# User:Tomruen/configuration

Example 8-cube. The diagonal elements are the k-face counts. The below diagonal rows are element counts of each k-face. The above diagonal rows are the k-figure element counts.

The configuration matrix shows the number of k-face elements along the diagonal, while the nondiagonal element show the incidence counts between all elements.[1] 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 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

Triangles
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
```

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

Regular 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)
v e f
v 4p/k q q
e 2 2pq/k 2
f p p 4q/k
With k=4-(p-2)(q-2)
```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

Symmetries of tetrahedra
Tetrahedra
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
```
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

The vertex figure can be seen as the top row, right of diagonal.
Semiregular polyhedra in tetrahedral family
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
```
Semiregular polyhedra in octahedral family
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
```
Semiregular polyhedra in icosahedron family
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

### Regular 4-polytopes

4D
{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]
v e f c
v f0 4q/k2 2qr\k2 4r\k2
e 2 f1 r r
f p p f2 2
c 4p/k1 2pq/k1 4q/k1 f3
With k1=4-(p-2)(q-2)
With k2=4-(q-2)(r-2)
f0=order([p,q,r])/order([q,r])
f1=order([p,q,r])/order([])/order([r])
f2=order([p,q,r])/order([])/order([p])
f3=order([p,q,r])/order([p,q])
```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-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 ```
D4 family
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

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

5D
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

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

6D regular
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

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

8D regular
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.

Truncated cuboctahedron
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

A4 k-face fk f0 f1 f2 f3 k-fig Notes
A3 ( ) f0 5 4 6 4 {3,3} A4/A3 = 5!/4! = 5
A2A1 { } f1 2 10 3 3 {3} A4/A2A1 = 5!/3!/2 = 10
A2A1 {3} f2 3 3 10 2 { }
A3 {3,3} f3 4 6 4 5 ( ) A4/A3 = 5!/4! = 5
##### rectified 5-cell

o3x3o3o - rap

A4 k-face fk f0 f1 f2 f3 k-fig Notes
A2A1 ( ) f0 10 6 3 6 3 2 {3}×{ } A4/A2A1 = 5!/3!/2 = 10
A1A1 { } f1 2 30 1 2 2 1 { }∨( ) A4/A1A1 = 5!/4 = 30
A2A1 {3} f2 3 3 10 * 2 0 { } A4/A2A1 = 5!/3!/2 = 10
A2 3 3 * 20 1 1 A4/A2 = 5!/3! = 20
A3 r{3,3} f3 6 12 4 4 5 * ( ) A4/A3 = 5!/4! = 5
{3,3} 4 6 0 4 * 5
##### Truncated 5-cell

x3x3o3o - tip

A4 k-face fk f0 f1 f2 f3 k-fig Notes
A2 ( ) f0 20 1 3 3 3 3 1 {3}∨( ) A4/A2 = 5!/3! = 20
A2A1 { } f1 2 10 * 3 0 3 0 {3} A4/A2A1 = 5!/3!/2 = 10
A1A1 2 * 30 1 2 2 1 { }∨( ) A4/A1A1 = 5!/4 = 30
A2A1 t{3} f2 6 3 3 10 * 2 0 { } A4/A2A1 = 5!/3!/2 = 10
A2 {3} 3 0 3 * 20 1 1 A4/A2 = 5!/3! = 20
A3 t{3,3} f3 12 6 12 4 4 5 * ( ) A4/A3 = 5!/4! = 5
{3,3} 4 0 6 0 4 * 5
##### 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

A4 k-face fk f0 f1 f2 f3 k-fig Notes
A1A1 ( ) f0 30 2 2 1 4 1 2 2 s{2,4} A4/A1A1 = 5!/4 = 30
{ } f1 2 30 * 1 2 0 2 1 { }∨( )
2 * 30 0 2 1 1 2
A2A1 {3} f2 3 3 0 10 * * 2 0 { } A4/A2A1 = 5!/3!/2 = 10
A2 t{3} 6 3 3 * 20 * 1 1 A4/A2 = 5!/3! = 20
A2A1 {3} 3 0 3 * * 10 0 2 A4/A2A1 = 5!/3!/2 = 10
A3 t{3,3} f3 12 12 6 4 4 0 5 * ( ) A4/A3 = 5!/4! = 5
12 6 12 0 4 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
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

F4 k-face fk f0 f1 f2 f3 k-fig Notes
B3 ( ) f0 24 8 12 6 {4,3} F4/B3 = 1152/48 = 24
A2A1 { } f1 2 96 3 3 {3} F4/A2A1 = 1152/3!/2 = 96
{3} f2 3 3 96 2 { }
B3 {3,4} f3 6 12 8 24 ( ) F4/B3 = 1152/48 = 24
##### Rectified 24-cell

o3x4o3o - rico

F4 k-face fk f0 f1 f2 f3 k-fig Notes
A2A1 ( ) f0 96 6 3 6 3 2 {3}×{ } F4/A2A1 = 1152/3!/2 = 96
A1A1 { } f1 2 288 1 2 2 1 { }∨( ) F4/A1A1 = 1152/4 = 288
A2A1 {3} f2 3 3 96 * 2 0 { } F4/A2A1 = 1152/3!/2 = 96
B2 {4} 4 4 * 144 1 1 F4/B2 = 1152/8 = 144
B3 r{4,3} f3 12 24 8 6 24 * ( ) F4/B3 = 1152/48 = 24
{4,3} 8 12 0 6 * 24
##### 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

F4 k-face fk f0 f1 f2 f3 k-fig Notes
A1A1 ( ) f0 288 2 2 1 4 1 2 2 s{2,4} F4/A1A1 = 288
{ } f1 2 288 * 1 2 0 2 1 { }∨( )
2 * 288 0 2 1 1 2
A2A1 {3} f2 3 3 0 96 * * 2 0 { } F4/A2A1 = 1152/6/2 = 96
B2 t{4} 8 4 4 * 144 * 1 1 F4/B2 = 1152/8 = 144
A2A1 {3} 3 0 3 * * 96 0 2 F4/A2A1 = 1152/6/2 = 96
B3 t{4,3} f3 24 24 12 8 6 0 24 * ( ) F4/B3 = 1152/48 = 24
24 12 24 0 6 8 * 24
##### 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

½F4 k-face fk f0 f1 f2 f3 k-fig Notes
demi( ) ( ) f0 96 3 6 3 9 3 3 1 4 I-3
( ) { } f1 2 144 * 0 2 2 1 1 2 { }||{ }
sefa( ) 2 * 288 1 2 0 2 0 1 { }∨( )
( ) {3} f2 3 0 3 96 * * 2 0 0 { }
sefa( ) 3 1 2 * 288 * 1 0 1
sefa( ) 3 3 0 * * 96 0 1 1
( ) {3,5} f3 12 6 24 8 12 0 24 * * ( )
( ) {3,3} 4 6 0 0 0 4 * 24 *
sefa( ) 4 3 3 0 3 1 * * 96

#### 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

H4 k-face fk f0 f1 f2 f3 k-fig Notes
H3 ( ) f0 120 12 30 20 {3,5} H4/H3 = 14400/120 = 120
A1H2 { } f1 2 720 5 5 {5} H4/H2A1 = 14400/10/2 = 720
A2A1 {3} f2 3 3 1200 2 { } H4/A2A1 = 14400/6/2 = 1200
A3 {3,3} f3 4 6 4 600 ( ) H4/A3 = 14400/24 = 600

### 120-cell

H4 k-face fk f0 f1 f2 f3 k-fig Notes
A3 ( ) f0 600 4 6 4 {3,3} H4/A3 = 14400/24 = 600
A1A2 { } f1 2 720 3 3 {3} H4/A2A1 = 14400/6/2 = 1200
H2A1 {5} f2 5 5 1200 2 { } H4/H2A1 = 14400/10/2 = 720
H3 {5,3} f3 20 30 12 120 ( ) H4/H3 = 14400/120 = 120

### 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