Talk:Coxeter–Dynkin diagram

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[edit] Graphics

Coxeter-Dynkins graphics on Wikipedia:

These component elements can be strung together to create linear diagrams.

There's two versions, one for whole numbers, and a taller one that leaves room for fractions.

Type ring node hole unused p q r s t u v w
Whole Image:CDW_ring.png Image:CDW_dot.png Image:CDW_hole.png Image:CDW_dash.png Image:CDW_p.png Image:CDW_q.png Image:CDW_r.png Image:CDW_s.png Image:CDW_t.png Image:CDW_u.png Image:CDW_v.png Image:CDW_w.png
Whole (svg) Image:CDW_ring.svg Image:CDW_dot.svg Image:CDW_hole.svg Image:CDW_dash.svg Image:CDW_p.svg Image:CDW_q.svg Image:CDW_r.svg Image:CDW_s.svg Image:CDW_t.svg Image:CDW_u.svg Image:CDW_v.svg
Fractional Image:CD_ring.png Image:CD_dot.png Image:CD_hole.png   Image:CD_p.png Image:CD_q.png Image:CD_r.png Image:CD_s.png Image:CD_t.png Image:CD_u.png Image:CD_v.png Image:CD_w.png Image:CD p-q.png

Attempted branching symbols that fit with tall fractional symbols:

b00 b10 b11 b00 b01 b10 b11 bsnub b33 bopen
Image:CD leftbranch-00.png Image:CD leftbranch-10.png Image:CD leftbranch-11.png Image:CD downbranch-00.png Image:CD downbranch-10.png Image:CD downbranch-01.png Image:CD downbranch-11.png Image:CD downbranch-snub.png Image:CD downbranch-33.png Image:CD downbranch-open.png Image:CD righttriangle-000.png Image:CD righttriangle-111.png Image:CD righttriangleopen 000.png Image:CD righttriangleopen 111.png Image:CD downbranch-11 open.png Image:CD downbranch-00-left-4.png Image:CD downbranch-00-left-5.png Image:CD downbranch-00-right-4.png Image:CD downbranch-00-right-5.png
Type ∞ 2 2b 3 3b 3/2 4 4/3 5 5/2 5/3 5/4 6 6/5 7 7/2 7/3 7/4 7/5 7/6
Whole Image:CDW_infin.png Image:CDW_2.png Image:CDW 2b.png Image:CDW_3.png Image:CDW_3b.png   Image:CDW_4.png   Image:CDW_5.png       Image:CDW_6.png   Image:CDW_7.png          
Fractional Image:CD_infin.png Image:CD_2.png Image:CD_2b.png Image:CD_3.png Image:CD_3b.png Image:CD_3-2.png Image:CD_4.png Image:CD_4-3.png Image:CD_5.png Image:CD_5-2.png Image:CD_5-3.png Image:CD_5-4.png Image:CD_6.png Image:CD_6-5.png Image:CD_7.png Image:CD_7-2.png Image:CD_7-3.png Image:CD_7-4.png Image:CD_7-5.png Image:CD_7-6.png
Type 8 8/3 8/5 8/7 9 9/2 9/4 9/5 9/7 10 10/3 10/7 10/9
Whole Image:CDW_8.png       Image:CDW_9.png         Image:CDW_10.png      
Fractional Image:CD_8.png Image:CD_8-3.png Image:CD_8-5.png Image:CD 8-7.png Image:CD_9.png Image:CD_9-2.png Image:CD_9-4.png Image:CD 9-7.png Image:CD 9-8.png Image:CD_10.png Image:CD_10-3.png Image:CD_10-7.png Image:CD_10-9.png
Type 11 11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9 11/10 12 12/5 12/7 12/11
Whole Image:CDW_11.png                   Image:CDW_12.png      
Fractional Image:CD_11.png Image:CD 11-2.png Image:CD 11-3.png Image:CD 11-4.png Image:CD 11-5.png Image:CD 11-6.png Image:CD 11-7.png Image:CD 11-8.png Image:CD 11-9.png Image:CD 11-10.png Image:CD_12.png Image:CD_12-5.png Image:CD_12-7.png Image:CD_12-11.png

Others:

  • B4 group
    • Image:CD_b4-0-100.png Image:CD_b4-0-110.png Image:CD_b4-0-111.png Image:CD_b4-1-000.png
    • Image:CD_b4-1-100.png Image:CD_b4-1-110.pngImage:CD_b4-1-111.pngImage:CD_b4-snub.png
  • Pn groups
    1. P3 Image:CD righttriangle-000.png
    2. P4 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD downbranch-00.png
    3. P5 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD righttriangleopen 000.png
    4. P6 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-00.png
    5. P7 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD righttriangleopen 000.png
    6. P8 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-00.png
    7. P9 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD righttriangleopen 000.png
    8. P10 Image:CD downbranch-00.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-open.pngImage:CD downbranch-33.pngImage:CD downbranch-00.png
  • P4 group
    • Image:CD_p4-1000.png Image:CD_p4-1100.pngImage:CD_p4-1010.pngImage:CD_p4-1110.pngImage:CD_p4-1111.png
  • Regular polyhedra truncations
    1. Image:Dynkins-100.png
    2. Image:Dynkins-010.png
    3. Image:Dynkins-001.png
    4. Image:Dynkins-110.png
    5. Image:Dynkins-011.png
    6. Image:Dynkins-101.png
    7. Image:Dynkins-111.png
    8. Image:Dynkins-sss.png
  • Regular polychora truncations
    1. Image:Dynkins-1000.png
    2. Image:Dynkins-0100.png
    3. Image:Dynkins-0010.png
    4. Image:Dynkins-0001.png
    5. Image:Dynkins-1100.png
    6. Image:Dynkins-0110.png
    7. Image:Dynkins-0011.png
    8. Image:Dynkins-1010.png
    9. Image:Dynkins-0101.png
    10. Image:Dynkins-1001.png
    11. Image:Dynkins-1110.png
    12. Image:Dynkins-0111.png
    13. Image:Dynkins-1101.png
    14. Image:Dynkins-1011.png
    15. Image:Dynkins-1111.png

[edit] by the way

Have the hyperbolic kaleidoscopes been enumerated? —Tamfang 07:01, 20 April 2007 (UTC)

Not given in wikipedia so far - Existence includes truncation permutations for the regular forms given at List of regular polytopes, but I don't know of any bifurcating graphs. Bitruncation lists some fun hyperbolic truncations. Tom Ruen 19:48, 20 April 2007 (UTC)

Wendy Krieger recently listed them in Talk:Polychoron. —Tamfang 08:21, 29 September 2007 (UTC)

There are an infinite number of groups with 3 nodes. In essence, {p,q,r:} (triangle marked p,q,r), except for those that occur in Euclidean and Spheric space. These are "finite extent" groups, because the symmetry has a finite linear size.

For bollochora, there are just the nine: 534, 353, 535, 53A, 3343:, 3353:, 3434:, 3435:, and 3535:

For bollotera, there are just five: 5333, 5334, 5335, 533A, amd 33334: .

No more exist in higher space (then 4dt or 5dp dt = dimensional-tilings, dp = dimensional polytopes.

For "finite content" groups, the cells have finite content, but one or more "horns" that stretch to the horizon. Because these converge logrithmically, there is little content in these horns.

If horotopic points (on the horizon) are permitted, then the number of wythoff-groups goes to 72, stretching to 9dt = 10dp. Unlike the "finite extent" groups, there is a lot of inter-relation between the groups.

If infinite cells are allowed, one gets every possible graph allowed. These are 'frieze-groups', in that there is transport in a small sector of the space, while the symmetry is bounded inside a laminotope (polytope bounded by unbounded face(t)s. A strip or layer is an example). One can 'cross' friezes to get symmetries, such as the octahedron-octahedral groups do. (these use groups like 8,3,4 ×8,3,4, 8,4,A × 8,4,A and 8,3,4,3 × 8,2,8,3.

--Wendy.krieger 07:58, 6 October 2007 (UTC)