User:Tomruen/List of Coxeter groups

This test article is my attempt to construct Coxeter groups by rank, including products. I subdivide them by their finite order, infinite, or hyperbolic infinite.

I name them by their group names, Coxeter's bracket notation names, and their Coxeter-Dynkin diagram graphs. I also give an example polytope or tessellation that has this symmetry.

= Finite Definite (spherical) =

Rank 1
(1-space)
 * A1: [ ] [[Image:CDW dot.png]]

Rank 2
(2-space)
 * A2: [3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (triangle)
 * B2: [4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] square)
 * H2: [5] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]] (pentagon)
 * I2(p) [p] [[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]] (p-gon)

(2-space prismatic)
 * A1xA1: [ ]x[ ] [[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (rectangle)

Rank 3
(3-space)
 * A3: [3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (tetrahedron)
 * B3: [4,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (cube/octahedron)
 * H3: [5,3] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (dodecahedron/icosahedron)

(3-space prismatic)
 * I2(p)xA1: [p]x[ ] [[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (p-gonal prism)
 * A1xA1xA1: [ ]x[ ]x[ ] [[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (cube/cuboid)

Rank 4
(4-space)
 * A4: [3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (5-cell)
 * B4: [4,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (tesseract/16-cell)
 * D4: [31,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]] (24-cell)
 * F4: [3,4,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (24-cell)
 * H4: [5,3,3] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (120-cell/600-cell)

(4-space prismatic)
 * A3xA1: [3,3]x[ ] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (tetrahedral prism)
 * B3xA1: [4,3]x[ ] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (tesseract/orthoplex)
 * H3xA1: [5,3]x[ ] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (dodecahedral prism)
 * I2(p)xA1xA1: [p]x[ ]x[ ] [[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (p-4 duoprism)
 * A1xA1xA1xA1: [ ]x[ ]x[ ]x[ ] [[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (tesseract/orthoplex)
 * I2(p)xI2(q) [p]x[q] [[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]] (p-q duoprism)

Rank 5
(5-space)
 * A5: [3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (hexateron)
 * B5: [4,3,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (Penteract)
 * D5: [32,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (Demipenteract)

(5-space prismatic)
 * A4xA1: [3,3,3]x[ ] [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * B4xA1: [4,3,3]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * F4xA1: [3,4,3]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * H4xA1: [5,3,3]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * D4xA1: [31,1,1]x[ ] - [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD_downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]]
 * A3xI2p: [3,3]x[p] - [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]]
 * B3xI2p: [4,3]x[p] - [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]]
 * H3xI2p: [5,3]x[p] - [[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]]
 * I2pxI2qxA1: [p]x[q]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW_dot.png]][[Image:CDW_q.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]]

Rank 6
(6-space)
 * A6: [3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (6-simplex)
 * B6: [4,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (6-orthoplex/6-hypercube)
 * D6: [33,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (6-orthoplex)
 * E6: [32,2,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (1_22 polytope)

Uniform prism

There are 6 categorical uniform prisms based the uniform 5-polytopes.

Uniform duoprism

There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower dimensional uniform polytopes. Five are formed as the product of a uniform polychoron with a regular polygon, and six are formed by the product of two uniform polyhedra:

Uniform triprisms

There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.

Rank 7
(7-space)
 * A7: [3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (8-simplex)
 * B7: [4,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (8-orthoplex/8-hypercube)
 * D7: [34,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (8-orthoplex)
 * E7: [33,2,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (3_21 polytope)

There are 16 uniform prismatic families based on the uniform 6-polytopes.

There are 18 uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.

There are 3 uniform triaprismatic families based on Cartesian products of uniform polyhedrons and two regular polygons.

Rank 8
(8-space)
 * A8: [3,3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (8-simplex)
 * B8: [4,3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (8-orthoplex/8-hypercube)
 * D8: [35,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (8-orthoplex)
 * E8: [34,2,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (4_21 polytope)

Four are based on the uniform 7-polytopes:

Three are based on the uniform 6-polytopes and regular polygons:

Fifteen are based on the product of the uniform polychora and uniform polyhedra:

Three are based on the uniform polyhedra and uniform duoprism:

There are 28 categorical uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.

There are 4 based on the uniform 6-polytopes and regular polygons:

There are 9 based on the uniform 5-polytopes and uniform polyhedra:

Finally there are 20 based on two uniform 4-polytopes:

There are 11 categorical uniform triaprismatic forms based on Cartesian products of lower dimensional uniform polytopes, for example these regular products:

Six are based on products of the uniform 4-polytopes and uniform duoprisms:

Five are based on triprism products of two uniform polyhedra and regular polygons:

There is one infinite family of uniform quadriprismatic figures based on Cartesian products of four regular polygons:

Rank 9
(9-space)
 * A9: [3,3,3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (9-simplex)
 * B9: [4,3,3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (9-orthoplex/9-hypercube)
 * D9: [36,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]] (9-orthoplex)

Rank 10
(10-space)
 * A10: [3,3,3,3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * B10: [4,3,3,3,3,3,3,3,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * D10: [37,1,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]]

= Euclidean compact =

Rank 2
(1-space)
 * I~1: [&infin;] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (apeirogon)

Rank 3
(2-space)
 * A~2: (3 3 3) [[Image:CD righttriangle-000.png]] (triangular tiling)
 * B~2: [4,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (square tiling)
 * H~2: [6,3] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (triangular tiling/hexagonal tiling)

Rank 4
(3-space)
 * A~3: (3 3 3 3) [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]] (quarter cubic honeycomb)
 * B~2: [4,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (cubic honeycomb)
 * D~3: [4,31,1] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 4.png]][[Image:CD dot.png]] (Alternated cubic honeycomb)

(2-space prismatic)
 * I~1xI~1: [&infin;]x[&infin;] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (square tiling)

Rank 5
(4-space)
 * A~4: (3 3 3 3 3) [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD righttriangleopen 000.png]]
 * B~4: [4,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (Tesseractic honeycomb)
 * C~4: [4,3,31,1] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]] (Demitesseractic honeycomb)
 * D~4: [31,1,1,1] [[File:CDT dot.png]][[File:CDT 3a.png]][[File:CDT branch000.png]][[File:CDT 3a.png]][[File:CDT dot.png]] (Demitesseractic honeycomb)
 * F~4: [3,4,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (Icositetrachoric honeycomb/Demitesseractic honeycomb)

(3-space prismatic)
 * A3xI~1: [3,3]x[&infin;] - [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * B3xI~1: [4,3]x[&infin;] - [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * H3xI~1: [5,3]x[&infin;] - [[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]

Rank 6
(5-space)
 * A~5: [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]]
 * B~5: [4,3,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * C~5: [4,32,31,1] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]]
 * D~5: [31,1,3,31,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]]

(4-space prismatic)
 * I~1xI~1xI2r: [&infin;] x [&infin;] x [r] = [4,4]x[r] - [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]]
 * B~3xI~1: [4,3,4]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * D~3xI~1: [4,31,1]x[&infin;] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]
 * A~3xI~1: [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]
 * A~2xA~2: [&Delta;]x[&Delta;] [[Image:CD righttriangle-000.png]][[Image:CDW 2.png]][[Image:CD righttriangle-000.png]]
 * A~2xB~2: [&Delta;]x[4,4] [[Image:CD righttriangle-000.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * A~2xH~2: [&Delta;]x[6,3] [[Image:CD righttriangle-000.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * B~2xB~2: [4,4]x[4,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * B~2xH~2: [4,4]x[6,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * H~2xH~2: [6,3]x[6,3] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * A4xI~1: [3,3,3]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (5-cell column)
 * B4xI~1: [4,3,3]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (tesseract/16-cell column)
 * H4xI~1: [5,3,3]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (120-cell/600-cell column)
 * F4xI~1: [3,4,3]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (24-cell column)
 * D4xI~1: [31,1,1]x[&infin;] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (24-cell column)

(3-space prismatic)
 * I~1xI~1xI~1: [&infin;] x [&infin;] x [&infin;] - [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]

Rank 7
(6-space)
 * A~6: (3 3 3 3 3 3) [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD righttriangleopen 000.png]]
 * B~6: [4,3,3,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * C~6: [4,32,31,1] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]]
 * D~6: [31,1,32,31,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]]
 * E~6: [32,2,2] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD 3b.png]][[Image:CD dot.png]]

(6-space prismatic)
 * A5xI~1: [3,3,3] x [&infin;] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * B5xI~1: [4,3,3] x [&infin;] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * D5xI~1: [32,1,1] x [&infin;] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]
 * A4xI~1xA1: [3,3,3] x [&infin;] x [ ] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * B4xI~1xA1: [4,3,3] x [&infin;] x [ ] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * F4xI~1xA1: [3,4,3] x [&infin;] x [ ] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * H4xI~1xA1: [5,3,3] x [&infin;] x [ ] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * D4xI~1xA1: [31,1,1] x [&infin;] x [ ] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * A3xI~1xI2q: [3,3] x [&infin;] x [q] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]]
 * B3xI~1xI2q: [4,3] x [&infin;] x [q] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]]
 * H3xI~1xI2q: [5,3] x [&infin;] x [q] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]]
 * I~1xI2qxI2rA1: [&infin;] x [q] x [r] x [ ] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]

(5-space prismatic)
 * A3xI~1xI~1: [3,3] x [&infin;] x [&infin;] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * B3xI~1xI~1: [4,3] x [&infin;] x [&infin;] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * H3xI~1xI~1: [5,3] x [&infin;] x [&infin;] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * I~1xI~1xI2rxA1: [&infin;] x [&infin;] x [r] x [ ] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]

(4-space prismatic)
 * I~1xI~1xI~1A1: [&infin;] x [&infin;] x [&infin;] x [ ] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]

Rank 8
(7-space)
 * A~7: (3 3 3 3 3 3 3) [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]]
 * B~7: [4,3,3,3,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * C~7: [4,33,31,1] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]]
 * D~7: [31,1,33,31,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]]
 * E~7: [33,3,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]]

Rank 9
(8-space)
 * A~9: (3 3 3 3 3 3 3 3) [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD righttriangleopen 000.png]]
 * B~8: [4,3,3,3,3,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * C~8: [4,34,31,1] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]]
 * D~8: [31,1,34,31,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]]
 * E~8: [35,2,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]] (E8 lattice)

Rank 10
(9-space)
 * A~9: (3 3 3 3 3 3 3 3 3) [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-open.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]]
 * B~9: [4,3,3,3,3,3,3,3,4] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]]
 * C~9: [4,35,31,1] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]]
 * D~9: [31,1,35,31,1] [[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD dot.png]][[Image:CD 3b.png]][[Image:CD downbranch-00.png]][[Image:CD 3b.png]][[Image:CD dot.png]]

= Euclidean noncompact =

Rank 3
(2-space)
 * I~1xA1: [&infin;]x[ ] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (apeirogonal prism)

Rank 4
(3-space)
 * B~2xA1: [4,4]x[ ] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (cubic prismatic slab)
 * H~2xA1: [6,3]x[ ] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] (triangular/hexagonal prismatic slab)
 * A~2xA1: (3 3 3 3)x[ ] [[Image:CD righttriangle-000.png]][[Image:CD 2.png]][[Image:CD dot.png]] (triangular prismatic slab)
 * I~1xA1xA1: [&infin;]x[ ]x[ ] [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]] = [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (4-&infin; semi-infinite duoprism)
 * I2(p)xI~1: [p]x[&infin;] [[Image:CDW dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]] (p-&infin; semiinfinite duoprism)

Rank 5
(4-space)
 * I2pxI~1xA1: [p]x[&infin;]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]]
 * D~3xA1: [4,31,1]x[ ] [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]]
 * A~3xA1: (3 3 3 3)x[ ] [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]][[Image:CD 2.png]][[Image:CD dot.png]]
 * A~2xI2p: (3 3 3)x[p] [[Image:CD righttriangle-000.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD p.png]][[Image:CD dot.png]]
 * B~2xI2p: [4,4]x[p] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CD p.png]][[Image:CDW dot.png]]
 * H~2xI2p: [6,3]x[p] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CD dot.png]][[Image:CDW p.png]][[Image:CDW dot.png]]

(3-space)
 * I~1xI~1xA1: [&infin;]x[&infin;]x[ ] - [[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW 2.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_2.png]][[Image:CDW_dot.png]]
 * B~2xI2p: [4,4]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * H~2xI2p: [6,3]x[&infin;] [[Image:CDW dot.png]][[Image:CDW 6.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * A~2xI2pxA1: [&Delta;]x[&infin;]x[ ] [[Image:CD righttriangle-000.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]
 * B~2xA1: [4,3,4]x[ ] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CD 2.png]][[Image:CD dot.png]]

Rank 6
(5-space)
 * A4xI~1: [3,3,3]x[&infin;] - [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * B4xI~1: [4,3,3]x[&infin;] - [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * F4xI~1: [3,4,3]x[&infin;] - [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * H4xI~1: [5,3,3]x[&infin;] - [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]]
 * D4xI~1: [31,1,1] x [&infin;] - [[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CD 2.png]][[Image:CD dot.png]][[Image:CD infin.png]][[Image:CD dot.png]]
 * A3xI~1xA1: [3,3] x [&infin;] x [ ] - [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * B3xI~1xA1: [4,3] x [&infin;] x [ ] - [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * H3xI~1xA1: [5,3] x [&infin;] x [ ] - [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * I~1xI2qxI2r: [&infin;] x [q] x [r] - [[Image:CDW dot.png]][[Image:CDW infin.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW q.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]][[Image:CDW r.png]][[Image:CDW dot.png]]


 * A~4xA1: (3 3 3 3)x[ ] [[Image:CD downbranch-00.png]][[Image:CD downbranch-33.png]][[Image:CD righttriangleopen 000.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * B~4xA1: [4,3,3,4]x[ ] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * C~4xA1: [4,3,31,1]x[ ] [[Image:CD dot.png]][[Image:CD 4.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]
 * D~4xA1: [31,1,1,1]x[ ] [[File:CDT dot.png]][[File:CDT 3a.png]][[File:CDT branch000.png]][[File:CDT 3a.png]][[File:CDT dot.png]][[Image:CDT 2a.png]][[Image:CDT dot.png]]
 * F~4xA1: [3,4,3,3]x[ ] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 2.png]][[Image:CDW dot.png]]

Rank 10
= Hyperbolic compact =

Rank 3
(2-space)
 * [p,q] [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW_q.png]][[Image:CDW_dot.png]], p,q>=3, p+q>9
 * [p,q,r:] [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW_q.png]][[Image:CDW_dot.png]][[Image:CDW_r.png]], p,q,r>=3, p+q+r>9

Rank 4
(3-space)
 * [3,5,3] [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]] (Order-3 icosahedral honeycomb)
 * [5,3,4] [[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]] (Order-5 cubic honeycomb/Order-4 dodecahedral honeycomb)
 * [5,3,5] [[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]] (Order-5 dodecahedral honeycomb)
 * [5,31,1] [[Image:CD dot.png]][[Image:CD 5.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]]
 * (4 3 3 3) [[Image:CD downbranch-00-left-4.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]]
 * (5 3 3 3) [[Image:CD downbranch-00-left-5.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00.png]]
 * (4 3 3 3) [[Image:CD downbranch-00-left-4.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00-right-4.png]]
 * (4 3 5 3) [[Image:CD downbranch-00-left-4.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00-right-5.png]]
 * (5 3 5 3) [[Image:CD downbranch-00-left-5.png]][[Image:CD downbranch-33.png]][[Image:CD downbranch-00-right-5.png]]

(2-space)
 * (p q r s) [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW_q.png]][[Image:CDW_dot.png]][[Image:CDW_r.png]][[Image:CDW_dot.png]][[Image:CDW_s.png]], p,q,r,s>=2, p+q+r+s>8

Rank 5
(4-space)
 * [5,3,3,3] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]] (Order-5 pentachoronic tetracomb/Order-3 hecatonicosachoronic tetracomb)
 * [5,3,3,4] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]] (Order-5 tesseractic tetracomb/Order-4 hecatonicosachoronic tetracomb)
 * [5,3,3,5] [[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 5.png]][[Image:CDW dot.png]] (Order-5 hecatonicosachoronic tetracomb)
 * [5,3,31,1] [[Image:CD dot.png]][[Image:CD 5.png]][[Image:CD dot.png]][[Image:CD 3.png]][[Image:CD downbranch-00.png]][[Image:CD 3.png]][[Image:CD dot.png]] (Order-5 demitesseractic tetracomb)
 * (4 3 3 3 3) [[Image:CDW 3b.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3b.png]]

Rank 6
Noncompact!

(5-space)
 * [3,4,3,3,3] [[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * [3,3,4,3,3][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]
 * [4,3,4,3,3][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]

Rank 10
= Hyperbolic noncompact =

Rank 3
(2-space)
 * [p,&infin;] [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]], p>=3
 * (p q &infin;) [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW_q.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]], p,q>=3, p+q>6
 * (p &infin; &infin;) [[Image:CDW_dot.png]][[Image:CDW_p.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]], p>=3
 * (&infin; &infin; &infin;) [[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]][[Image:CDW_dot.png]][[Image:CDW_infin.png]]

Rank 4
(3-space)
 * [6,3,3] [[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]]
 * [4,4,3] [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]]
 * [3,6,3] [[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]]
 * [6,3,4] [[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]
 * [4,4,4] [[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]][[Image:CDW_4.png]][[Image:CDW_dot.png]]
 * [6,3,5] [[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_5.png]][[Image:CDW_dot.png]]
 * [6,3,6] [[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]][[Image:CDW_3.png]][[Image:CDW_dot.png]][[Image:CDW_6.png]][[Image:CDW_dot.png]]

Rank 5
(4-space)
 * [4,3,4,3] [[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]][[Image:CDW 4.png]][[Image:CDW dot.png]][[Image:CDW 3.png]][[Image:CDW dot.png]]

Finite Coxeter groups
Families of convex uniform polytopes are defined by Coxeter groups.

Note: (Alternate names as Simple Lie groups also given)
 * 1) An forms the simplex polytope family. (Same An)
 * 2) Bn is the family of demihypercubes, beginning at n=4 with the 16-cell, and n=5 with the demipenteract. (Also named Dn)
 * 3) Cn forms the hypercube polytope family. (Same Cn)
 * 4) D2n forms the regular polygons. (Also named I 1 n)
 * E6,E7,E8 are the generators of the Gosset Semiregular polytopes (Same E6,E7,E8)
 * 1) F4 is the 24-cell polychoron family. (Same F4)
 * 2) G3 is the dodecahedron/icosahedron polyhedron family. (Also named H 3 )
 * 3) G4 is the 120-cell/600-cell polychoron family. (Also named H 4 )