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

Further information: Regular polygon

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

Further information: Triangle § By_lengths_of_sides

Triangles

1,1	2,2	3,3
Equilateral {3}	Isosceles { }∨( )	Scalene ( )∨( )∨( )
(v:3; e:3)	(v:2+1; e:2+1)	(v:1+1+1; e:1+1+1)
| A | a --+---+--- A | 3 | 2 --+---+--- a | 2 | 3	| A B | a b --+-----+----- A | 2 * | 1 1 B | * 1 | 2 0 --+-----+----- a | 1 1 | 2 * b | 2 0 | * 1	| A B C | a b c --+-------+------- A | 1 * * | 0 1 1 B | * 1 * | 1 0 1 C | * * 1 | 1 1 0 --+-------+------- a | 0 1 1 | 1 * * b | 1 0 1 | * 1 * c | 1 1 0 | * * 1

Quadrilateral

Further information: Square § Symmetry

Quadrilaterals by symmetry

Quadrilaterals

1,1	1,2	2,1	2,2	2,3	3,2	4,4
Square {4}	Rectangle { }×{ }	Rhombus { }+{ }	Parallelogram	Isosceles trapezoid { }||{ }	Kite	General
(v:4; e:4)	(v:4; e:2+2)	(v:2+2; e:4)	(v:2+2; e:2+2)	(v:2+2; e:1+1+2)	(v:1+1+2; e:2+2)	(v:1+1+1+1; e:1+1+1+1)
| A | a --+---+-- A | 4 | 2 --+---+-- a | 2 | 4	| A | a b --+---+---- A | 4 | 1 1 --+---+---- a | 2 | 2 * b | 2 | * 2	| A B | a --+-----+-- A | 2 * | 2 B | * 2 | 2 --+-----+-- a | 1 1 | 4	| A B | a b --+-----+---- A | 2 * | 1 1 B | * 2 | 1 1 --+-----+---- a | 1 1 | 2 * b | 1 1 | * 2	| A B | a b c --+-----+------ A | 2 * | 1 0 1 B | * 2 | 0 1 1 --+-----+------ a | 2 0 | 1 * * b | 0 2 | * 1 * c | 1 1 | * * 2	| A B C | a b --+-------+---- A | 1 * * | 2 0 B | * 1 * | 0 2 C | * * 2 | 1 1 --+-------+---- a | 1 0 1 | 2 * b | 0 1 1 | * 2	| A B C D | a b c d --+---------+-------- A | 1 * * * | 1 0 0 1 B | * 1 * * | 1 1 0 0 C | * * 1 * | 0 1 1 0 D | * * * 1 | 0 0 1 1 --+---------+-------- a | 1 1 0 0 | 1 * * * b | 0 1 1 0 | * 1 * * c | 0 0 1 1 | * * 1 * d | 1 0 0 1 | * * * 1

Polyhedra

Further information: Platonic solid and Kepler–Poinsot polyhedron

Regular polyhedra

Platonic solid {p,q}	Tetrahedron {3,3} (v:4; e:6; f:4)	Icosahedron {3,5} (v:12; e:30; f:20)	Dodecahedron {5,3} (v:20; e:30; f:12)	Stellated dodecahedron {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 {3,4} (v:6; e:12; f:8)	Cube {4,3} (v:8; e:12; f:6)	Great icosahedron {3,5/2} (v:12; e:30; f:20)	Great stellated dodecahedron {5/2,3} (v:20; e:30; f:12)	Great dodecahedron {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

Further information: Tetrahedron § Isometries_of_irregular_tetrahedra

Symmetries of tetrahedra

Tetrahedra

1,1,1	1,2,1	1,3,1	2,3,2	2,4,2
Regular (v:4; e:6; f:4)	tetragonal disphenoid (v:4; e:2+4; f:4)	Rhombic disphenoid (v:4; e:2+2+2; f:4)	Digonal disphenoid (v:2+2; e:4+1+1; f:2+2)	Phyllic disphenoid (v:2+2; e:2+2+1+1; f:2+2)
A| 4 | 3 | 3 ---+---+---+-- a| 2 | 6 | 2 ---+---+---+-- aaa| 3 | 3 | 4	A| 4 | 2 1 | 3 ---+---+-----+-- a| 2 | 4 * | 2 b| 2 | * 2 | 2 ---+---+-----+-- aab| 3 | 2 1 | 4	A| 4 | 1 1 1 | 3 ----+---+-------+-- a| 2 | 2 * * | 2 b| 2 | * 2 * | 2 c| 2 | * * 2 | 2 ----+---+-------+-- abc| 3 | 1 1 1 | 4	A| 2 * | 2 1 0 | 2 1 B| * 2 | 2 0 1 | 1 2 ---+-----+-------+---- a| 1 1 | 4 * * | 1 1 b| 2 0 | * 1 * | 2 0 c| 0 2 | * * 1 | 0 2 ---+-----+-------+---- aab| 2 1 | 2 1 0 | 2 * aac| 1 2 | 2 0 1 | * 2	A| 2 * | 1 0 1 1 | 1 2 B| * 2 | 1 1 1 0 | 2 1 ---+-----+---------+---- a| 1 1 | 2 * * * | 1 1 b| 1 1 | * 2 * * | 1 1 c| 0 2 | * * 1 * | 2 0 d| 2 0 | * * * 1 | 0 2 ---+-----+---------+---- abc| 1 2 | 1 1 1 0 | 2 * bcd| 2 1 | 1 1 0 1 | * 2
2,2,2	3,4,3		4,6,4
Triangular pyramid (v:3+1; e:3+3; f:3+1)	Mirrored spheroid (v:2+1+1; e:2+2+1+1; f:2+1+1)		No symmetry (v:1+1+1+1; e:1+1+1+1+1+1; f:1+1+1+1)
A| 3 * | 2 1 | 2 1 B| * 1 | 0 3 | 3 0 ---+-----+-----+---- a| 2 0 | 3 * | 1 1 b| 1 1 | * 3 | 2 0 ---+-----+-----+---- abb| 2 1 | 1 2 | 3 * aaa| 3 0 | 3 0 | * 1	A| 1 * * | 2 0 1 0 | 2 1 0 B| * 1 * | 0 2 1 0 | 2 0 1 C| * * 2 | 1 1 0 1 | 1 1 1 ---+-------+---------+------ a| 1 0 1 | 2 * * * | 1 1 0 b| 0 1 1 | * 2 * * | 1 0 1 c| 1 1 0 | * * 1 * | 2 0 0 d| 0 0 2 | * * * 1 | 0 1 1 ---+-------+---------+------ ABC| 1 1 1 | 1 1 1 0 | 2 * * ACC| 1 0 2 | 2 0 0 1 | * 1 * BCC| 0 1 2 | 0 2 0 1 | * * 1		A | 1 0 0 0 | 1 1 1 0 0 0 | 1 1 1 0 B | 0 1 0 0 | 1 0 0 1 1 0 | 1 1 0 1 C | 0 0 1 0 | 0 1 0 1 0 1 | 1 0 1 1 D | 0 0 0 1 | 0 0 1 0 1 1 | 0 1 1 1 ----+---------+-------------+-------- a | 1 1 0 0 | 1 0 0 0 0 0 | 1 1 0 0 b | 1 0 1 0 | 0 1 0 0 0 0 | 1 0 1 0 c | 1 0 0 1 | 0 0 1 0 0 0 | 0 1 1 0 d | 0 1 1 0 | 0 0 0 1 0 0 | 1 0 0 1 e | 0 1 0 1 | 0 0 0 0 1 0 | 0 1 0 1 f | 0 0 1 1 | 0 0 0 0 0 1 | 0 0 1 1 ----+---------+-------------+-------- ABC | 1 1 1 0 | 1 1 0 1 0 0 | 1 0 0 0 ABD | 1 1 0 1 | 1 0 1 0 1 0 | 0 1 0 0 ACD | 1 0 1 1 | 0 1 1 0 0 1 | 0 0 1 0 BCD | 0 1 1 1 | 0 0 0 1 1 1 | 0 0 0 1

Uniform polyhedra

The vertex figure can be seen as the top row, right of diagonal.

Further information: Archimedean solid

Semiregular polyhedra in tetrahedral family

Tetratetrahedron (v:6; e:12; f:4+4)	Truncated tetrahedron (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 (v:12; e:12+12; f:4+6+4)	Truncated tetratetrahedron (v:12; e:12+12; f:4+6+4)	Snub tetrahedron (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 (v:12; e:24; f:8+6)	Truncated cube (v:24; e:12+24; f:8+6)	Truncated octahedron (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 (v:24; e:24+24; f:8+12+6)	Truncated cuboctahedron (v:48; e:24+24+24; f:8+12+6)	Snub cube (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 (v:30; e:60; f:20+12)	Truncated dodecahedron (v:60; e:30+60; f:20+12)	Truncated icosahedron (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 (v:60; e:60+60; f:20+30+12)	Truncated icosidodecahedron (v:120; e:60+60+60; f:20+30+12)	Snub dodecahedron (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

Further information: Regular 4-polytope and Regular 4-polytope § Regular star (Schläfli–Hess) 4-polytopes

4D

{p,q,r}	5-cell {3,3,3}	16-cell {3,3,4}	600-cell {3,3,5}	120-cell {5,3,3}
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}	tesseract {4,3,3}	grand 600-cell {3,3,5/2}	great grand stellated 120-cell {5/2,3,3}
	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}	icosahedral 120-cell {3,5,5/2}	small stellated 120-cell {5/2,5,3}	great 120-cell {5,5/2,5}
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}	great icosahedral 120-cell {3,5/2,5}	great grand 120-cell {5,5/2,3}	grand stellated 120-cell {5/2,5,5/2}
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

Further information: 5-cell § Irregular_5-cell

5-cells

5-cell {3,3,3}	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

Further information: Uniform 4-polytope

A4 family

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}

24-cell {31,1,1}

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

B4 family

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

F4 family

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

H4 family

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

Further information: 5-polytope

5D

5-simplex {3,3,3,3}	5-orthoplex {3,3,3,4}	5-cube {4,3,3,3}
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}
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}	r{3,3,3,3}	2r{3,3,3,3}	2r{4,3,3,3}
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

Further information: 6-polytope

6D regular

6-simplex {3,3,3,3,3}	6-orthoplex {3,3,3,3,4}	6-cube {4,3,3,3,3}
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}	221 {3,3,32,1}	122 {3,32,2}
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}	birectified 6-simplex r{3,3,3,3,3}	rectified 1_22 r{3,32,2}
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

Further information: 7-polytope

7-simplex {3,3,3,3,3,3}	7-orthoplex {3,3,3,3,3,4}	7-cube {4,3,3,3,3,3}
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}	321 {3,3,3,32,1}
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}	132 {3,33,2}
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}	042 2r{3,3,3,3,3,3}
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}
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}
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

Further information: 8-polytope

8D regular

8-simplex {3,3,3,3,3,3,3}	8-orthoplex {3,3,3,3,3,3,4}	8-cube {4,3,3,3,3,3,3}
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}	421 {3,3,3,3,32,1}
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}	142 {3,34,2}
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}	052 2r{3,3,3,3,3,3,3}
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}
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}
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		{3,3}	f3	4	6	4	241920	10	30	20	10	5	5	021	E8/A3A4 = 192x10!/4!/5! = 241920
A4A2A1		{3,3,3}	f4	5	10	10	5	483840	6	6	3	2	3	{3}×{ }	E8/A4A2A1 = 192x10!/5!/3!/2 = 483840
A5A1		{3,3,3,3}	f5	6	15	20	15	6	483840	2	1	1	2	{ }∨( )	E8/A5A1 = 192x10!/6!/2 = 483840
A6		{3,3,3,3,3}	f6	7	21	35	35	21	7	138240	*	1	1	{ }	E8/A6 = 192x10!/7! = 138240
A6A1				7	21	35	35	21	7	*	69120	0	2		E8/A6A1 = 192x10!/7!/2 = 69120
A7		{3,3,3,3,3,3}	f7	8	28	56	70	56	28	8	0	17280	*	( )	E8/A7 = 192x10!/8! = 17280
D7		{3,3,3,3,3,4}		14	84	280	560	672	448	64	64	*	2160		E8/D7 = 192x10!/2^6/7! = 2160

2_41

Example on 2_41 polytope:

E8		k-face	fk	f0	f1	f2	f3	f4		f5		f6		f7		k-fig	notes
D7		( )	f0	2160	64	672	2240	560	2240	280	1344	84	448	14	64	h{4,3,3,3,3,3}	E8/D7 = 192*10!/64/7! = 2160
A6A1		{ }	f1	2	69120	21	105	35	140	35	105	21	42	7	7	r{3,3,3,3,3}	E8/A6A1 = 192*10!/7!/2 = 69120
A4A2A1		{3}	f2	3	3	483840	10	5	20	10	20	10	10	5	2	{}×{3,3,3}	E8/A4A2A1 = 192*10!/5!/3!/2 = 483840
A3A3		{3,3}	f3	4	6	4	1209600	1	4	4	6	6	4	4	1	{3,3}∨( )	E8/A3A3 = 192*10!/4!/4! = 1209600
A4A3		{3,3,3}	f4	5	10	10	5	241920	*	4	0	6	0	4	0	{3,3}	E8/A4A3 = 192*10!/5!/4! = 241920
A4A2				5	10	10	5	*	967680	1	3	3	3	3	1	{3}∨( )	E8/A4A2 = 192*10!/5!/3! = 967680
D5A2		{3,3,31,1}	f5	10	40	80	80	16	16	60480	*	3	0	3	0	{3}	E8/D5A2 = 192*10!/16/5!/2 = 40480
A5A1		{3,3,3,3}		6	15	20	15	0	6	*	483840	1	2	2	1	{ }∨( )	E8/A5A1 = 192*10!/6!/2 = 483840
E6A1		{3,3,32,1}	f6	27	216	720	1080	216	432	27	72	6720	*	2	0	{ }	E8/E6A1 = 192*10!/72/6! = 6720
A6		{3,3,3,3,3}		7	21	35	35	0	21	0	7	*	138240	1	1		E8/A6 = 192*10!/7! = 138240
E7		{3,3,33,1}	f7	126	2016	10080	20160	4032	12096	756	4032	56	576	240	*	( )	E8/E7 = 192*10!/72!/8! = 240
A7		{3,3,3,3,3,3}		8	28	56	70	0	56	0	28	0	8	*	17280		E8/A7 = 192*10!/8! = 17280

1_42

Example on 1_42 polytope:

E8		k-face	fk	f0	f1	f2	f3		f4			f5			f6			f7		k-fig	notes
A7		( )	f0	17280	56	420	280	560	70	280	420	56	168	168	28	56	28	8	8	2r{36}
A4A2A1		{ }	f1	2	483840	15	15	30	5	30	30	10	30	15	10	15	3	5	3	{3}×{3,3,3}
A3A2A1		{3}	f2	3	3	2419200	2	4	1	8	6	4	12	4	6	8	1	4	2	{3.3}∨{ }
A3A3		{3,3}	f3	4	6	4	1209600	*	1	4	0	4	6	0	6	4	0	4	1	{3,3}∨( )
A3A2A1				4	6	4	*	2419200	0	2	3	1	6	3	3	6	1	3	2	{3}∨{ }
A4A3		{3,32,0}	f4	5	10	10	5	0	241920	*	*	4	0	0	6	0	0	4	0	{3,3}
D4A2		{3,31,1}		8	24	32	8	8	*	604800	*	1	3	0	3	3	0	3	1	{3}∨( )
A4A1A1		{3,32,0}		5	10	10	0	5	*	*	1451520	0	2	2	1	4	1	2	2	{ }∨{ }
D5A2		{3,32,1}	f5	16	80	160	80	40	16	10	0	60480	*	*	3	0	0	3	0	{3}
D5A1				16	80	160	40	80	0	10	16	*	181440	*	1	2	0	2	1	{ }∨( )
A5A1		{3,33,0}		6	15	20	0	15	0	0	6	*	*	483840	0	2	1	1	2
E6A1		{3,32,2}	f6	72	720	2160	1080	1080	216	270	216	27	27	0	6720	*	*	2	0	{ }
D6		{3,33,1}		32	240	640	160	480	0	60	192	0	12	32	*	30240	*	1	1
A6A1		{3,34,0}		7	21	35	0	35	0	0	21	0	0	7	*	*	69120	0	2
E7		{3,33,2}	f7	576	10080	40320	20160	30240	4032	7560	12096	756	1512	2016	56	126	0	240	*	( )
D7		{3,34,1}		64	672	2240	560	2240	0	280	1344	0	84	448	0	14	64	*	2160

0_421

Example on rectified 1_42 polytope:

E8		k-face	fk	f0	f1	f2			f3					f4						f5						f6					f7			k-fig
A4A2A1		( )	f0	483840	30	30	15	60	10	15	60	30	60	5	20	30	60	30	30	10	20	30	30	15	6	10	10	15	6	3	5	2	3	{3,3,3}×{3,3}×{}
A3A1A1		{ }	f1	2	7257600	2	1	4	1	2	8	4	6	1	4	8	12	6	4	4	6	12	8	4	1	6	4	8	2	1	4	1	2
A3A2		{3}	f2	3	3	4838400	*	*	1	1	4	0	0	1	4	4	6	0	0	4	6	6	4	0	0	6	4	4	1	0	4	1	1
A3A2A1				3	3	*	2419200	*	0	2	0	4	0	1	0	8	0	6	0	4	0	12	0	4	0	6	0	8	0	1	4	0	2
A2A2A1				3	3	*	*	9676800	0	0	2	1	3	0	1	2	6	3	3	1	3	6	6	3	1	3	3	6	2	1	3	1	2
A3A3		0200	f3	4	6	4	0	0	1209600	*	*	*	*	1	4	0	0	0	0	4	6	0	0	0	0	6	4	0	0	0	4	1	0
		0110		6	12	4	4	0	*	1209600	*	*	*	1	0	4	0	0	0	4	0	6	0	0	0	6	0	4	0	0	4	0	1
A3A2				6	12	4	0	4	*	*	4838400	*	*	0	1	1	3	0	0	1	3	3	3	0	0	3	3	3	1	0	3	1	1
A3A2A1				6	12	0	4	4	*	*	*	2419200	*	0	0	2	0	3	0	1	0	6	0	3	0	3	0	6	0	1	3	0	2
A3A1A1		0200		4	6	0	0	4	*	*	*	*	7257600	0	0	0	2	1	2	0	1	2	4	2	1	1	2	4	2	1	2	1	2
A4A3		0210	f4	10	30	20	10	0	5	5	0	0	0	241920	*	*	*	*	*	4	0	0	0	0	0	6	0	0	0	0	4	0	0
A4A2				10	30	20	0	10	5	0	5	0	0	*	967680	*	*	*	*	1	3	0	0	0	0	3	3	0	0	0	3	1	0
D4A2		0111		24	96	32	32	32	0	8	8	8	0	*	*	604800	*	*	*	1	0	3	0	0	0	3	0	3	0	0	3	0	1
A4A1		0210		10	30	10	0	20	0	0	5	0	5	*	*	*	2903040	*	*	0	1	1	2	0	0	1	2	2	1	0	2	1	1
A4A1A1				10	30	0	10	20	0	0	0	5	5	*	*	*	*	1451520	*	0	0	2	0	2	0	1	0	4	0	1	2	0	2
A4A1		0300		5	10	0	0	10	0	0	0	0	5	*	*	*	*	*	2903040	0	0	0	2	1	1	0	1	2	2	1	1	1	2
D5A2		0211	f5	80	480	320	160	160	80	80	80	40	0	16	16	10	0	0	0	60480	*	*	*	*	*	3	0	0	0	0	3	0	0	{3}
A5A1		0220		20	90	60	0	60	15	0	30	0	15	0	6	0	6	0	0	*	483840	*	*	*	*	1	2	0	0	0	2	1	0	{ }∨()
D5A1		0211		80	480	160	160	320	0	40	80	80	80	0	0	10	16	16	0	*	*	181440	*	*	*	1	0	2	0	0	2	0	1
A5		0310		15	60	20	0	60	0	0	15	0	30	0	0	0	6	0	6	*	*	*	967680	*	*	0	1	1	1	0	1	1	1	( )∨( )∨()
A5A1				15	60	0	20	60	0	0	0	15	30	0	0	0	0	6	6	*	*	*	*	483840	*	0	0	2	0	1	1	0	2	{ }∨()
		0400		6	15	0	0	20	0	0	0	0	15	0	0	0	0	0	6	*	*	*	*	*	483840	0	0	0	2	1	0	1	2
E6A1		0221	f6	720	6480	4320	2160	4320	1080	1080	2160	1080	1080	216	432	270	432	216	0	27	72	27	0	0	0	6720	*	*	*	*	2	0	0	{ }
A6		0320		35	210	140	0	210	35	0	105	0	105	0	21	0	42	0	21	0	7	0	7	0	0	*	138240	*	*	*	1	1	0
D6		0311		240	1920	640	640	1920	0	160	480	480	960	0	0	60	192	192	192	0	0	12	32	32	0	*	*	30240	*	*	1	0	1
A6		0410		21	105	35	0	140	0	0	35	0	105	0	0	0	21	0	42	0	0	0	7	0	7	*	*	*	138240	*	0	1	1
A6A1				21	105	0	35	140	0	0	0	35	105	0	0	0	0	21	42	0	0	0	0	7	7	*	*	*	*	69120	0	0	2
E7		0321	f7	10080	120960	80640	40320	120960	20160	20160	60480	30240	60480	4032	12096	7560	24192	12096	12096	756	4032	1512	4032	2016	0	56	576	126	0	0	240	*	*	( )
A7		0420		56	420	280	0	560	70	0	280	0	420	0	56	0	168	0	168	0	28	0	56	0	28	0	8	0	8	0	*	17280	*
D7		0411		672	6720	2240	2240	8960	0	560	2240	2240	6720	0	0	280	1344	1344	2688	0	0	84	448	448	448	0	0	14	64	64	*	*	2160