3x3o3o . ♦ 20 | 90 | 60 60 | 15 30 15 | 0 6 6 0 | * 56 * | 1 1 . . o3x3o3o3o ♦ 15 | 60 | 20 60 | 0 15 30 | 0 0 6 6 | * * 28 | 0 2 --------------+----+-----+---------+-------------+---------------+----------+---- o3o3o3x3o3o . ♦ 35 | 210 | 210 140 | 105 105 35 | 21 42 21 0 | 7 7 0 | 8 * . o3o3x3o3o3o ♦ 35 | 210 | 140 210 | 35 105 105 | 0 21 42 21 | 0 7 7 | * 8 | |
| 0321 r{3,33,2} [73] | |
o3o3x3o *c3o3o3o - rolin . . . . . . . | 10080 ♦ 24 | 24 12 36 | 8 12 36 18 24 | 4 12 18 24 12 6 | 6 8 12 6 3 | 4 2 3 -----------------+-------+--------+--------------------+-------------------------------+-----------------------------------+-------------------------+----------- . . x . . . . | 2 | 120960 | 2 1 3 | 1 2 6 3 3 | 1 3 6 6 3 1 | 3 3 6 2 1 | 3 1 2 -----------------+-------+--------+--------------------+-------------------------------+-----------------------------------+-------------------------+----------- . o3x . . . . | 3 | 3 | 80640 * * | 1 1 3 0 0 | 1 3 3 3 0 0 | 3 3 3 1 0 | 3 1 1 . . x3o . . . | 3 | 3 | * 40320 * | 0 2 0 3 0 | 1 0 6 0 3 0 | 3 0 6 0 1 | 3 0 2 . . x . *c3o . . | 3 | 3 | * * 120960 | 0 0 2 1 2 | 0 1 2 4 2 1 | 1 2 4 2 1 | 2 1 2 -----------------+-------+--------+--------------------+-------------------------------+-----------------------------------+-------------------------+----------- o3o3x . . . . ♦ 4 | 6 | 4 0 0 | 20160 * * * * | 1 3 0 0 0 0 | 3 3 0 0 0 | 3 1 0 . o3x3o . . . ♦ 6 | 12 | 4 4 0 | * 20160 * * * | 1 0 3 0 0 0 | 3 0 3 0 0 | 3 0 1 . o3x . *c3o . . ♦ 6 | 12 | 4 0 4 | * * 60480 * * | 0 1 1 2 0 0 | 1 2 2 1 0 | 2 1 1 . . x3o *c3o . . ♦ 6 | 12 | 0 4 4 | * * * 30240 * | 0 0 2 0 2 0 | 1 0 4 0 1 | 2 0 2 . . x . *c3o3o . ♦ 4 | 6 | 0 0 4 | * * * * 60480 | 0 0 0 2 1 1 | 0 1 2 2 1 | 1 1 2 -----------------+-------+--------+--------------------+-------------------------------+-----------------------------------+-------------------------+----------- o3o3x3o . . . ♦ 10 | 30 | 20 10 0 | 5 5 0 0 0 | 4032 * * * * * | 3 0 0 0 0 | 3 0 0 o3o3x . *c3o . . ♦ 10 | 30 | 20 0 10 | 5 0 5 0 0 | * 12096 * * * * | 1 2 0 0 0 | 2 1 0 . o3x3o *c3o . . ♦ 24 | 96 | 32 32 32 | 0 8 8 8 0 | * * 7560 * * * | 1 0 2 0 0 | 2 0 1 . o3x . *c3o3o . ♦ 10 | 30 | 10 0 20 | 0 0 5 0 5 | * * * 24192 * * | 0 1 1 1 0 | 1 1 1 . . x3o *c3o3o . ♦ 10 | 30 | 0 10 20 | 0 0 0 5 5 | * * * * 12096 * | 0 0 2 0 1 | 1 0 2 . . x . *c3o3o3o ♦ 5 | 10 | 0 0 10 | 0 0 0 0 5 | * * * * * 12096 | 0 0 0 2 1 | 0 1 2 -----------------+-------+--------+--------------------+-------------------------------+-----------------------------------+-------------------------+----------- o3o3x3o *c3o . . ♦ 80 | 480 | 320 160 160 | 80 80 80 40 0 | 16 16 10 0 0 0 | 756 * * * * | 2 0 0 o3o3x . *c3o3o . ♦ 20 | 90 | 60 0 60 | 15 0 30 0 15 | 0 6 0 6 0 0 | * 4032 * * * | 1 1 0 . o3x3o *c3o3o . ♦ 80 | 480 | 160 160 320 | 0 40 80 80 80 | 0 0 10 16 16 0 | * * 1512 * * | 1 0 1 . o3x . *c3o3o3o ♦ 15 | 60 | 20 0 60 | 0 0 15 0 30 | 0 0 0 6 0 6 | * * * 4032 * | 0 1 1 . . x3o *c3o3o3o ♦ 15 | 60 | 0 20 60 | 0 0 0 15 30 | 0 0 0 0 6 6 | * * * * 2016 | 0 0 2 -----------------+-------+--------+--------------------+-------------------------------+-----------------------------------+-------------------------+----------- o3o3x3o *c3o3o . ♦ 720 | 6480 | 4320 2160 4320 | 1080 1080 2160 1080 1080 | 216 432 270 432 216 0 | 27 72 27 0 0 | 56 * * o3o3x . *c3o3o3o ♦ 35 | 210 | 140 0 210 | 35 0 105 0 105 | 0 21 0 42 0 21 | 0 7 0 7 0 | * 576 * . o3x3o *c3o3o3o ♦ 240 | 1920 | 640 640 1920 | 0 160 480 480 960 | 0 0 60 192 192 192 | 0 0 12 32 32 | * * 126 | |
8D
| 8-simplex {3,3,3,3,3,3,3} [74] | 8-orthoplex {3,3,3,3,3,3,4} [75] | 8-cube {4,3,3,3,3,3,3} [76] |
|---|---|---|
x3o3o3o3o3o3o3o . . . . . . . . | 9 ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 ----------------+---+----+----+-----+-----+----+----+-- x . . . . . . . | 2 | 36 ♦ 7 | 21 | 35 | 35 | 21 | 7 ----------------+---+----+----+-----+-----+----+----+-- x3o . . . . . . | 3 | 3 | 84 ♦ 6 | 15 | 20 | 15 | 6 ----------------+---+----+----+-----+-----+----+----+-- x3o3o . . . . . ♦ 4 | 6 | 4 | 126 ♦ 5 | 10 | 10 | 5 ----------------+---+----+----+-----+-----+----+----+-- x3o3o3o . . . . ♦ 5 | 10 | 10 | 5 | 126 ♦ 4 | 6 | 4 ----------------+---+----+----+-----+-----+----+----+-- x3o3o3o3o . . . ♦ 6 | 15 | 20 | 15 | 6 | 84 | 3 | 3 ----------------+---+----+----+-----+-----+----+----+-- x3o3o3o3o3o . . ♦ 7 | 21 | 35 | 35 | 21 | 7 | 36 | 2 ----------------+---+----+----+-----+-----+----+----+-- x3o3o3o3o3o3o . ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 | 9 |
x3o3o3o3o3o3o4o . . . . . . . . | 16 ♦ 14 | 84 | 280 | 560 | 672 | 448 | 128 ----------------+----+-----+-----+------+------+------+------+---- x . . . . . . . | 2 | 112 ♦ 12 | 60 | 160 | 240 | 192 | 64 ----------------+----+-----+-----+------+------+------+------+---- x3o . . . . . . | 3 | 3 | 448 ♦ 10 | 40 | 80 | 80 | 32 ----------------+----+-----+-----+------+------+------+------+---- x3o3o . . . . . ♦ 4 | 6 | 4 | 1120 ♦ 8 | 24 | 32 | 16 ----------------+----+-----+-----+------+------+------+------+---- x3o3o3o . . . . ♦ 5 | 10 | 10 | 5 | 1792 ♦ 6 | 12 | 8 ----------------+----+-----+-----+------+------+------+------+---- x3o3o3o3o . . . ♦ 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4 ----------------+----+-----+-----+------+------+------+------+---- x3o3o3o3o3o . . ♦ 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2 ----------------+----+-----+-----+------+------+------+------+---- x3o3o3o3o3o3o . ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256 |
o3o3o3o3o3o3o4x . . . . . . . . | 256 ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 ----------------+-----+------+------+------+------+-----+-----+--- . . . . . . . x | 2 | 1024 ♦ 7 | 21 | 35 | 35 | 21 | 7 ----------------+-----+------+------+------+------+-----+-----+--- . . . . . . o4x | 4 | 4 | 1792 ♦ 6 | 15 | 20 | 15 | 6 ----------------+-----+------+------+------+------+-----+-----+--- . . . . . o3o4x ♦ 8 | 12 | 6 | 1792 ♦ 5 | 10 | 10 | 5 ----------------+-----+------+------+------+------+-----+-----+--- . . . . o3o3o4x ♦ 16 | 32 | 24 | 8 | 1120 ♦ 4 | 6 | 4 ----------------+-----+------+------+------+------+-----+-----+--- . . . o3o3o3o4x ♦ 32 | 80 | 80 | 40 | 10 | 448 | 3 | 3 ----------------+-----+------+------+------+------+-----+-----+--- . . o3o3o3o3o4x ♦ 64 | 192 | 240 | 160 | 60 | 12 | 112 | 2 ----------------+-----+------+------+------+------+-----+-----+--- . o3o3o3o3o3o4x ♦ 128 | 448 | 672 | 560 | 280 | 84 | 14 | 16 |
Uniform 8D
| 8-demicube h{4,3,3,3,3,3,3} [77] | 421 {3,3,3,3,32,1} [78] |
|---|---|
x3o3o *b3o3o3o3o3o . . . . . . . . | 128 ♦ 28 | 168 | 56 280 | 70 280 | 56 168 | 28 56 | 8 8 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x . . . . . . . | 2 | 1792 ♦ 12 | 6 30 | 15 40 | 20 30 | 15 12 | 6 2 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x3o . . . . . . | 3 | 3 | 7168 | 1 5 | 5 10 | 10 10 | 10 5 | 5 1 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x3o3o . . . . . ♦ 4 | 6 | 4 | 1792 * ♦ 5 0 | 10 0 | 10 0 | 5 0 x3o . *b3o . . . . ♦ 4 | 6 | 4 | * 8960 | 1 4 | 4 6 | 6 4 | 4 1 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x3o3o *b3o . . . . ♦ 8 | 24 | 32 | 8 8 | 1120 * ♦ 4 0 | 6 0 | 4 0 x3o . *b3o3o . . . ♦ 5 | 10 | 10 | 0 5 | * 7168 | 1 3 | 3 3 | 3 1 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x3o3o *b3o3o . . . ♦ 16 | 80 | 160 | 40 80 | 10 16 | 448 * | 3 0 | 3 0 x3o . *b3o3o3o . . ♦ 6 | 15 | 20 | 0 15 | 0 6 | * 3584 | 1 2 | 2 1 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x3o3o *b3o3o3o . . ♦ 32 | 240 | 640 | 160 480 | 60 192 | 12 32 | 112 * | 2 0 x3o . *b3o3o3o3o . ♦ 7 | 21 | 35 | 0 35 | 0 21 | 0 7 | * 1024 | 1 1 -------------------+-----+------+------+-----------+-----------+----------+----------+------- x3o3o *b3o3o3o3o . ♦ 64 | 672 | 2240 | 560 2240 | 280 1344 | 84 448 | 14 64 | 16 * x3o . *b3o3o3o3o3o ♦ 8 | 28 | 56 | 0 70 | 0 56 | 0 28 | 0 8 | * 128 |
o3o3o3o *c3o3o3o3x . . . . . . . . | 240 ♦ 56 | 756 | 4032 | 10080 | 12096 | 4032 2016 | 576 126 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- . . . . . . . x | 2 | 6720 ♦ 27 | 216 | 720 | 1080 | 432 216 | 72 27 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- . . . . . . o3x | 3 | 3 | 60480 ♦ 16 | 80 | 160 | 80 40 | 16 10 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- . . . . . o3o3x ♦ 4 | 6 | 4 | 241920 ♦ 10 | 30 | 20 10 | 5 5 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- . . . . o3o3o3x ♦ 5 | 10 | 10 | 5 | 483840 ♦ 6 | 6 3 | 2 3 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- . . o . *c3o3o3o3x ♦ 6 | 15 | 20 | 15 | 6 | 483840 | 2 1 | 1 2 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- . o3o . *c3o3o3o3x ♦ 7 | 21 | 35 | 35 | 21 | 7 | 138240 * | 1 1 . . o3o *c3o3o3o3x ♦ 7 | 21 | 35 | 35 | 21 | 7 | * 69120 | 0 2 -------------------+-----+------+-------+--------+--------+--------+--------------+----------- o3o3o . *c3o3o3o3x ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 0 | 17280 * . o3o3o *c3o3o3o3x ♦ 14 | 84 | 280 | 560 | 672 | 448 | 64 64 | * 2160 |
| 241 {3,3,34,1} [79] | 142 {3,34,2} [80] |
x3o3o3o *c3o3o3o3o . . . . . . . . | 2160 ♦ 64 | 672 | 2240 | 560 2240 | 280 1344 | 84 448 | 14 64 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x . . . . . . . | 2 | 69120 ♦ 21 | 105 | 35 140 | 35 105 | 21 42 | 7 7 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x3o . . . . . . | 3 | 3 | 483840 ♦ 10 | 5 20 | 10 20 | 10 10 | 5 2 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x3o3o . . . . . ♦ 4 | 6 | 4 | 1209600 | 1 4 | 4 6 | 6 4 | 4 1 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x3o3o3o . . . . ♦ 5 | 10 | 10 | 5 | 241920 * | 4 0 | 6 0 | 4 0 x3o3o . *c3o . . . ♦ 5 | 10 | 10 | 5 | * 967680 | 1 3 | 3 3 | 3 1 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x3o3o3o *c3o . . . ♦ 10 | 40 | 80 | 80 | 16 16 | 60480 * | 3 0 | 3 0 x3o3o . *c3o3o . . ♦ 6 | 15 | 20 | 15 | 0 6 | * 483840 | 1 2 | 2 1 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x3o3o3o *c3o3o . . ♦ 27 | 216 | 720 | 1080 | 216 432 | 27 72 | 6720 * | 2 0 x3o3o . *c3o3o3o . ♦ 7 | 21 | 35 | 35 | 0 21 | 0 7 | * 138240 | 1 1 -------------------+------+-------+--------+---------+---------------+--------------+-------------+--------- x3o3o3o *c3o3o3o . ♦ 126 | 2016 | 10080 | 20160 | 4032 12096 | 756 4032 | 56 576 | 240 * x3o3o . *c3o3o3o3o ♦ 8 | 28 | 56 | 70 | 0 56 | 0 28 | 0 8 | * 17280 |
o3o3o3x *c3o3o3o3o . . . . . . . . | 17280 ♦ 56 | 420 | 280 560 | 70 280 420 | 56 168 168 | 28 56 28 | 8 8 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- . . . x . . . . | 2 | 483840 ♦ 15 | 15 30 | 5 30 30 | 10 30 15 | 10 15 3 | 5 3 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- . . o3x . . . . | 3 | 3 | 2419200 | 2 4 | 1 8 6 | 4 12 4 | 6 8 1 | 4 2 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- . o3o3x . . . . ♦ 4 | 6 | 4 | 1209600 * | 1 4 0 | 4 6 0 | 6 4 0 | 4 1 . . o3x *c3o . . . ♦ 4 | 6 | 4 | * 2419200 | 0 2 3 | 1 6 3 | 3 6 1 | 3 2 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- o3o3o3x . . . . ♦ 5 | 10 | 10 | 5 0 | 241920 * * | 4 0 0 | 6 0 0 | 4 0 . o3o3x *c3o . . . ♦ 8 | 24 | 32 | 8 8 | * 604800 * | 1 3 0 | 3 3 0 | 3 1 . . o3x *c3o3o . . ♦ 5 | 10 | 10 | 0 5 | * * 1451520 | 0 2 2 | 1 4 1 | 2 2 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- o3o3o3x *c3o . . . ♦ 16 | 80 | 160 | 80 40 | 16 10 0 | 60480 * * | 3 0 0 | 3 0 . o3o3x *c3o3o . . ♦ 16 | 80 | 160 | 40 80 | 0 10 16 | * 181440 * | 1 2 0 | 2 1 . . o3x *c3o3o3o . ♦ 6 | 15 | 20 | 0 15 | 0 0 6 | * * 483840 | 0 2 1 | 1 2 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- o3o3o3x *c3o3o . . ♦ 72 | 720 | 2160 | 1080 1080 | 216 270 216 | 27 27 0 | 6720 * * | 2 0 . o3o3x *c3o3o3o . ♦ 32 | 240 | 640 | 160 480 | 0 60 192 | 0 12 32 | * 30240 * | 1 1 . . o3x *c3o3o3o3o ♦ 7 | 21 | 35 | 0 35 | 0 0 21 | 0 0 7 | * * 69120 | 0 2 -------------------+-------+--------+---------+-----------------+------------------------+---------------------+------------------+--------- o3o3o3x *c3o3o3o . ♦ 576 | 10080 | 40320 | 20160 30240 | 4032 7560 12096 | 756 1512 2016 | 56 126 0 | 240 * . o3o3x *c3o3o3o3o ♦ 64 | 672 | 2240 | 560 2240 | 0 280 1344 | 0 84 448 | 0 14 64 | * 2160 |
| 061 r{3,3,3,3,3,3,3} [81] | 052 2r{3,3,3,3,3,3,3} [82] |
o3x3o3o3o3o3o3o - rene . . . . . . . . | 36 ♦ 14 | 7 42 | 21 70 | 35 70 | 35 42 | 21 14 | 7 2 ----------------+----+-----+--------+---------+---------+--------+-------+---- . x . . . . . . | 2 | 252 | 1 6 | 6 15 | 15 20 | 20 15 | 15 6 | 6 1 ----------------+----+-----+--------+---------+---------+--------+-------+---- o3x . . . . . . | 3 | 3 | 84 * ♦ 6 0 | 15 0 | 20 0 | 15 0 | 6 0 . x3o . . . . . | 3 | 3 | * 504 | 1 5 | 5 10 | 10 10 | 10 5 | 5 1 ----------------+----+-----+--------+---------+---------+--------+-------+---- o3x3o . . . . . ♦ 6 | 12 | 4 4 | 126 * ♦ 5 0 | 10 0 | 10 0 | 5 0 . x3o3o . . . . ♦ 4 | 6 | 0 4 | * 630 | 1 4 | 4 6 | 6 4 | 4 1 ----------------+----+-----+--------+---------+---------+--------+-------+---- o3x3o3o . . . . ♦ 10 | 30 | 10 20 | 5 5 | 126 * ♦ 4 0 | 6 0 | 4 0 . x3o3o3o . . . ♦ 5 | 10 | 0 10 | 0 5 | * 504 | 1 3 | 3 3 | 3 1 ----------------+----+-----+--------+---------+---------+--------+-------+---- o3x3o3o3o . . . ♦ 15 | 60 | 20 60 | 15 30 | 6 6 | 84 * | 3 0 | 3 0 . x3o3o3o3o . . ♦ 6 | 15 | 0 20 | 0 15 | 0 6 | * 252 | 1 2 | 2 1 ----------------+----+-----+--------+---------+---------+--------+-------+---- o3x3o3o3o3o . . ♦ 21 | 105 | 35 140 | 35 105 | 21 42 | 7 7 | 36 * | 2 0 . x3o3o3o3o3o . ♦ 7 | 21 | 0 35 | 0 35 | 0 21 | 0 7 | * 72 | 1 1 ----------------+----+-----+--------+---------+---------+--------+-------+---- o3x3o3o3o3o3o . ♦ 28 | 168 | 56 280 | 70 280 | 56 168 | 28 56 | 8 8 | 9 * . x3o3o3o3o3o3o ♦ 8 | 28 | 0 56 | 0 70 | 0 56 | 0 28 | 0 8 | * 9 |
o3o3x3o3o3o3o3o - brene . . . . . . . . | 84 ♦ 18 | 18 45 | 6 45 60 | 15 60 45 | 20 45 18 | 15 18 3 | 6 3 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- . . x . . . . . | 2 | 756 | 2 5 | 1 10 10 | 5 20 10 | 10 20 5 | 10 10 1 | 5 2 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- . o3x . . . . . | 3 | 3 | 504 * | 1 5 0 | 5 10 0 | 10 10 0 | 10 5 0 | 5 1 . . x3o . . . . | 3 | 3 | * 1260 | 0 2 4 | 1 8 6 | 6 12 4 | 6 8 1 | 4 2 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- o3o3x . . . . . ♦ 4 | 6 | 4 0 | 126 * * ♦ 5 0 0 | 10 0 0 | 10 0 0 | 5 0 . o3x3o . . . . ♦ 6 | 12 | 4 4 | * 630 * | 1 4 0 | 4 6 0 | 6 4 0 | 4 1 . . x3o3o . . . ♦ 4 | 6 | 0 4 | * * 1260 | 0 2 3 | 1 6 3 | 3 6 1 | 3 2 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- o3o3x3o . . . . ♦ 10 | 30 | 20 10 | 5 5 0 | 126 * * ♦ 4 0 0 | 6 0 0 | 4 0 . o3x3o3o . . . ♦ 10 | 30 | 10 20 | 0 5 5 | * 504 * | 1 3 0 | 3 3 0 | 3 1 . . x3o3o3o . . ♦ 5 | 10 | 0 10 | 0 0 5 | * * 756 | 0 2 2 | 1 4 1 | 2 2 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- o3o3x3o3o . . . ♦ 20 | 90 | 60 60 | 15 30 15 | 6 6 0 | 84 * * | 3 0 0 | 3 0 . o3x3o3o3o . . ♦ 15 | 60 | 20 60 | 0 15 30 | 0 6 6 | * 252 * | 1 2 0 | 2 1 . . x3o3o3o3o . ♦ 6 | 15 | 0 20 | 0 0 15 | 0 0 6 | * * 252 | 0 2 1 | 1 2 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- o3o3x3o3o3o . . ♦ 35 | 210 | 140 210 | 35 105 105 | 21 42 21 | 7 7 0 | 36 * * | 2 0 . o3x3o3o3o3o . ♦ 21 | 105 | 35 140 | 0 35 105 | 0 21 42 | 0 7 7 | * 72 * | 1 1 . . x3o3o3o3o3o ♦ 7 | 21 | 0 35 | 0 0 35 | 0 0 21 | 0 0 7 | * * 36 | 0 2 ----------------+----+-----+----------+--------------+-------------+------------+----------+---- o3o3x3o3o3o3o . ♦ 56 | 420 | 280 560 | 70 280 420 | 56 168 168 | 28 56 28 | 8 8 0 | 9 * . o3x3o3o3o3o3o ♦ 28 | 168 | 56 280 | 0 70 280 | 0 56 168 | 0 28 56 | 0 8 8 | * 9 |
| 043 3r{3,3,3,3,3,3,3} [83] | |
o3o3o3x3o3o3o3o - trene . . . . . . . . | 126 ♦ 20 | 30 40 | 20 60 40 | 5 40 60 20 | 10 40 30 4 | 10 20 6 | 5 4 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- . . . x . . . . | 2 | 1260 | 3 4 | 3 12 6 | 1 12 18 4 | 4 18 12 1 | 6 12 3 | 4 3 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- . . o3x . . . . | 3 | 3 | 1260 * | 2 4 0 | 1 8 6 0 | 4 12 4 0 | 6 8 1 | 4 2 . . . x3o . . . | 3 | 3 | * 1680 | 0 3 3 | 0 3 9 3 | 1 9 9 1 | 3 9 3 | 3 3 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- . o3o3x . . . . ♦ 4 | 6 | 4 0 | 630 * * | 1 4 0 0 | 4 6 0 0 | 6 4 0 | 4 1 . . o3x3o . . . ♦ 6 | 12 | 4 4 | * 1260 * | 0 2 3 0 | 1 6 3 0 | 3 6 1 | 3 2 . . . x3o3o . . ♦ 4 | 6 | 0 4 | * * 1260 | 0 0 3 2 | 0 3 6 1 | 1 6 3 | 2 3 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- o3o3o3x . . . . ♦ 5 | 10 | 10 0 | 5 0 0 | 126 * * * ♦ 4 0 0 0 | 6 0 0 | 4 0 . o3o3x3o . . . ♦ 10 | 30 | 20 10 | 5 5 0 | * 504 * * | 1 3 0 0 | 3 3 0 | 3 1 . . o3x3o3o . . ♦ 10 | 30 | 10 20 | 0 5 5 | * * 756 * | 0 2 2 0 | 1 4 1 | 2 2 . . . x3o3o3o . ♦ 5 | 10 | 0 10 | 0 0 5 | * * * 504 | 0 0 3 1 | 0 3 3 | 1 3 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- o3o3o3x3o . . . ♦ 15 | 60 | 60 20 | 30 15 0 | 6 6 0 0 | 84 * * * | 3 0 0 | 3 0 . o3o3x3o3o . . ♦ 20 | 90 | 60 60 | 15 30 15 | 0 6 6 0 | * 252 * * | 1 2 0 | 2 1 . . o3x3o3o3o . ♦ 15 | 60 | 20 60 | 0 15 30 | 0 0 6 6 | * * 252 * | 0 2 1 | 1 2 . . . x3o3o3o3o ♦ 6 | 15 | 0 20 | 0 0 15 | 0 0 0 6 | * * * 84 | 0 0 3 | 0 3 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- o3o3o3x3o3o . . ♦ 35 | 210 | 210 140 | 105 105 35 | 21 42 21 0 | 7 7 0 0 | 36 * * | 2 0 . o3o3x3o3o3o . ♦ 35 | 210 | 140 210 | 35 105 105 | 0 21 42 21 | 0 7 7 0 | * 72 * | 1 1 . . o3x3o3o3o3o ♦ 21 | 105 | 35 140 | 0 35 105 | 0 0 21 42 | 0 0 7 7 | * * 36 | 0 2 ----------------+-----+------+-----------+---------------+-----------------+---------------+----------+---- o3o3o3x3o3o3o . ♦ 70 | 560 | 560 560 | 280 420 280 | 56 168 168 56 | 28 56 28 0 | 8 8 0 | 9 * . o3o3x3o3o3o3o ♦ 56 | 420 | 280 560 | 70 280 420 | 0 56 168 168 | 0 28 56 28 | 0 8 8 | * 9 | |
| 0421 r{3,34,2} [84] | |
o3o3x3o *c3o3o3o3o - buffy . . . . . . . . | 483840 ♦ 30 | 30 15 60 | 10 15 60 30 60 | 5 20 30 60 30 30 | 10 20 30 30 15 6 | 10 10 15 6 3 | 5 2 3 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- . . x . . . . . | 2 | 7257600 | 2 1 4 | 1 2 8 4 6 | 1 4 8 12 6 4 | 4 6 12 8 4 1 | 6 4 8 2 1 | 4 1 2 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- . o3x . . . . . | 3 | 3 | 4838400 * * | 1 1 4 0 0 | 1 4 4 6 0 0 | 4 6 6 4 0 0 | 6 4 4 1 0 | 4 1 1 . . x3o . . . . | 3 | 3 | * 2419200 * | 0 2 0 4 0 | 1 0 8 0 6 0 | 4 0 12 0 4 0 | 6 0 8 0 1 | 4 0 2 . . x . *c3o . . . | 3 | 3 | * * 9676800 | 0 0 2 1 3 | 0 1 2 6 3 3 | 1 3 6 6 3 1 | 3 3 6 2 1 | 3 1 2 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- o3o3x . . . . . ♦ 4 | 6 | 4 0 0 | 1209600 * * * * | 1 4 0 0 0 0 | 4 6 0 0 0 0 | 6 4 0 0 0 | 4 1 0 . o3x3o . . . . ♦ 6 | 12 | 4 4 0 | * 1209600 * * * | 1 0 4 0 0 0 | 4 0 6 0 0 0 | 6 0 4 0 0 | 4 0 1 . o3x . *c3o . . . ♦ 6 | 12 | 4 0 4 | * * 4838400 * * | 0 1 1 3 0 0 | 1 3 3 3 0 0 | 3 3 3 1 0 | 3 1 1 . . x3o *c3o . . . ♦ 6 | 12 | 0 4 4 | * * * 2419200 * | 0 0 2 0 3 0 | 1 0 6 0 3 0 | 3 0 6 0 1 | 3 0 2 . . x . *c3o3o . . ♦ 4 | 6 | 0 0 4 | * * * * 7257600 | 0 0 0 2 1 2 | 0 1 2 4 2 1 | 1 2 4 2 1 | 2 1 2 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- o3o3x3o . . . . ♦ 10 | 30 | 20 10 0 | 5 5 0 0 0 | 241920 * * * * * ♦ 4 0 0 0 0 0 | 6 0 0 0 0 | 4 0 0 o3o3x . *c3o . . . ♦ 10 | 30 | 20 0 10 | 5 0 5 0 0 | * 967680 * * * * | 1 3 0 0 0 0 | 3 3 0 0 0 | 3 1 0 . o3x3o *c3o . . . ♦ 24 | 96 | 32 32 32 | 0 8 8 8 0 | * * 604800 * * * | 1 0 3 0 0 0 | 3 0 3 0 0 | 3 0 1 . o3x . *c3o3o . . ♦ 10 | 30 | 10 0 20 | 0 0 5 0 5 | * * * 2903040 * * | 0 1 1 2 0 0 | 1 2 2 1 0 | 2 1 1 . . x3o *c3o3o . . ♦ 10 | 30 | 0 10 20 | 0 0 0 5 5 | * * * * 1451520 * | 0 0 2 0 2 0 | 1 0 4 0 1 | 2 0 2 . . x . *c3o3o3o . ♦ 5 | 10 | 0 0 10 | 0 0 0 0 5 | * * * * * 2903040 | 0 0 0 2 1 1 | 0 1 2 2 1 | 1 1 2 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- o3o3x3o *c3o . . . ♦ 80 | 480 | 320 160 160 | 80 80 80 40 0 | 16 16 10 0 0 0 | 60480 * * * * * | 3 0 0 0 0 | 3 0 0 o3o3x . *c3o3o . . ♦ 20 | 90 | 60 0 60 | 15 0 30 0 15 | 0 6 0 6 0 0 | * 483840 * * * * | 1 2 0 0 0 | 2 1 0 . o3x3o *c3o3o . . ♦ 80 | 480 | 160 160 320 | 0 40 80 80 80 | 0 0 10 16 16 0 | * * 181440 * * * | 1 0 2 0 0 | 2 0 1 . o3x . *c3o3o3o . ♦ 15 | 60 | 20 0 60 | 0 0 15 0 30 | 0 0 0 6 0 6 | * * * 967680 * * | 0 1 1 1 0 | 1 1 1 . . x3o *c3o3o3o . ♦ 15 | 60 | 0 20 60 | 0 0 0 15 30 | 0 0 0 0 6 6 | * * * * 483840 * | 0 0 2 0 1 | 1 0 2 . . x . *c3o3o3o3o ♦ 6 | 15 | 0 0 20 | 0 0 0 0 15 | 0 0 0 0 0 6 | * * * * * 483840 | 0 0 0 2 1 | 0 1 2 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- o3o3x3o *c3o3o . . ♦ 720 | 6480 | 4320 2160 4320 | 1080 1080 2160 1080 1080 | 216 432 270 432 216 0 | 27 72 27 0 0 0 | 6720 * * * * | 2 0 0 o3o3x . *c3o3o3o . ♦ 35 | 210 | 140 0 210 | 35 0 105 0 105 | 0 21 0 42 0 21 | 0 7 0 7 0 0 | * 138240 * * * | 1 1 0 . o3x3o *c3o3o3o . ♦ 240 | 1920 | 640 640 1920 | 0 160 480 480 960 | 0 0 60 192 192 192 | 0 0 12 32 32 0 | * * 30240 * * | 1 0 1 . o3x . *c3o3o3o3o ♦ 21 | 105 | 35 0 140 | 0 0 35 0 105 | 0 0 0 21 0 42 | 0 0 0 7 0 7 | * * * 138240 * | 0 1 1 . . x3o *c3o3o3o3o ♦ 21 | 105 | 0 35 140 | 0 0 0 35 105 | 0 0 0 0 21 42 | 0 0 0 0 7 7 | * * * * 69120 | 0 0 2 -------------------+--------+---------+-------------------------+-----------------------------------------+----------------------------------------------+------------------------------------------+--------------------------------+--------------- o3o3x3o *c3o3o3o . ♦ 10080 | 120960 | 80640 40320 120960 | 20160 20160 60480 30240 60480 | 4032 12096 7560 24192 12096 12096 | 756 4032 1512 4032 2016 0 | 56 576 126 0 0 | 240 * * o3o3x . *c3o3o3o3o ♦ 56 | 420 | 280 0 560 | 70 0 280 0 420 | 0 56 0 168 0 168 | 0 28 0 56 0 28 | 0 8 0 8 0 | * 17280 * . o3x3o *c3o3o3o3o ♦ 672 | 6720 | 2240 2240 8960 | 0 560 2240 2240 6720 | 0 0 280 1344 1344 2688 | 0 0 84 448 448 448 | 0 0 14 64 64 | * * 2160 | |
8-honeycombs
o3o3o3o *c3o3o3o3o3x - goh
o3o3o3o *c3o3o3o3o3x (N → ∞) . . . . . . . . . | N ♦ 240 | 6720 | 60480 | 241920 | 483840 | 483840 | 138240 69120 | 17280 2160 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . . . . . . . . x | 2 | 120N ♦ 56 | 756 | 4032 | 10080 | 12096 | 4032 2016 | 576 126 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . . . . . . . o3x | 3 | 3 | 2240N ♦ 27 | 216 | 720 | 1080 | 432 216 | 72 27 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . . . . . . o3o3x ♦ 4 | 6 | 4 | 15120N ♦ 16 | 80 | 160 | 80 40 | 16 10 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . . . . . o3o3o3x ♦ 5 | 10 | 10 | 5 | 48384N ♦ 10 | 30 | 20 10 | 5 5 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . . . . o3o3o3o3x ♦ 6 | 15 | 20 | 15 | 6 | 80640N ♦ 6 | 6 3 | 2 3 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . . o . *c3o3o3o3o3x ♦ 7 | 21 | 35 | 35 | 21 | 7 | 69120N | 2 1 | 1 2 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- . o3o . *c3o3o3o3o3x ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 | 17280N * | 1 1 . . o3o *c3o3o3o3o3x ♦ 8 | 28 | 56 | 70 | 56 | 28 | 8 | * 8640N | 0 2 ---------------------+----+------+-------+--------+--------+--------+--------+--------------+----------- o3o3o . *c3o3o3o3o3x ♦ 9 | 36 | 84 | 126 | 126 | 84 | 36 | 8 0 | 1920N * . o3o3o *c3o3o3o3o3x ♦ 16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 128 128 | * 135N
Computation
The f-vector values, seen on the diagonal, are computed by systematically removing nodes (mirrors) from the Kaleidoscope. The element of a given set of removals is defined by the set of nodes connected to at least one ringed nodes. The number of elements of that type is computed from the full order of the Coxeter group divided by the order of the remaining mirrors. If groups of mirrors are not connected, the order is the product of all such connected groups remaining.
Polyhedra
Truncated cuboctahedron
Example truncated cuboctahedron, with all mirrors active, all 1+3+3+1 fundamental domain simplex positions contain elements.
| B3 | |
k-face | fk | f0 | f1 | f2 | k-fig | Notes | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
 | ( ) | f0 | 48 | 1 | 1 | 1 | 1 | 1 | 1 | ( )∨( )∨( ) | B3 = 48 | |
| A1 | | { } | f1 | 2 | 24 | * | * | 1 | 1 | 0 | { } | B3/A1 = 24 |
| A1 | | 2 | * | 24 | * | 1 | 0 | 1 | B3/A1 = 24 | |||
| A1 | | 2 | * | * | 24 | 0 | 1 | 1 | B3/A1 = 24 | |||
| A2 | | {6} | f2 | 6 | 3 | 3 | 0 | 8 | * | * | ( ) | B3/A2 = 8*6/6 = 8 |
| A1A1 | | {4} | 4 | 2 | 0 | 2 | * | 12 | * | B3/A1/A1 = 48/4 = 12 | ||
| B2 | | {8} | 8 | 0 | 4 | 4 | * | * | 6 | B3/B2 = 48/8 = 6 | ||
4-polytopes
5-cell family
5-cell
x3o3o3o - pen
rectified 5-cell
o3x3o3o - rap
Truncated 5-cell
x3x3o3o - tip
Cantellated 5-cell
x3o3x3o - srip
| A4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1A1 | | ( ) | f0 | 30 | 2 | 4 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | Irr {3}×{ } | A4/A1A1 = 5!/4 = 30 |
| A2A1 | | { } | f1 | 2 | 30 | * | 1 | 2 | 0 | 0 | 2 | 1 | 0 | { }∨( ) | A4/A2A1 = 5!/3!/2 = 30 |
| A1 | |
2 | * | 60 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | ( )∨( )∨( ) | A4/A1 = 5!/2 = 60 | ||
| A2A1 | | {3} | f2 | 3 | 3 | 0 | 10 | * | * | * | 2 | 0 | 0 | { } | A4/A2A1 = 5!/3!/2 = 10 |
| A1A1 | | { }×{ } | 4 | 2 | 2 | * | 30 | * | * | 1 | 1 | 0 | A4/A1A1 = 5!/4 = 30 | ||
| A2 | | {3} | 3 | 0 | 3 | * | * | 20 | * | 1 | 0 | 1 | A4/A2 = 5!/3! = 20 | ||
| A2 | |
3 | 0 | 3 | * | * | * | 20 | 0 | 1 | 1 | A4/A2 = 5!/3! =20 | |||
| A3 | | rr{3,3} | f3 | 12 | 12 | 12 | 4 | 6 | 4 | 0 | 5 | * | * | ( ) | A4/A3 = 5!/4! = 5 |
| A2A1 | | {3}×{ } | 6 | 3 | 6 | 0 | 3 | 0 | 2 | * | 10 | * | A4/A2A1 = 5!/3!/2 = 10 | ||
| A3 | | r{3,3} | 6 | 0 | 12 | 0 | 0 | 4 | 4 | * | * | 5 | A4/A3 = 5!/4! = 5 | ||
runcinated 5-cell
x3o3o3x - spid
| A4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A2 | | ( ) | f0 | 20 | 3 | 3 | 3 | 6 | 3 | 1 | 3 | 3 | 1 | s{2,6} | A4/A2 = 5!/3! = 20 |
| A1A1 | | { } | f1 | 2 | 30 | * | 2 | 2 | 0 | 1 | 2 | 1 | 0 | { }×{ } | A4/A1A1 = 5!/4 = 30 |
|
2 | * | 30 | 0 | 2 | 2 | 0 | 1 | 2 | 1 | |||||
| A2 | | {3} | f2 | 3 | 3 | 0 | 20 | * | * | 1 | 1 | 0 | 0 | { } | A4/A2 = 5!/3! =20 |
| A1A1 | | { }×{ } | 4 | 2 | 2 | * | 30 | * | 0 | 1 | 1 | 0 | A4/A1A1 = 5!/4 = 30 | ||
| A2 | | {3} | 3 | 0 | 3 | * | * | 20 | 0 | 0 | 1 | 1 | A4/A2 = 5!/3! = 20 | ||
| A3 | | {3,3} | f3 | 4 | 6 | 0 | 4 | 0 | 0 | 5 | * | * | * | ( ) | A4/A3 = 5!/4! = 5 |
| A2A1 | | {3}×{ } | 6 | 6 | 3 | 2 | 3 | 0 | * | 10 | * | * | A4/A2A1 = 5!/3!/2 = 10 | ||
|
6 | 3 | 6 | 0 | 3 | 2 | * | * | 10 | * | |||||
| A3 | | {3,3} | 4 | 0 | 6 | 0 | 0 | 4 | * | * | * | 5 | A4/A3 = 5!/4! = 5 | ||
Bitruncated 5-cell
o3x3x3o - deca
Runcitruncated 5-cell
x3x3o3x - prip
| A4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1 | | ( ) | f0 | 60 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | irr. { }×{ }∨( ) | A4/A1 = 5!/2 = 60 |
| A1A1 | | { } | f1 | 2 | 30 | * | * | 2 | 2 | 0 | 0 | 0 | 1 | 2 | 1 | 0 | { }×{ } | A4/A1A1 = 5!/4 = 30 |
| A1 | |
2 | * | 60 | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | ( )∨( )∨( ) | A4/A1 = 5!/2 = 60 | ||
|
| 2 | * | * | 60 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | |||||
| A2 | | t{3} | f2 | 6 | 3 | 3 | 0 | 20 | * | * | * | * | 1 | 1 | 0 | 0 | { } | A4/A2 = 5!/3! = 20 |
| A1A1 | | { }×{ } | 4 | 2 | 0 | 2 | * | 30 | * | * | * | 0 | 1 | 1 | 0 | A4/A1A1 = 5!/4 = 30 | ||
| A2 | | {3} | 3 | 0 | 3 | 0 | * | * | 20 | * | * | 1 | 0 | 0 | 1 | A4/A2 = 5!/3! = 20 | ||
| A1A1 | | { }×{ } | 4 | 0 | 2 | 2 | * | * | * | 30 | * | 0 | 1 | 0 | 1 | A4/A1A1 = 5!/4 = 30 | ||
| A2 | | {3} | 3 | 0 | 0 | 3 | * | * | * | * | 20 | 0 | 0 | 1 | 1 | A4/A2 = 5!/3! = 20 | ||
| A3 | | t{3,3} | f3 | 12 | 6 | 12 | 0 | 4 | 0 | 4 | 0 | 0 | 5 | * | * | * | ( ) | A4/A3 = 5!/4! = 5 |
| A2A1 | | t{3}×{ } | 12 | 6 | 6 | 6 | 2 | 3 | 0 | 3 | 0 | * | 10 | * | * | A4/A2A1 = 5!/3!/2 = 10 | ||
| {3}×{ } | 6 | 3 | 0 | 6 | 0 | 3 | 0 | 0 | 2 | * | * | 10 | * | ||||
| A3 | | rr{3,3} | 12 | 0 | 12 | 12 | 0 | 0 | 4 | 6 | 4 | * | * | * | 5 | A4/A3 = 5!/4! = 5 | ||
Runcitruncated 5-cell
x3x3o3x - prip
| A4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1 | | ( ) | f0 | 60 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | irr. { }×{ }∨( ) | A4/A1 = 5!/2 = 60 |
| A1A1 | | { } | f1 | 2 | 30 | * | * | 2 | 2 | 0 | 0 | 0 | 1 | 2 | 1 | 0 | { }×{ } | A4/A1A1 = 5!/4 = 30 |
| A1 | |
2 | * | 60 | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | { }∨( ) | A4/A1 = 5!/2 = 60 | ||
|
| 2 | * | * | 60 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | |||||
| A2 | | t{3} | f2 | 6 | 3 | 3 | 0 | 20 | * | * | * | * | 1 | 1 | 0 | 0 | { } | A4/A2 = 5!/3! = 20 |
| A1A1 | | { }×{ } | 4 | 2 | 0 | 2 | * | 30 | * | * | * | 0 | 1 | 1 | 0 | A4/A1A1 = 5!/4 = 30 | ||
| A2 | | {3} | 3 | 0 | 3 | 0 | * | * | 20 | * | * | 1 | 0 | 0 | 1 | A4/A2 = 5!/3! = 20 | ||
| A1A1 | | { }×{ } | 4 | 0 | 2 | 2 | * | * | * | 30 | * | 0 | 1 | 0 | 1 | A4/A1A1 = 5!/4 = 30 | ||
| A3 | | {3} | 3 | 0 | 0 | 3 | * | * | * | * | 20 | 0 | 0 | 1 | 1 | A4/A1A1 = 5!/4 = 30 | ||
| A3 | | t{3,3} | f3 | 12 | 6 | 12 | 0 | 4 | 0 | 4 | 0 | 0 | 5 | * | * | * | ( ) | A4/A3 = 5!/4! = 5 |
| A2A1 | | t{3}×{ } | 12 | 6 | 6 | 6 | 2 | 3 | 0 | 3 | 0 | * | 10 | * | * | A4/A2A1 = 5!/3!/2 = 10 | ||
| {3}×{ } | 6 | 3 | 0 | 6 | 0 | 3 | 0 | 0 | 2 | * | * | 10 | * | ||||
| A3 | | rr{3,3} | 12 | 0 | 12 | 12 | 0 | 0 | 4 | 6 | 4 | * | * | * | 5 | A4/A3 = 5!/4! = 5 | ||
Omnitruncated 5-cell
x3x3x3x - gippid
| A4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ( ) | f0 | 120 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | irr {3,3} | A4 = 5! = 120 | |
| A1 | | { } | f1 | 2 | 60 | * | * | * | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | { }∨( ) | A4/A1 = 5!/2 = 60 |
|
2 | * | 60 | * | * | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | |||||
|
2 | * | * | 60 | * | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | |||||
|
2 | * | * | * | 60 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | |||||
| A2 | | t{3} | f2 | 6 | 3 | 3 | 0 | 0 | 20 | * | * | * | * | * | 1 | 1 | 0 | 0 | { } | A4/A2 = 5!/3! = 20 |
| A1A1 | | { }×{ } | 4 | 2 | 0 | 2 | 0 | * | 30 | * | * | * | * | 1 | 0 | 1 | 0 | A4/A1A1 = 5!/4 = 30 | ||
|
4 | 2 | 0 | 0 | 2 | * | * | 30 | * | * | * | 0 | 1 | 1 | 0 | |||||
| A2 | | t{3} | 6 | 0 | 3 | 3 | 0 | * | * | * | 20 | * | * | 1 | 0 | 0 | 1 | A4/A2 = 5!/3! = 20 | ||
| A1A1 | | { }×{ } | 4 | 0 | 2 | 0 | 2 | * | * | * | * | 30 | * | 0 | 1 | 0 | 1 | A4/A1A1 = 5!/4 = 30 | ||
| A2 | | t{3} | 6 | 0 | 0 | 3 | 3 | * | * | * | * | * | 20 | 0 | 0 | 1 | 1 | A4/A2 = 5!/3! = 20 | ||
| A3 | | tr{3,3} | f3 | 24 | 12 | 12 | 12 | 0 | 4 | 6 | 0 | 4 | 0 | 0 | 5 | * | * | * | ( ) | A4/A3 = 5!/4! = 5 |
| A2A1 | | t{3}×{ } | 12 | 6 | 6 | 0 | 6 | 2 | 0 | 3 | 0 | 3 | 0 | * | 10 | * | * | A4/A2A1 = 5!/3!/2 = 10 | ||
|
12 | 6 | 0 | 6 | 6 | 0 | 3 | 3 | 0 | 0 | 2 | * | * | 10 | * | |||||
| A3 | | tr{3,3} | 24 | 0 | 12 | 12 | 12 | 0 | 0 | 0 | 4 | 6 | 4 | * | * | * | 5 | A4/A3 = 5!/4! = 5 | ||
24-cell family
24-cell
x3o4o3o - ico
Rectified 24-cell
o3x4o3o - rico
Truncated 24-cell
x3x4o3o - tico
| F4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A2 | | ( ) | f0 | 192 | 1 | 3 | 3 | 3 | 3 | 1 | {3}∨( ) | F4/A2 = 1152/3! = 192 |
| A2A1 | | { } | f1 | 2 | 96 | * | 3 | 0 | 3 | 0 | {3} | F4/A2A1 = 1152/3!/2 = 96 |
| A1A1 | |
2 | * | 288 | 1 | 2 | 2 | 1 | { }∨( ) | F4/A1A1 = 1152/4 = 288 | ||
| A2A1 | | t{3} | f2 | 6 | 3 | 3 | 96 | * | 2 | 0 | { } | F4/A2A1 = 1152/3!/2 = 96 |
| B2 | | {4} | 4 | 0 | 4 | * | 144 | 1 | 1 | F4/B2 = 1152/8 = 144 | ||
| B3 | | t{3,4} | f3 | 24 | 12 | 24 | 8 | 6 | 24 | * | ( ) | F4/B3 = 1152/48 = 24 |
| {4,3} | 8 | 0 | 12 | 0 | 6 | * | 24 | ||||
Cantellated 24-cell
x3o4x3o - sric
| F4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1A1 | | ( ) | f0 | 288 | 2 | 4 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | irr {3}×{ } | F4/A1A1 = 1152/4 = 288 |
| { } | f1 | 2 | 288 | * | 1 | 2 | 0 | 0 | 2 | 1 | 0 | { }∨( ) | ||
| A1 | |
2 | * | 576 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | ( )∨( )∨( ) | F4/A1 = 1152/2 = 576 | ||
| A2A1 | |
f2 | 3 | 3 | 0 | 96 | * | * | * | 2 | 0 | 0 | { } | F4/A2A1 = 1152/3!/2 = 96 | |
| A1A1 | | { }×{ } | 4 | 2 | 2 | * | 288 | * | * | 1 | 1 | 0 | F4/A1A1 = 1152/4 = 288 | ||
| B2 | | {4} | 4 | 0 | 4 | * | * | 144 | * | 1 | 0 | 1 | F4/B2 = 1152/8 = 144 | ||
| A2 | | {3} | 3 | 0 | 3 | * | * | * | 192 | 0 | 1 | 1 | F4/A2 = 1152/3! = 192 | ||
| B3 | | rr{4,3} | f3 | 24 | 24 | 24 | 8 | 12 | 6 | 0 | 24 | * | * | ( ) | F4/B3 = 1152/48 = 24 |
| A2A1 | | {3}×{ } | 6 | 3 | 6 | 0 | 3 | 0 | 2 | * | 96 | * | F4/A2A1 = 1152/3!/2 = 96 | ||
| B3 | | r{4,3} | 12 | 0 | 24 | 0 | 0 | 6 | 8 | * | * | 24 | F4/B3 = 1152/48 = 24 | ||
Runcinated 24-cell
x3o4o3x - spic
| F4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B2 | | ( ) | f0 | 144 | 4 | 4 | 4 | 8 | 4 | 1 | 4 | 4 | 1 | elong s{2,8} | F4/B2 = 1152/8 = 144 |
| A1A1 | | { } | f1 | 2 | 288 | * | 2 | 2 | 0 | 1 | 2 | 1 | 0 | { }∨( ) | F4/A1A1 = 1152/4 = 288 |
|
2 | * | 288 | 0 | 2 | 2 | 0 | 1 | 2 | 1 | |||||
| A2 | | {3} | f2 | 3 | 3 | 0 | 192 | * | * | 1 | 1 | 0 | 0 | { } | F4/A1 = 1152/3! = 192 |
| A1A1 | | { }×{ } | 4 | 2 | 2 | * | 288 | * | 0 | 1 | 1 | 0 | F4/A1A1 = 1152/4 = 288 | ||
| A2 | | {3} | 3 | 0 | 3 | * | * | 192 | 0 | 0 | 1 | 1 | F4/A2 = 1152/3! = 192 | ||
| B3 | | {3,4} | f3 | 6 | 12 | 0 | 8 | 0 | 0 | 24 | * | * | * | ( ) | F4/B3 = 1152/48 = 24 |
| A2A1 | | {3}×{ } | 6 | 6 | 3 | 2 | 3 | 0 | * | 96 | * | * | F4/A2A1 = 1152/3!/2 = 96 | ||
|
6 | 3 | 6 | 0 | 3 | 2 | * | * | 96 | * | |||||
| B3 | | {3,4} | 6 | 0 | 12 | 0 | 0 | 8 | * | * | * | 24 | F4/B3 = 1152/48 = 24 | ||
Bitruncated 24-cell
o3x4x3o - cont
Cantitruncated 24-cell
x3x4x3o - grico
| F4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1 | | f0 | 576 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | F4/A1 = 1152/2 = 576 | ||
| A1A1 | | f1 | 2 | 288 | * | * | 1 | 2 | 0 | 0 | 2 | 1 | 0 | F4/A1A1A1A1 = 1152/4 = 288 | ||
| 2 | * | 288 | * | 1 | 0 | 2 | 0 | 2 | 0 | ||||||
| A1 | | 2 | * | * | 576 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | F4/A1 = 1152/2 = 576 | |||
| A2A1 | | f2 | 6 | 3 | 3 | 0 | 96 | * | * | * | 2 | 0 | 0 | F4/A2A1 = 1152/3!/2 = 96 | ||
| A1A1 | | 4 | 2 | 0 | 2 | * | 288 | * | * | 1 | 1 | 0 | F4/A1A1 = 1152/4 = 288 | |||
| B2 | | 8 | 0 | 4 | 4 | * | * | 144 | * | 1 | 0 | 1 | F4/B2 = 1152/8 = 144 | |||
| A2 | | 3 | 0 | 0 | 3 | * | * | * | 192 | 0 | 1 | 1 | F4/A2 = 1152/3! = 192 | |||
| B3 | | f3 | 48 | 24 | 24 | 24 | 8 | 12 | 6 | 0 | 24 | * | * | F4/B3 = 1152/48 = 24 | ||
| A2A1 | | 6 | 3 | 0 | 6 | 0 | 3 | 0 | 2 | * | 96 | * | F4/A2A1 = 1152/3!/2 = 96 | |||
| B3 | | 24 | 0 | 12 | 24 | 0 | 0 | 6 | 8 | * | * | 24 | F4/B3 = 1152/48 = 24 | |||
Runcitruncated 24-cell
x3x4o3x - prico
| F4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A1 | | f0 | 576 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | F4 = 1152 | ||
| A1A1 | | f1 | 2 | 288 | * | * | 2 | 2 | 0 | 0 | 0 | 1 | 2 | 1 | 0 | F4/A1 = 1152/2 = 288 | ||
| A1 | | 2 | * | 576 | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | F4/A1 = 1152/2 = 576 | |||
| 2 | * | * | 576 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | |||||
| A2 | | f2 | 6 | 3 | 3 | 0 | 192 | * | * | * | * | 1 | 1 | 0 | 0 | F4/A2 = 1152/3! = 192 | ||
| A1A1 | | 4 | 2 | 0 | 2 | * | 288 | * | * | * | 0 | 1 | 1 | 0 | F4/A1A1 = 1152/4 = 288 | |||
| B2 | | 4 | 0 | 4 | 0 | * | * | 144 | * | * | 1 | 0 | 0 | 1 | F4/B2 = 1152/8 = 144 | |||
| A1A1 | | 4 | 0 | 2 | 2 | * | * | * | 288 | * | 0 | 1 | 0 | 1 | F4/A1A1 = 1152/4 = 288 | |||
| A2 | | 3 | 0 | 0 | 3 | * | * | * | * | 192 | 0 | 0 | 1 | 1 | F4/A2 = 1152/3! = 192 | |||
| B3 | | f3 | 24 | 12 | 24 | 0 | 8 | 0 | 6 | 0 | 0 | 24 | * | * | * | F4/B3 = 1152/48 = 24 | ||
| A2A1 | | 12 | 6 | 6 | 6 | 2 | 3 | 0 | 3 | 0 | * | 96 | * | * | F4/A2A1 = 1152/3!/2 = 96 | |||
| 6 | 3 | 0 | 6 | 0 | 3 | 0 | 0 | 2 | * | * | 96 | * | |||||
| B3 | | 24 | 0 | 24 | 24 | 0 | 0 | 6 | 12 | 8 | * | * | * | 24 | F4/B3 = 1152/48 = 24 | |||
Omnitruncated 24-cell
x3x4x3x - gippic
| F4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ( ) | f0 | 1152 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Irr. {3,3} | F4 = 1152 | |
| A1 | | { } | f1 | 2 | 576 | * | * | * | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | ( )∨( )∨( ) | F4/A1 = 1152/2 = 576 |
| 2 | * | 576 | * | * | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | |||||
| 2 | * | * | 576 | * | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | |||||
| 2 | * | * | * | 576 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | |||||
| A2 | | t{3} | f2 | 6 | 3 | 3 | 0 | 0 | 192 | * | * | * | * | * | 1 | 1 | 0 | 0 | { } | F4/A2 = 1152/3! = 192 |
| A1A1 | | { }×{ } | 4 | 2 | 0 | 2 | 0 | * | 288 | * | * | * | * | 1 | 0 | 1 | 0 | F4/A1A1 = 1152/4 = 288 | ||
| 4 | 2 | 0 | 0 | 2 | * | * | 288 | * | * | * | 0 | 1 | 1 | 0 | |||||
| B2 | | t{4} | 8 | 0 | 4 | 4 | 0 | * | * | * | 144 | * | * | 1 | 0 | 0 | 1 | F4/B2 = 1152/8 = 144 | ||
| A1A1 | | { }×{ } | 4 | 0 | 2 | 0 | 2 | * | * | * | * | 288 | * | 0 | 1 | 0 | 1 | F4/A1A1 = 1152/4 = 288 | ||
| A2 | | t{3} | 6 | 0 | 0 | 3 | 3 | * | * | * | * | * | 192 | 0 | 0 | 1 | 1 | F4/A2 = 1152/3! = 192 | ||
| B3 | | tr{4,3} | f3 | 48 | 24 | 24 | 24 | 0 | 8 | 12 | 0 | 6 | 0 | 0 | 24 | * | * | * | ( ) | F4/B3 = 1152/48 = 24 |
| A2A1 | | t{3}×{ } | 12 | 6 | 6 | 0 | 6 | 2 | 0 | 3 | 0 | 3 | 0 | * | 96 | * | * | F4/A2A1 = 1152/3!/2 = 96 | ||
|
12 | 6 | 0 | 6 | 6 | 0 | 3 | 3 | 0 | 0 | 2 | * | * | 96 | * | |||||
| B3 | | tr{4,3} | 48 | 0 | 24 | 24 | 24 | 0 | 0 | 0 | 6 | 12 | 8 | * | * | * | 24 | F4/B3 = 1152/ 48 = 24 | ||
Snub 24-cell
Example: snub 24-cell
Omnitruncated tesseract
Example on omnitruncated tesseract. An omnitruncated 4-polytope will have 2^4-1 or 15 types of elements.
| B4 | |
k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ( ) | f0 | 384 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | {3,3} | B4 = 384 | |
| A1 | | { } | f1 | 2 | 192 | * | * | * | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | ( )∨( )∨( ) | B4/A1 = 192 |
| A1 | | { } | 2 | * | 192 | * | * | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | B4/A1 = 192 | ||
| A1 | | { } | 2 | * | * | 192 | * | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | B4/A1 = 192 | ||
| A1 | | { } | 2 | * | * | * | 192 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | B4/A1 = 192 | ||
| A2 | | {6} | f2 | 6 | 3 | 3 | 0 | 0 | 64 | * | * | * | * | * | 1 | 1 | 0 | 0 | { } | B4/A2 = 64 |
| A1A1 | | {4} | 4 | 2 | 0 | 2 | 0 | * | 96 | * | * | * | * | 1 | 0 | 1 | 0 | B4/A1A1 = 96 | ||
| A1A1 | | {4} | 4 | 2 | 0 | 0 | 2 | * | * | 96 | * | * | * | 0 | 1 | 1 | 0 | B4/A1A1 = 96 | ||
| A2 | | {6} | 6 | 0 | 3 | 3 | 0 | * | * | * | 64 | * | * | 1 | 0 | 0 | 1 | B4/A2 = 64 | ||
| A1A1 | | {4} | 4 | 0 | 2 | 0 | 2 | * | * | * | * | 96 | * | 0 | 1 | 0 | 1 | B4/A1A1 = 96 | ||
| B2 | | {8} | 8 | 0 | 0 | 4 | 4 | * | * | * | * | * | 48 | 0 | 0 | 1 | 1 | B4/B2 = 48 | ||
| A3 | | tr{3,3} | f3 | 24 | 12 | 12 | 12 | 0 | 4 | 6 | 0 | 4 | 0 | 0 | 16 | * | * | * | ( ) | B4/A3 = 16 |
| A2A1 | | {6}×{ } | 12 | 6 | 6 | 0 | 6 | 2 | 0 | 3 | 0 | 3 | 0 | * | 32 | * | * | B4/A2A1 = 32 | ||
| B2A1 | | {8}×{ } | 16 | 8 | 0 | 8 | 8 | 0 | 4 | 4 | 0 | 0 | 2 | * | * | 24 | * | B4/B2A1 = 24 | ||
| B3 | | tr{4,3} | 48 | 0 | 24 | 24 | 24 | 0 | 0 | 0 | 8 | 12 | 6 | * | * | * | 8 | B4/B3 = 8 | ||
600-cell
120-cell
5-polytopes
0_31
Example rectified 5-simplex
| A5 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | k-fig | notes | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A3A1 | | ( ) | f0 | 15 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | { }×{3,3} | | A5/A3A1 = 6!/4!/2 = 15 |
| A2A1 | | { } | f1 | 2 | 60 | 1 | 3 | 3 | 3 | 3 | 1 | ( )∨{3} | | A5/A2A1 = 6!/3!/2 = 60 |
| A2A2 | | r{3} | f2 | 3 | 3 | 20 | * | 3 | 0 | 3 | 0 | {3} | | A5/A2A2 = 6!/3!/3! =20 |
| A2A1 | | {3} | 3 | 3 | * | 60 | 1 | 2 | 2 | 1 | ( )×{ } | | A5/A2A1 = 6!/3!/2 = 60 | |
| A3A1 | | r{3,3} | f3 | 6 | 12 | 4 | 4 | 15 | * | 2 | 0 | { } | | A5/A3A1 = 6!/4!/2 = 15 |
| A3 | | {3,3} | 4 | 6 | 0 | 4 | * | 30 | 1 | 1 | ( )∨( ) | | A5/A3 = 6!/4! = 30 | |
| A4 | | r{3,3,3} | f4 | 10 | 30 | 10 | 20 | 5 | 5 | 6 | * | ( ) | | A5/A4 = 6!/5! = 6 |
| A4 | | {3,3,3} | 5 | 10 | 0 | 10 | 0 | 5 | * | 6 | | A5/A4 = 6!/5! = 6 | ||
0_22
Example birectified 5-simplex
| A5 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | k-fig | notes | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A2A2 | | ( ) | f0 | 20 | 9 | 9 | 9 | 3 | 9 | 3 | 3 | 3 | {3}×{3} | | A5/A2A2 = 6!/3!/3! = 20 |
| A1A1A1 | | { } | f1 | 2 | 90 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }∨{ } | | A5/A1A1A1 = 6!/8 = 90 |
| A2A1 | | {3} | f2 | 3 | 3 | 60 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | | A5/A2A1 = 6!/3!/2 = 60 |
| A2A1 | |
3 | 3 | * | 60 | 0 | 2 | 1 | 1 | 2 | ( )∨{ } | | |||
| A3A1 | | {3,3} | f3 | 4 | 6 | 4 | 0 | 15 | * | * | 2 | 0 | { } | | A5/A3A1 = 6!/4!/2 = 15 |
| A3 | | r{3,3} | 6 | 12 | 4 | 4 | * | 30 | * | 1 | 1 | | A5/A3 = 6!/4! = 30 | ||
| A3A1 | | {3,3} | 4 | 6 | 0 | 4 | * | * | 15 | 0 | 2 | | A5/A3A1 = 6!/4!/2 = 15 | ||
| A4 | | r{3,3,3} | f4 | 10 | 30 | 20 | 10 | 5 | 5 | 0 | 6 | * | ( ) | | A5/A4 = 6!/5! = 6 |
| A4 | |
10 | 30 | 10 | 20 | 0 | 5 | 5 | * | 6 | |||||
1_21
Example 5-demicube:
| D5 |     | k-face | fk | f0 | f1 | f2 | f3 | f4 | k-fig | notes | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A4 |  | ( ) | f0 | 16 | 10 | 30 | 10 | 20 | 5 | 5 | r{3,3,3} | D5/A4 = 16*5!/5! = 16 |
| A2A1A1 |  | { } | f1 | 2 | 80 | 6 | 3 | 6 | 3 | 2 | {3}×{ } | D5/A2A1A1 = 16*5!/3!/4 = 80 |
| A2A1 |  | {3} | f2 | 3 | 3 | 160 | 1 | 2 | 2 | 1 | { }∨( ) | D5/A2A1 = 16*5!/3!/2 = 160 |
| A3A1 | | h{4,3} | f3 | 4 | 6 | 4 | 40 | * | 2 | 0 | { } | D5/A3A1 = 16*5!/4!/2 = 40 |
| A3 | | {3,3} | 4 | 6 | 4 | * | 80 | 1 | 1 | D5/A3 = 16*5!/4! = 80 | ||
| D4 | | h{4,3,3} | f4 | 8 | 24 | 32 | 8 | 8 | 10 | * | ( ) | D5/D4 = 16*5!/8/4! = 10 |
| A4 | | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 16 | D5/A4 = 16*5!/5! = 16 | ||
6-polytopes
1_31
Example 6-demicube
| D6 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-fig | notes | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A4 | | ( ) | f0 | 32 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | r{3,3,3,3} | D6/A4 = 32*6!/5! = 32 |
| A3A1A1 | | { } | f1 | 2 | 240 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {}×{3,3} | D6/A3A1A1 = 32*6!/4!/4 = 240 |
| A3A2 | | {3} | f2 | 3 | 3 | 640 | 1 | 3 | 3 | 3 | 3 | 1 | {3}∨( ) | D6/A3A2 = 32*6!/4!/3! = 640 |
| A3A1 | | h{4,3} | f3 | 4 | 6 | 4 | 160 | * | 3 | 0 | 3 | 0 | {3} | D6/A3A1 = 32*6!/4!/2 = 160 |
| A3A2 | | {3,3} | 4 | 6 | 4 | * | 480 | 1 | 2 | 2 | 1 | {}∨( ) | D6/A3A2 = 32*6!/4!/3! = 480 | |
| D4A1 | | h{4,3,3} | f4 | 8 | 24 | 32 | 8 | 8 | 60 | * | 2 | 0 | { } | D6/D4A1 = 32*6!/8/4!/2 = 60 |
| A4 | | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 192 | 1 | 1 | D6/A4 = 32*6!/5! = 192 | ||
| D5 | | h{4,3,3,3} | f5 | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 12 | * | ( ) | D6/D5 = 32*6!/16/5! = 12 |
| A5 | | {3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 32 | D6/A5 = 32*6!/6! = 32 | ||
2_21
Example on 2_21 polytope:
| E6 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-fig | notes | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| D5 | | ( ) | f0 | 27 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | h{4,3,3,3} | E6/D5 = 51840/1920 = 27 |
| A4A1 | | { } | f1 | 2 | 216 | 10 | 30 | 20 | 10 | 5 | 5 | r{3,3,3} | E6/A4A1 = 51840/120/2 = 216 |
| A2A2A1 | | {3} | f2 | 3 | 3 | 720 | 6 | 6 | 3 | 2 | 3 | {3}×{ } | E6/A2A2A1 = 51840/6/6/2 = 720 |
| A3A1 | | {3,3} | f3 | 4 | 6 | 4 | 1080 | 2 | 1 | 1 | 2 | { }∨( ) | E6/A3A1 = 51840/24/2 = 1080 |
| A4 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 432 | * | 1 | 1 | { } | E6/A4 = 51840/120 = 432 |
| A4A1 | |
5 | 10 | 10 | 5 | * | 216 | 0 | 2 | E6/A4A1 = 51840/120/2 = 216 | |||
| A5 | | {3,3,3,3} | f5 | 6 | 15 | 20 | 15 | 6 | 0 | 72 | * | ( ) | E6/A5 = 51840/720 = 72 |
| D5 | | {3,3,3,4} | 10 | 40 | 80 | 80 | 16 | 16 | * | 27 | E6/D5 = 51840/1920 = 27 | ||
1_22
Example on 1_22 polytope:
| E6 |  | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-fig | notes | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A5 | | ( ) | f0 | 72 | 20 | 90 | 60 | 60 | 15 | 15 | 30 | 6 | 6 | r{3,3,3} |  | E6/A5 = 72*6!/6! = 72 |
| A2A2A1 |  | { } | f1 | 2 | 720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}×{3} | | E6/A2A2A1 = 72*6!/3!/3!/2 = 720 |
| A2A1A1 |  | {3} | f2 | 3 | 3 | 2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | { }∨{ } | | E6/A2A1A1 = 72*6!/3!/4 = 2160 |
| A3A1 |  | {3,3} | f3 | 4 | 6 | 4 | 1080 | * | 1 | 0 | 2 | 2 | 1 | { }∨( ) | | E6/A3A1 = 72*6!/4!/2 = 1080 |
 |
4 | 6 | 4 | * | 1080 | 0 | 1 | 2 | 1 | 2 | | |||||
| A4A1 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 0 | 216 | * | * | 2 | 0 | { } | | E6/A4A1 = 72*6!/5!/2 = 216 |
|
5 | 10 | 10 | 0 | 5 | * | 216 | * | 0 | 2 | ||||||
| D4 | | {3,3,4} | 8 | 24 | 32 | 8 | 8 | * | * | 270 | 1 | 1 | | E6/D4 = 72*6!/8/4! = 270 | ||
| D5 | | h{4,3,3,3} | f5 | 16 | 80 | 160 | 80 | 40 | 16 | 0 | 10 | 27 | * | ( ) | | E6/D5 = 72*6!/16/5! = 27 |
| 16 | 80 | 160 | 40 | 80 | 0 | 16 | 10 | * | 27 | ||||||
0_221
Example Rectified 1_22 polytope
| E6 |  | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-fig | notes | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A2A2A1 | | ( ) | f0 | 720 | 18 | 18 | 18 | 9 | 6 | 18 | 9 | 6 | 9 | 6 | 3 | 6 | 9 | 3 | 2 | 3 | 3 | {3}×{3}×{ } | E6/A2A2A1 = 72*6!/3!/3!/2 = 720 |
| A1A1A1 | | { } | f1 | 2 | 6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }∨{ }∨( ) | E6/A1A1A1 = 72*6!/8 = 6480 |
| A2A1 | | {3} | f2 | 3 | 3 | 4320 | * | * | 1 | 2 | 1 | 0 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 2 | 1 | Sphenoid | E6/A2A1 = 72*6!/3!/2 = 4320 |
|
3 | 3 | * | 4320 | * | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | |||||
| A2A1A1 | |
3 | 3 | * | * | 2160 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 0 | 2 | 2 | { }∨{ } | E6/A2A1A1 = 72*6!/3!/4 = 2160 | ||
| A2A1 | | {3,3} | f3 | 4 | 6 | 4 | 0 | 0 | 1080 | * | * | * | * | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }∨( ) | E6/A2A1 = 72*6!/3!/2 = 1080 |
| A3 | | r{3,3} | 6 | 12 | 4 | 4 | 0 | * | 2160 | * | * | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | {3} | E6/A3 = 72*6!/4! = 2160 | |
| A3A1 | |
6 | 12 | 4 | 0 | 4 | * | * | 1080 | * | * | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | { }∨( ) | E6/A3A1 = 72*6!/4!/2 = 1080 | ||
| {3,3} | 4 | 6 | 0 | 4 | 0 | * | * | * | 1080 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | ||||
| r{3,3} | 6 | 12 | 0 | 4 | 4 | * | * | * | * | 1080 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | ||||
| A4 | | r{3,3,3} | f4 | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 432 | * | * | * | * | 1 | 1 | 0 | { } | E6/A4 = 72*6!/5! = 432 |
| A4A1 | |
10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 216 | * | * | * | 0 | 2 | 0 | E6/A4A1 = 72*6!/5!/2 = 216 | |||
| A4 | |
10 | 30 | 10 | 20 | 0 | 0 | 5 | 0 | 5 | 0 | * | * | 432 | * | * | 1 | 0 | 1 | E6/A4 = 72*6!/5! = 432 | |||
| D4 | | h{4,3,3} | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 0 | 8 | * | * | * | 270 | * | 0 | 1 | 1 | E6/D4 = 72*6!/8/4! = 270 | ||
| A4A1 | | r{3,3,3} | 10 | 30 | 0 | 20 | 10 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 216 | 0 | 0 | 2 | E6/A4A1 = 72*6!/5!/2 = 216 | ||
| A5 | | 2r{3,3,3,3} | f5 | 20 | 90 | 60 | 60 | 0 | 15 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | 72 | * | * | ( ) | E6/A5 = 72*6!/6! = 72 |
| D5 | | rh{4,3,3,3} | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 0 | 40 | 16 | 16 | 0 | 10 | 0 | * | 27 | * | E6/D5 = 72*6!/16/5! = 27 | ||
|
80 | 480 | 160 | 320 | 160 | 0 | 80 | 40 | 80 | 80 | 0 | 0 | 16 | 10 | 16 | * | * | 27 | |||||
Omnitruncated 6-simplex
Example: Omnitruncated 6-simplex BIG TEST!
| A6 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | k-fig | notes | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5040 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | ||||||
| 2 | 5040 | * | * | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 0 | 0 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 0 | 1 | 2 | 2 | ||||||
| 2 | * | 5040 | * | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 1 | 0 | 1 | 0 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 0 | 1 | 2 | 1 | 2 | ||||||
| 2 | * | * | 5040 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 2 | 2 | 1 | ||||||
| 6 | 3 | 3 | 0 | 1680 | * | * | * | * | * | * | * | * | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 2 | ||||||
| 4 | 2 | 0 | 2 | * | 2520 | * | * | * | * | * | * | * | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 2 | 1 | ||||||
| 4 | 2 | 0 | 2 | * | * | 2520 | * | * | * | * | * | * | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 2 | 1 | ||||||
| 4 | 2 | 2 | 0 | * | * | * | 2520 | * | * | * | * | * | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 2 | ||||||
| 4 | 4 | 0 | 0 | * | * | * | * | 1260 | * | * | * | * | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 1 | 0 | 1 | 0 | 0 | 2 | 2 | ||||||
| 6 | 0 | 3 | 3 | * | * | * | * | * | 1680 | * | * | * | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 2 | 1 | 1 | ||||||
| 4 | 0 | 2 | 2 | * | * | * | * | * | * | 2520 | * | * | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 2 | 1 | 1 | ||||||
| 4 | 0 | 4 | 0 | * | * | * | * | * | * | * | 1260 | * | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 0 | 1 | 2 | 0 | 2 | ||||||
| 6 | 0 | 0 | 6 | * | * | * | * | * | * | * | * | 840 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | 2 | 2 | 0 | ||||||
| 24 | 12 | 12 | 12 | 4 | 6 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 420 | * | * | * | * | * | * | * | * | * | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | ||||||
| 12 | 6 | 6 | 6 | 2 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 0 | * | 840 | * | * | * | * | * | * | * | * | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | ||||||
| 12 | 6 | 12 | 0 | 2 | 0 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | * | * | 840 | * | * | * | * | * | * | * | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 2 | ||||||
| 12 | 12 | 6 | 0 | 2 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | * | * | * | 840 | * | * | * | * | * | * | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 2 | ||||||
| 12 | 6 | 0 | 12 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 2 | * | * | * | * | 840 | * | * | * | * | * | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 2 | 0 | ||||||
| 8 | 4 | 4 | 4 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | * | * | * | * | * | 1260 | * | * | * | * | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | ||||||
| 8 | 8 | 0 | 4 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | * | * | * | * | * | * | 1260 | * | * | * | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 2 | 1 | ||||||
| 12 | 6 | 6 | 6 | 0 | 0 | 3 | 3 | 0 | 2 | 0 | 0 | 0 | * | * | * | * | * | * | * | 840 | * | * | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | ||||||
| 24 | 0 | 12 | 24 | 0 | 0 | 0 | 0 | 0 | 4 | 6 | 0 | 4 | * | * | * | * | * | * | * | * | 420 | * | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 2 | 1 | 0 | ||||||
| 12 | 0 | 12 | 6 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 3 | 0 | * | * | * | * | * | * | * | * | * | 8 | 40 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 2 | 0 | 1 | |||||
| 120 | 60 | 60 | 120 | 20 | 30 | 30 | 0 | 0 | 20 | 30 | 0 | 20 | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 5 | 0 | 84 | * | * | * | * | * | * | * | * | 1 | 1 | 0 | ||||||
| 48 | 24 | 48 | 24 | 8 | 12 | 0 | 12 | 0 | 8 | 12 | 12 | 0 | 2 | 0 | 4 | 0 | 0 | 6 | 0 | 0 | 0 | 4 | * | 210 | * | * | * | * | * | * | * | 1 | 0 | 1 | ||||||
| 48 | 48 | 24 | 24 | 8 | 12 | 12 | 12 | 12 | 8 | 0 | 0 | 0 | 2 | 0 | 0 | 4 | 0 | 0 | 6 | 4 | 0 | 0 | * | * | 210 | * | * | * | * | * | * | 0 | 1 | 1 | ||||||
| 36 | 18 | 36 | 18 | 6 | 0 | 9 | 9 | 0 | 6 | 9 | 9 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 3 | * | * | * | 280 | * | * | * | * | * | 1 | 0 | 1 | ||||||
| 24 | 24 | 12 | 12 | 4 | 6 | 6 | 6 | 6 | 0 | 6 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 3 | 3 | 0 | 0 | 0 | * | * | * | * | 420 | * | * | * | * | 0 | 1 | 1 | ||||||
| 36 | 36 | 36 | 0 | 12 | 0 | 0 | 18 | 9 | 0 | 0 | 9 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | * | * | * | * | * | 140 | * | * | * | 0 | 0 | 2 | ||||||
| 48 | 24 | 24 | 48 | 0 | 12 | 12 | 12 | 0 | 8 | 12 | 0 | 8 | 0 | 0 | 0 | 0 | 4 | 6 | 0 | 4 | 2 | 0 | * | * | * | * | * | * | 210 | * | * | 1 | 1 | 0 | ||||||
| 24 | 24 | 0 | 24 | 0 | 12 | 12 | 0 | 6 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 6 | 0 | 0 | 0 | * | * | * | * | * | * | * | 210 | * | 0 | 2 | 0 | ||||||
| 120 | 0 | 120 | 120 | 0 | 0 | 0 | 0 | 0 | 40 | 60 | 30 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 20 | * | * | * | * | * | * | * | * | 42 | 2 | 0 | 0 | ||||||
| 720 | 360 | 720 | 720 | 120 | 180 | 180 | 180 | 0 | 240 | 360 | 180 | 120 | 30 | 60 | 60 | 0 | 60 | 90 | 0 | 60 | 60 1 | 20 | 6 | 15 | 0 | 20 | 0 | 0 | 15 | 0 | 6 | 14 | * | * | ||||||
| 240 | 240 | 120 | 240 | 40 | 120 | 120 | 60 | 60 | 40 | 60 | 0 | 40 | 10 | 20 | 0 | 20 | 40 | 30 | 60 | 20 | 10 | 0 | 2 | 0 | 5 | 0 | 10 | 0 | 5 | 10 | 0 | * | 42 | * | ||||||
| 144 | 144 | 144 | 72 | 48 | 36 | 36 | 72 | 36 | 24 | 36 | 36 | 0 | 6 | 12 | 24 | 24 | 0 | 18 | 18 | 12 | 0 | 12 | 0 | 3 | 3 | 4 | 6 | 4 | 0 | 0 | 0 | * | * | 70 | ||||||
7-polytopes
1_41
Example on 7-demicube:
| D7 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | k-fig | notes | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A6 | | ( ) | f0 | 64 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | r{3,3,3,3,3} | D7/A6 = 64*7!/7! = 64 |
| A4A1A1 | | { } | f1 | 2 | 672 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | { }×{3,3,3} | D7/A4A1A1 = 64*7!/5!/4 = 672 |
| A3A2 | | {3} | f2 | 3 | 3 | 2240 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}∨( ) | D7/A3A2 = 64*7!/4!/3! = 2240 |
| A3A3 | | h{4,3} | f3 | 4 | 6 | 4 | 560 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | D7/A3A3 = 64*7!/4!/4! = 560 |
| A3A2 | | {3,3} | 4 | 6 | 4 | * | 2240 | 1 | 3 | 3 | 3 | 3 | 1 | {3}∨( ) | D7/A3A2 = 64*7!/4!/3! = 2240 | |
| D4A2 | | h{4,3,3} | f4 | 8 | 24 | 32 | 8 | 8 | 280 | * | 3 | 0 | 3 | 0 | {3} | D7/D4A2 = 64*7!/8/4!/2 = 280 |
| A4A1 | | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 1344 | 1 | 2 | 2 | 1 | { }∨( ) | D7/A4A1 = 64*7!/5!/2 = 1344 | |
| D5A1 | | h{4,3,3,3} | f5 | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 84 | * | 2 | 0 | { } | D7/D5A1 = 64*7!/16/5!/2 = 84 |
| A5 | | {3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 448 | 1 | 1 | D7/A5 = 64*7!/6! = 448 | ||
| D6 | | h{4,3,3,3,3} | f6 | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 14 | * | ( ) | D7/D6 = 64*7!/32/6! = 14 |
| A6 | | {3,3,3,3,3} | 7 | 21 | 35 | 0 | 35 | 0 | 21 | 0 | 7 | * | 64 | D7/A6 = 64*7!/7! = 64 | ||
3_21
Example on 3_21 polytope:
| E7 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | k-fig | notes | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| E6 | | ( ) | f0 | 56 | 27 | 216 | 720 | 1080 | 432 | 216 | 72 | 27 | 221 | E7/E6 = 72x8!/72x6! = 56 |
| D5A1 | | { } | f1 | 2 | 756 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 5-demicube | E7/D5A1 = 72x8!/16/5!/2 = 756 |
| A4A2 | | {3} | f2 | 3 | 3 | 4032 | 10 | 30 | 20 | 10 | 5 | 5 | rectified 5-cell | E7/A4A2 = 72x8!/5!/2 = 4032 |
| A3A2A1 | | {3,3} | f3 | 4 | 6 | 4 | 10080 | 6 | 6 | 3 | 2 | 3 | triangular prism | E7/A3A2A1 = 72x8!/4!/3!/2 = 10080 |
| A4A1 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 12096 | 2 | 1 | 1 | 2 | isosceles triangle | E7/A4A1 = 72x8!/5!/2 = 12096 |
| A5A1 | | {3,3,3,3} | f5 | 6 | 15 | 20 | 15 | 6 | 4032 | * | 1 | 1 | { } | E7/A5A1 = 72x8!/6!/2 = 4032 |
| A5 | |
6 | 15 | 20 | 15 | 6 | * | 2016 | 0 | 2 | E7/A5 = 72x8!/6! = 2016 | |||
| A6 | | {3,3,3,3,3} | f6 | 7 | 21 | 35 | 35 | 21 | 10 | 0 | 576 | * | ( ) | E7/A6 = 72x8!/7! = 576 |
| D6 | | {3,3,3,3,4} | 12 | 60 | 160 | 240 | 192 | 32 | 32 | * | 126 | E7/D6 = 72x8!/32/6! = 126 | ||
2_31
Example on 2_31 polytope:
| E7 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | k-fig | notes | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| D6 | | ( ) | f0 | 126 | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 6-demicube | E7/D6 = 72x8!/32/6! = 126 |
| A5A1 | | { } | f1 | 2 | 2016 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | rectified 5-simplex | E7/A5A1 = 72x8!/6!/2 = 2016 |
| A3A2A1 | | {3} | f2 | 3 | 3 | 10080 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | tetrahedral prism | E7/A3A2A1 = 72x8!/4!/3!/2 = 10080 |
| A3A2 | | {3,3} | f3 | 4 | 6 | 4 | 20160 | 1 | 3 | 3 | 3 | 3 | 1 | tetrahedron | E7/A3A2 = 72x8!/4!/3! = 20160 |
| A4A2 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 4032 | * | 3 | 0 | 3 | 0 | {3} | E7/A4A2 = 72x8!/5!/3! = 4032 |
| A4A1 | |
5 | 10 | 10 | 5 | * | 12096 | 1 | 2 | 2 | 1 | Isosceles triangle | E7/A4A1 = 72x8!/5!/2 = 12096 | ||
| D5A1 | | {3,3,3,4} | f5 | 10 | 40 | 80 | 80 | 16 | 16 | 756 | * | 2 | 0 | { } | E7/D5A1 = 72x8!/32/5! = 756 |
| A5 | | {3,3,3,3} | 6 | 15 | 20 | 15 | 0 | 6 | * | 4032 | 1 | 1 | E7/A5 = 72x8!/6! = 72*8*7 = 4032 | ||
| E6 | | {3,3,32,1} | f6 | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 56 | * | ( ) | E7/E6 = 72x8!/72x6! = 8*7 = 56 |
| A6 | | {3,3,3,3,3} | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 576 | E7/A6 = 72x8!/7! = 72×8 = 576 | ||
1_32
Example on 1_32 polytope:
| E7 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | k-fig | notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A6 | | ( ) | f0 | 576 | 35 | 210 | 140 | 210 | 35 | 105 | 105 | 21 | 42 | 21 | 7 | 7 | 2r{3,3,3,3,3} | E7/A6 = 72*8!/7! = 576 |
| A3A2A1 | | { } | f1 | 2 | 10080 | 12 | 12 | 18 | 4 | 12 | 12 | 6 | 12 | 3 | 4 | 3 | {3,3}×{3} | E7/A3A2A1 = 72*8!/4!/3!/2 = 10080 |
| A2A2A1 | | {3} | f2 | 3 | 3 | 40320 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | { }∨{3} | E7/A2A2A1 = 72*8!/3!/3!/2 = 40320 |
| A3A2 | | {3,3} | f3 | 4 | 6 | 4 | 20160 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}∨( ) | E7/A3A2 = 72*8!/4!/3! = 20160 |
| A3A1A1 | |
4 | 6 | 4 | * | 30240 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | Phyllic disphenoid | E7/A3A1A1 = 72*8!/4!/4 = 30240 | ||
| A4A2 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 0 | 4032 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | E7/A4A2 = 72*8!/5!/3! = 4032 |
| D4A1 | | {3,3,4} | 8 | 24 | 32 | 8 | 8 | * | 7560 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | E7/D4A1 = 72*8!/8/4!/2 = 7560 | |
| A4A1 | | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | * | 12096 | 0 | 2 | 1 | 1 | 2 | E7/A4A1 = 72*8!/5!/2 = 12096 | ||
| D5A1 | | h{4,3,3,3} | f5 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 756 | * | * | 2 | 0 | { } | E7/D5A1 = 72*8!/16/5!/2 = 756 |
| D5 | |
16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 1512 | * | 1 | 1 | E7/D5 = 72*8!/16/5! = 1512 | |||
| A5A1 | | {3,3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 2016 | 0 | 2 | E7/A5A1 = 72*8!/6!/2 = 2016 | ||
| E6 | | {3,32,2} | f6 | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 56 | * | ( ) | E7/E6 = 72*8!/72/6! = 56 |
| D6 | | h{4,3,3,3,3} | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | * | 126 | E7/D6 = 72*8!/32/6! = 126 | ||
0_321
Example on rectified 1_32 polytope:
| E7 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | k-fig | notes | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A3A2A1 | | ( ) | f0 | 10080 | 24 | 24 | 12 | 36 | 8 | 12 | 36 | 18 | 24 | 4 | 12 | 18 | 24 | 12 | 6 | 6 | 8 | 12 | 6 | 3 | 4 | 2 | 3 | {3,3}×{3}×{ } | E7/A3A2A1 = 72*8!/4!/3!/2 = 10080 |
| A2A1A1 | | { } | f1 | 2 | 120960 | 2 | 1 | 3 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | ( )∨{3}∨{ } | E7/A2A1A1 = 72*8!/3!/4 = 120960 |
| A2A2 | | 01 | f2 | 3 | 3 | 80640 | * | * | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | {3}∨( )∨( ) | E7/A2A2 = 72*8!/3!/3! = 80640 |
| A2A2A1 | |
3 | 3 | * | 40320 | * | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | {3}∨{ } | E7/A2A2A1 = 72*8!/3!/3!/2 = 40320 | ||
| A2A1A1 | |
3 | 3 | * | * | 120960 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | { }∨{ }∨( ) | E7/A2A1A1 = 72*8!/3!/4 = 120960 | ||
| A3A2 | | 02 | f3 | 4 | 6 | 4 | 0 | 0 | 20160 | * | * | * | * | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | {3}∨( ) | E7/A3A2 = 72*8!/4!/3! = 20160 |
| 011 | 6 | 12 | 4 | 4 | 0 | * | 20160 | * | * | * | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | ||||
| A3A1 | |
6 | 12 | 4 | 0 | 4 | * | * | 60480 | * | * | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | Sphenoid | E7/A3A1 = 72*8!/4!/2 = 60480 | ||
| A3A1A1 | |
6 | 12 | 0 | 4 | 4 | * | * | * | 30240 | * | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | { }∨{ } | E7/A3A1A1 = 72*8!/4!/2/2 = 30240 | ||
| A3A1 | | 02 | 4 | 6 | 0 | 0 | 4 | * | * | * | * | 60480 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | Sphenoid | E7/A3A1 = 72*8!/4!/2 = 60480 | |
| A4A2 | | 021 | f4 | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 4032 | * | * | * | * | * | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} | E7/A4A2 = 72*8!/5!/3! = 4032 |
| A4A1 | |
10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 12096 | * | * | * | * | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | { }∨() | E7/A4A1 = 72*8!/5!/2 = 12096 | ||
| D4A1 | | 0111 | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | * | * | 7560 | * | * | * | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | E7/D4A1 = 72*8!/8/4!/2 = 7560 | ||
| A4 | | 021 | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | * | * | * | 24192 | * | * | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )∨( )∨( ) | E7/A4 = 72*8!/5! = 34192 | |
| A4A1 | |
10 | 30 | 0 | 10 | 20 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 12096 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }∨() | E7/A4A1 = 72*8!/5!/2 = 12096 | ||
| 03 | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | * | * | * | * | * | 12096 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | ||||
| D5A1 | | 0211 | f5 | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | 756 | * | * | * | * | 2 | 0 | 0 | { } | E7/D5A1 = 72*8!/16/5!/2 = 756 |
| A5 | | 022 | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | * | 4032 | * | * | * | 1 | 1 | 0 | E7/A5 = 72*8!/6! = 4032 | ||
| D5 | | 0211 | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | * | * | 1512 | * | * | 1 | 0 | 1 | E7/D5 = 72*8!/16/5! = 1512 | ||
| A5 | | 031 | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | * | * | * | 4032 | * | 0 | 1 | 1 | E7/A5 = 72*8!/6! = 4032 | ||
| A5A1 | |
15 | 60 | 0 | 20 | 60 | 0 | 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 | 6 | 6 | * | * | * | * | 2016 | 0 | 0 | 2 | E7/A5A1 = 72*8!/6!/2 = 2016 | |||
| E6 | | 0221 | f6 | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | 56 | * | * | ( ) | E7/E6 = 72*8!/72/6! = 56 |
| A6 | | 032 | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | * | 576 | * | E7/A6 = 72*8!/7! = 576 | ||
| D6 | | 0311 | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | * | * | 126 | E7/D6 = 72*8!/32/6! = 126 | ||
8-polytopes
8-cube
Example on 8-cube. A regular n-polytope will have n types of elements, one for each dimension.
| B8 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-fig | notes |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A7 | | ( ) | f0 | 256 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | {3,3,3,3,3,3} | B8/A7 = 2^8*8!/8! = 256 |
| A6A1 | | { } | f1 | 2 | 1024 | 7 | 21 | 35 | 35 | 21 | 7 | {3,3,3,3,3} | B8/A6A1 = 2^8*8!/7!/2 = 1024 |
| A5B2 | | {4} | f2 | 4 | 4 | 1792 | 6 | 15 | 20 | 15 | 6 | {3,3,3,3} | B8/A5B2 = 2^8*8!/6!/4/2 = 1792 |
| A4B3 | | {4,3} | f3 | 8 | 12 | 6 | 1792 | 5 | 10 | 10 | 5 | {3,3,3} | B8/A4B3 = 2^8*8!/5!/8/3! = 1792 |
| A3B4 | | {4,3,3} | f4 | 16 | 32 | 24 | 8 | 1120 | 4 | 6 | 4 | {3,3} | B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120 |
| A2B5 | | {4,3,3,3} | f5 | 32 | 80 | 80 | 40 | 10 | 448 | 3 | 3 | {3} | B8/A2B5 = 2^8*8!/3!/2^5/5! = 448 |
| A1B6 | | {4,3,3,3,3} | f6 | 64 | 192 | 240 | 160 | 60 | 12 | 112 | 2 | { } | B8/A1B6 = 2^8*8!/2/2^6/6!= 112 |
| B7 | | {4,3,3,3,3,3} | f7 | 128 | 448 | 672 | 560 | 280 | 84 | 14 | 16 | ( ) | B8/B7 = 2^8*8!/2^7/7! = 16 |
8-orthoplex
| B8 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-fig | notes |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B7 | | ( ) | f0 | 16 | 14 | 84 | 280 | 560 | 672 | 448 | 128 | {3,3,3,3,3,4} | B8/B7 = 2^8*8!/2^7/7! = 16 |
| A1B6 | | { } | f1 | 2 | 112 | 12 | 60 | 160 | 240 | 192 | 64 | {3,3,3,3,4} | B8/A1B6 = 2^8*8!/2/2^6/6! = 112 |
| A2B5 | | {3} | f2 | 3 | 3 | 448 | 10 | 40 | 80 | 80 | 32 | {3,3,3,4} | B8/A2B5 = 2^8*8!/3!/2^5/5! = 448 |
| A3B4 | | {3,3} | f3 | 4 | 6 | 4 | 1120 | 8 | 24 | 32 | 16 | {3,3,4} | B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120 |
| A4B3 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 1792 | 6 | 12 | 8 | {3,4} | B8/A4B3 = 2^8*8!/5!/8/3! = 1792 |
| A5B2 | | {3,3,3,3} | f5 | 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4 | {4} | B8/A5B2 = 2^8*8!/6!/4/2 = 1792 |
| A6A1 | | {3,3,3,3,3} | f6 | 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2 | { } | B8/A6A1 = 2^8*8!/7!/2 = 1024 |
| A7 | | {3,3,3,3,3,3} | f7 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256 | ( ) | B8/A7 = 2^8*8!/8! = 256 |
4_21
Example on 4_21 polytope:
| E8 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-fig | notes | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| E7 | | ( ) | f0 | 240 | 56 | 756 | 4032 | 10080 | 12096 | 4032 | 2016 | 576 | 126 | 321 | E8/E7 = 192x10!/72x8! = 240 |
| A1E6 | | { } | f1 | 2 | 6720 | 27 | 216 | 720 | 1080 | 432 | 216 | 72 | 27 | 221 | E8/A1E6 = 192x10!/2/72x6! = 6720 |
| A2D5 | | {3} | f2 | 3 | 3 | 60480 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 121 | E8/A2D5 = 192x10!/6/2^4/5! = 60480 |
| A3A4 | | {3,3} | f3 | 4 | 6 | 4 | 241920 | 10 | 30 | 20 | 10 | 5 | 5 | 021 | E8/A3A4 = 192x10!/4!/5! = 241920 |
| A4A2A1 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 483840 | 6 | 6 | 3 | 2 | 3 | {3}×{ } | E8/A4A2A1 = 192x10!/5!/3!/2 = 483840 |
| A5A1 | | {3,3,3,3} | f5 | 6 | 15 | 20 | 15 | 6 | 483840 | 2 | 1 | 1 | 2 | { }∨( ) | E8/A5A1 = 192x10!/6!/2 = 483840 |
| A6 | | {3,3,3,3,3} | f6 | 7 | 21 | 35 | 35 | 21 | 7 | 138240 | * | 1 | 1 | { } | E8/A6 = 192x10!/7! = 138240 |
| A6A1 | |
7 | 21 | 35 | 35 | 21 | 7 | * | 69120 | 0 | 2 | E8/A6A1 = 192x10!/7!/2 = 69120 | |||
| A7 | | {3,3,3,3,3,3} | f7 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 0 | 17280 | * | ( ) | E8/A7 = 192x10!/8! = 17280 |
| D7 | | {3,3,3,3,3,4} | 14 | 84 | 280 | 560 | 672 | 448 | 64 | 64 | * | 2160 | E8/D7 = 192x10!/2^6/7! = 2160 | ||
2_41
Example on 2_41 polytope:
| E8 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-fig | notes | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| D7 | | ( ) | f0 | 2160 | 64 | 672 | 2240 | 560 | 2240 | 280 | 1344 | 84 | 448 | 14 | 64 | h{4,3,3,3,3,3} | 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} | E8/A6A1 = 192*10!/7!/2 = 69120 |
| A4A2A1 | | {3} | f2 | 3 | 3 | 483840 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | {}×{3,3,3} | E8/A4A2A1 = 192*10!/5!/3!/2 = 483840 |
| A3A3 | | {3,3} | f3 | 4 | 6 | 4 | 1209600 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}∨( ) | E8/A3A3 = 192*10!/4!/4! = 1209600 |
| A4A3 | | {3,3,3} | f4 | 5 | 10 | 10 | 5 | 241920 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | E8/A4A3 = 192*10!/5!/4! = 241920 |
| A4A2 | |
5 | 10 | 10 | 5 | * | 967680 | 1 | 3 | 3 | 3 | 3 | 1 | {3}∨( ) | E8/A4A2 = 192*10!/5!/3! = 967680 | ||
| D5A2 | | {3,3,31,1} | f5 | 10 | 40 | 80 | 80 | 16 | 16 | 60480 | * | 3 | 0 | 3 | 0 | {3} | E8/D5A2 = 192*10!/16/5!/2 = 40480 |
| A5A1 | | {3,3,3,3} | 6 | 15 | 20 | 15 | 0 | 6 | * | 483840 | 1 | 2 | 2 | 1 | { }∨( ) | E8/A5A1 = 192*10!/6!/2 = 483840 | |
| E6A1 | | {3,3,32,1} | 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} | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 138240 | 1 | 1 | E8/A6 = 192*10!/7! = 138240 | ||
| E7 | | {3,3,33,1} | 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} | 8 | 28 | 56 | 70 | 0 | 56 | 0 | 28 | 0 | 8 | * | 17280 | E8/A7 = 192*10!/8! = 17280 | ||
1_42
Example on 1_42 polytope:
| E8 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-fig | notes | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A7 | | ( ) | f0 | 17280 | 56 | 420 | 280 | 560 | 70 | 280 | 420 | 56 | 168 | 168 | 28 | 56 | 28 | 8 | 8 | 2r{36} | [note 1] |
| A4A2A1 | | { } | f1 | 2 | 483840 | 15 | 15 | 30 | 5 | 30 | 30 | 10 | 30 | 15 | 10 | 15 | 3 | 5 | 3 | {3}×{3,3,3} | [note 2] |
| A3A2A1 | | {3} | f2 | 3 | 3 | 2419200 | 2 | 4 | 1 | 8 | 6 | 4 | 12 | 4 | 6 | 8 | 1 | 4 | 2 | {3.3}∨{ } | [note 3] |
| A3A3 | | {3,3} | f3 | 4 | 6 | 4 | 1209600 | * | 1 | 4 | 0 | 4 | 6 | 0 | 6 | 4 | 0 | 4 | 1 | {3,3}∨( ) | [note 4] |
| A3A2A1 | |
4 | 6 | 4 | * | 2419200 | 0 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | {3}∨{ } | [note 5] | ||
| A4A3 | | {3,32,0} | f4 | 5 | 10 | 10 | 5 | 0 | 241920 | * | * | 4 | 0 | 0 | 6 | 0 | 0 | 4 | 0 | {3,3} | [note 6] |
| D4A2 | | {3,31,1} | 8 | 24 | 32 | 8 | 8 | * | 604800 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}∨( ) | [note 7] | |
| A4A1A1 | | {3,32,0} | 5 | 10 | 10 | 0 | 5 | * | * | 1451520 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }∨{ } | [note 8] | |
| D5A2 | | {3,32,1} | f5 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 60480 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | [note 9] |
| D5A1 | |
16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 181440 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | [note 10] | ||
| A5A1 | | {3,33,0} | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 483840 | 0 | 2 | 1 | 1 | 2 | [note 11] | ||
| E6A1 | | {3,32,2} | f6 | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 6720 | * | * | 2 | 0 | { } | [note 12] |
| D6 | | {3,33,1} | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | * | 30240 | * | 1 | 1 | [note 13] | ||
| A6A1 | | {3,34,0} | 7 | 21 | 35 | 0 | 35 | 0 | 0 | 21 | 0 | 0 | 7 | * | * | 69120 | 0 | 2 | [note 14] | ||
| E7 | | {3,33,2} | f7 | 576 | 10080 | 40320 | 20160 | 30240 | 4032 | 7560 | 12096 | 756 | 1512 | 2016 | 56 | 126 | 0 | 240 | * | ( ) | [note 15] |
| D7 | | {3,34,1} | 64 | 672 | 2240 | 560 | 2240 | 0 | 280 | 1344 | 0 | 84 | 448 | 0 | 14 | 64 | * | 2160 | [note 16] | ||
Notes
- E8/A7 = 192*10!/8! = 17280
- E8/A4A2A1 = 192*10!/5!/4 = 483840
- E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
- E8/A3A3 = 192*10!/4!/4! = 1209600
- E8/A3A2A1 = 192*10!/4!/3!/2 = 2419200
- E8/A4A3 = 192*10!/4!/4! = 241920
- E8/D4A2 = 192*10!/8/4!/3! = 604800
- E8/A4A1A1 = 192*10!/5!/4 = 1451520
- E8/D5A2 = 192*10!/16/5!/3! = 40480
- E8/D5A1 = 192*10!/16/5!/2 = 181440
- E8/A5A1 = 192*10!/6!/2 = 483840
- E8/E6A1 = 192*10!/72/6!/2 = 6720
- E8/D6 = 192*10!/32/6! = 30240
- E8/A6A1 = 192*10!/7!/2 = 69120
- E8/E7 = 192*10!/72/8! = 240
- E8/D7 = 192*10!/64/7! = 2160
0_421
Example on rectified 1_42 polytope:
| E8 | | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-fig | ||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A4A2A1 | | ( ) | f0 | 483840 | 30 | 30 | 15 | 60 | 10 | 15 | 60 | 30 | 60 | 5 | 20 | 30 | 60 | 30 | 30 | 10 | 20 | 30 | 30 | 15 | 6 | 10 | 10 | 15 | 6 | 3 | 5 | 2 | 3 | {3,3,3}×{3,3}×{} |
| A3A1A1 | | { } | f1 | 2 | 7257600 | 2 | 1 | 4 | 1 | 2 | 8 | 4 | 6 | 1 | 4 | 8 | 12 | 6 | 4 | 4 | 6 | 12 | 8 | 4 | 1 | 6 | 4 | 8 | 2 | 1 | 4 | 1 | 2 | |
| A3A2 | | {3} | f2 | 3 | 3 | 4838400 | * | * | 1 | 1 | 4 | 0 | 0 | 1 | 4 | 4 | 6 | 0 | 0 | 4 | 6 | 6 | 4 | 0 | 0 | 6 | 4 | 4 | 1 | 0 | 4 | 1 | 1 | |
| A3A2A1 | |
3 | 3 | * | 2419200 | * | 0 | 2 | 0 | 4 | 0 | 1 | 0 | 8 | 0 | 6 | 0 | 4 | 0 | 12 | 0 | 4 | 0 | 6 | 0 | 8 | 0 | 1 | 4 | 0 | 2 | |||
| A2A2A1 | |
3 | 3 | * | * | 9676800 | 0 | 0 | 2 | 1 | 3 | 0 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | |||
| A3A3 | | 0200 | f3 | 4 | 6 | 4 | 0 | 0 | 1209600 | * | * | * | * | 1 | 4 | 0 | 0 | 0 | 0 | 4 | 6 | 0 | 0 | 0 | 0 | 6 | 4 | 0 | 0 | 0 | 4 | 1 | 0 | |
| 0110 | 6 | 12 | 4 | 4 | 0 | * | 1209600 | * | * | * | 1 | 0 | 4 | 0 | 0 | 0 | 4 | 0 | 6 | 0 | 0 | 0 | 6 | 0 | 4 | 0 | 0 | 4 | 0 | 1 | |||
| A3A2 | |
6 | 12 | 4 | 0 | 4 | * | * | 4838400 | * | * | 0 | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | |||
| A3A2A1 | |
6 | 12 | 0 | 4 | 4 | * | * | * | 2419200 | * | 0 | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | |||
| A3A1A1 | | 0200 | 4 | 6 | 0 | 0 | 4 | * | * | * | * | 7257600 | 0 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | ||
| A4A3 | | 0210 | f4 | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 241920 | * | * | * | * | * | 4 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | |
| A4A2 | |
10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 967680 | * | * | * | * | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | |||
| D4A2 | | 0111 | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | * | * | 604800 | * | * | * | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | ||
| A4A1 | | 0210 | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | * | * | * | 2903040 | * | * | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | ||
| A4A1A1 | |
10 | 30 | 0 | 10 | 20 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 1451520 | * | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | |||
| A4A1 | | 0300 | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | * | * | * | * | * | 2903040 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | ||
| D5A2 | | 0211 | f5 | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | 60480 | * | * | * | * | * | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} |
| A5A1 | | 0220 | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | * | 483840 | * | * | * | * | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | { }∨() | |
| D5A1 | | 0211 | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | * | * | 181440 | * | * | * | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | ||
| A5 | | 0310 | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | * | * | * | 967680 | * | * | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )∨( )∨() | |
| A5A1 | |
15 | 60 | 0 | 20 | 60 | 0 | 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 | 6 | 6 | * | * | * | * | 483840 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }∨() | ||
| 0400 | 6 | 15 | 0 | 0 | 20 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 6 | * | * | * | * | * | 483840 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | |||
| E6A1 | | 0221 | f6 | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | 0 | 6720 | * | * | * | * | 2 | 0 | 0 | { } |
| A6 | | 0320 | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | 0 | * | 138240 | * | * | * | 1 | 1 | 0 | ||
| D6 | | 0311 | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | 0 | * | * | 30240 | * | * | 1 | 0 | 1 | ||
| A6 | | 0410 | 21 | 105 | 35 | 0 | 140 | 0 | 0 | 35 | 0 | 105 | 0 | 0 | 0 | 21 | 0 | 42 | 0 | 0 | 0 | 7 | 0 | 7 | * | * | * | 138240 | * | 0 | 1 | 1 | ||
| A6A1 | |
21 | 105 | 0 | 35 | 140 | 0 | 0 | 0 | 35 | 105 | 0 | 0 | 0 | 0 | 21 | 42 | 0 | 0 | 0 | 0 | 7 | 7 | * | * | * | * | 69120 | 0 | 0 | 2 | |||
| E7 | | 0321 | f7 | 10080 | 120960 | 80640 | 40320 | 120960 | 20160 | 20160 | 60480 | 30240 | 60480 | 4032 | 12096 | 7560 | 24192 | 12096 | 12096 | 756 | 4032 | 1512 | 4032 | 2016 | 0 | 56 | 576 | 126 | 0 | 0 | 240 | * | * | ( ) |
| A7 | | 0420 | 56 | 420 | 280 | 0 | 560 | 70 | 0 | 280 | 0 | 420 | 0 | 56 | 0 | 168 | 0 | 168 | 0 | 28 | 0 | 56 | 0 | 28 | 0 | 8 | 0 | 8 | 0 | * | 17280 | * | ||
| D7 | | 0411 | 672 | 6720 | 2240 | 2240 | 8960 | 0 | 560 | 2240 | 2240 | 6720 | 0 | 0 | 280 | 1344 | 1344 | 2688 | 0 | 0 | 84 | 448 | 448 | 448 | 0 | 0 | 14 | 64 | 64 | * | * | 2160 | ||