wendy wrote:I have been spending some time with John Conway's solution (of sorts).
As i pointed out to him, it's a hopeless mess, but it contains all of the possible solutions. It contains a lot of garbage too.
Anyway, time is spent to see if one can turn the orbifold into something that can be decorated, in the manner that the coxeter-dynkin symbol supports the wythoff mirror construction.
It's awfully complex here, because there are 'active regions' which can interact with one or more elements, and the 'join' operator works here. That is, you can open up any vertex figure, and insert a different vertex figure (Conway's × and o operators are of this kind). I even have not found if there is a formula for the active content of a given group, although this is known to be quite large except for three-element groups (zB 7 * 7).
The laws of symmetry don't really work for four operators or more. The group "2 2 2 2" is a subgroup of both {3,6} and {4,4}, because the group corresponds to a rotated rectangle, with digonals on all four sides.
I have now completed all possible configurations for 3-5 vertices and work on 6.
This, for example, is a result for the weird configuration AABABB:
- Code: Select all
AABABB
a b b a a
b a A b b : A1 xP1 -H1,G2,I,C2
a b b a b
b a A a b : B1 xU1 -J1,K
a b b b a
b a A b a : C1 xX1 -L,E2,H2,A2
a b b b a
b a A a b : D x Z -M,J2
b a b a a
b a A b b : E1 xQ1 -H1,G2,I,C2
b a b a b
b a A a b : F xV -J1,K
b a b b a
b a A b a : G1 xW2 -L,E2,H2,A2
b a b b a
b a A a b : B2 xU2 -M,J2
a b b a a
a b A b b : H1 xR1 -H1,G2,I,C2
a b b a b
a b A a b : G2 xW1 -J1,K
a b b b a
a b A b a : I xY -L,E2,H2,A2
a b b b a
a b A a b : C2 xX2 -M,J2
b b a b a
a b A b b : J1 xJ1 -A1,B1,C1,D
b b a b b
a b A b a : K xM -E1,F,G1,B2
a a b a a
b b A b b : L xS -H1,G2,I,C2
a a b a b
b b A a b : E2 xQ2 -J1,K
a a b b a
b b A b a : H2 xR2 -L,E2,H2,A2
a a b b a
b b A a b : A2 xP2 -M,J2
a b a b a
b b A b b : M xK -A1,B1,C1,D
a b a b b
b b A b a : J2 xJ2 -E1,F,G1,B2
b a b a a
a a B a b : N1 xN1 -X1,W2,Y,C2
b a b a b
a a B a a : O xT -Z,U2,X2,P2
b b a a b
a a B b a : P1 xA1 -N1,O
b b a a b
a a B a b : Q1 xE1 -P1,Q1,R1,S
b b a b a
a a B b a : R1 xH1 -T,N2
b b a b b
a a B a a : S xL -U1/V/W1/Q2
a a b a a
b a B a b : T xO -X1,W2,Y,C2
a a b a b
b a B a a : N2 xN2 -Z,U2,X2,P2
b a a a b
b a B b a : U1 xB1 -N1,O
b a a a b
b a B a b : V xF -P1,Q1,R1,S
b a a b a
b a B b a : W1 xG2 -T,N2
b a a b b
b a B a a : Q2 xE2 -U1/V/W1/Q2
a b a a b
a b B b a : X1 xC1 -N1,O
a b a a b
a b B a b : W2 xG1 -P1,Q1,R1,S
a b a b a
a b B b a : Y xI -T,N2
a b a b b
a b B a a : R2 xH2 -U1/V/W1/Q2
b a a a b
a b B b a : Z xD -N1,O
b a a a b
a b B a b : U2 xB2 -P1,Q1,R1,S
b a a b a
a b B b a : X2 xC2 -T,N2
b a a b b
a b B a a : P2 xA2 -U1/V/W1/Q2
A1/P1-N1/N1-W2/G1-A2/P2-W1/G2-J1/J1 | G1/W2-P1/A1-G2/W1-N2/N2-P2/A2-J2/J2 (A1G2J1,G1A2J2|N1W2P1,N2P2W1)
A1/P1-N1/N1-W2/G1-A2/P2-W1/G2-K/M | G1/W2-P1/A1-G2/W1-N2/N2-P2/A2-M/K (A1G2KG1A2M|N1W2P1,N2P2W1)
A1/P1-N1/N1-Y/I-A2/P2-V/F-J1/J1 | F/V-P1/A1-I/Y-N2/N2-P2/A2-J2/J2 (A1IA2J2FJ1|N1YN2P2VP1)
A1/P1-N1/N1-Y/I-A2/P2-V/F-K/M | F/V-P1/A1-I/Y-N2/N2-P2/A2-M/K (A1IA2M,FK|N1YN2P2VP1)
A1/P1-O/T-W2/G1-A2/P2-W1/G2-J1/J1 | G1/W2-P1/A1-G2/W1-T/O-P2/A2-J2/J2 (A1G2J1,G1A2J2|OP2W1TW2P1)
A1/P1-O/T-W2/G1-A2/P2-W1/G2-K/M | G1/W2-P1/A1-G2/W1-T/O-P2/A2-M/K (A1G2KG1A2M|OP2W1TW2P1)
A1/P1-O/T-Y/I-A2/P2-V/F-J1/J1 | F/V-P1/A1-I/Y-T/O-P2/A2-J2/J2 (A1IA2J2FJ1|OP2VP1,TY)
A1/P1-O/T-Y/I-A2/P2-V/F-K/M | F/V-P1/A1-I/Y-T/O-P2/A2-M/K (A1IA2M,FK|OP2VP1,TY)
B1/U1-N1/N1-Y/I-L/S-U1/B1-J1/J1 | B2/U2-S/L-I/Y-N2/N2-U2/B2-J2/J2 (B1J1,B2J2,IL|N1YN2U2SU1)
B1/U1-N1/N1-Y/I-L/S-U1/B1-K/M | B2/U2-S/L-I/Y-N2/N2-U2/B2-M/K (B1KB2M,IL|N1YN2U2SU1)
B1/U1-N1/N1-R2/H2-H2/R2-U1/B1-J1/J1 | B2/U2-R1/H1-H1/R1-N2/N2-U2/B2-J2/J2 (B1J1,H2|N1R2U1)/(B2J2,H1|R1N2U2)
B1/U1-N1/N1-R2/H2-H2/R2-U1/B1-K/M | B2/U2-R1/H1-H1/R1-N2/N2-U2/B2-M/K (B1KB2M,H1,H2|N1R2U1,R1N2U2)
B1/U1-O/T-Y/I-L/S-U1/B1-J1/J1 | B2/U2-S/L-I/Y-T/O-U2/B2-J2/J2 (B1J1,B2J2,IL|OU2SU1,TY)
B1/U1-O/T-Y/I-L/S-U1/B1-K/M | B2/U2-S/L-I/Y-T/O-U2/B2-M/K (B1KB2M,IL|OU2SU1,TY)
B1/U1-O/T-R2/H2-H2/R2-U1/B1-J1/J1 | B2/U2-R1/H1-H1/R1-T/O-U2/B2-J2/J2 (B1J1,B2J2,H1,H2|OU2R1TR2U1)
B1/U1-O/T-R2/H2-H2/R2-U1/B1-K/M | B2/U2-R1/H1-H1/R1-T/O-U2/B2-M/K (B1KB2M,H1,H2|OU2R1TR2U1)
C1/X1-N1/N1-X1/C1-L/S-V/F-J1/J1 | F/V-S/L-C2/X2-N2/N2-X2/C2-J2/J2 (C1LC2J2FJ1|N1X1,SV,N2X2)
C1/X1-N1/N1-X1/C1-L/S-V/F-K/M | F/V-S/L-C2/X2-N2/N2-X2/C2-M/K (C1LC2M,FK|N1X1,SV,N2X2)
C1/X1-N1/N1-X1/C1-E2/Q2-Q2/E2-J1/J1 | E1/Q1-Q1/E1-C2/X2-N2/N2-X2/C2-J2/J2 (C1E2J1|N1X1,Q2)/(E1C2J2|Q1,N2X2)
C1/X1-N1/N1-X1/C1-E2/Q2-Q2/E2-K/M | E1/Q1-Q1/E1-C2/X2-N2/N2-X2/C2-M/K (C1E2KE1C2M|N1X1,Q1,N2X2,Q2)
C1/X1-O/T-X1/C1-L/S-V/F-J1/J1 | F/V-S/L-C2/X2-T/O-X2/C2-J2/J2 (C1LC2J2FJ1|OX2TX1,SV)
C1/X1-O/T-X1/C1-L/S-V/F-K/M | F/V-S/L-C2/X2-T/O-X2/C2-M/K (C1LC2M,FK|OX2TX1,SV)
C1/X1-O/T-X1/C1-E2/Q2-Q2/E2-J1/J1 | E1/Q1-Q1/E1-C2/X2-T/O-X2/C2-J2/J2 (C1E2J1,E1C2J2|OX2TX1,Q1,Q2)
C1/X1-O/T-X1/C1-E2/Q2-Q2/E2-K/M | E1/Q1-Q1/E1-C2/X2-T/O-X2/C2-M/K (C1E2KE1C2M|OX2TX1,Q1,Q2)
D/Z-N1/N1-W2/G1-L/S-W1/G2-J1/J1 | G1/W2-S/L-G2/W1-N2/N2-Z/D-J2/J2 (DJ2G1LG2J1|N1W2SW1N2Z)
D/Z-N1/N1-W2/G1-L/S-W1/G2-K/M | G1/W2-S/L-G2/W1-N2/N2-Z/D-M/K (DM,G1LG2K|N1W2SW1N2Z)
D/Z-N1/N1-Y/I-L/S-V/F-J1/J1 | F/V-S/L-I/Y-N2/N2-Z/D-J2/J2 (DJ2FJ1,IL|N1YN2Z,SV)
D/Z-N1/N1-Y/I-L/S-V/F-K/M | F/V-S/L-I/Y-N2/N2-Z/D-M/K (DM,FK,IL|N1YN2Z,SV)
D/Z-N1/N1-Y/I-E2/Q2-Q2/E2-J1/J1 | E1/Q1-Q1/E1-I/Y-N2/N2-Z/D-J2/J2 (DJ2E1IE2J1|N1YN2Z,Q1,Q2)
D/Z-N1/N1-Y/I-E2/Q2-Q2/E2-K/M | E1/Q1-Q1/E1-I/Y-N2/N2-Z/D-M/K (DM,E1IE2K|N1YN2Z,Q1,Q2)
D/Z-N1/N1-R2/H2-H2/R2-V/F-J1/J1 | F/V-R1/H1-H1/R1-N2/N2-Z/D-J2/J2 (DJ2FJ1,H1,H2|N1R2VR1N2Z)
D/Z-N1/N1-R2/H2-H2/R2-V/F-K/M | F/V-R1/H1-H1/R1-N2/N2-Z/D-M/K (DM,FK,H1,H2|N1R2VR1N2Z)
D/Z-O/T-W2/G1-L/S-W1/G2-J1/J1 | G1/W2-S/L-G2/W1-T/O-Z/D-J2/J2 (DJ2G1LG2J1|OZ,SW1TW2)
D/Z-O/T-W2/G1-L/S-W1/G2-K/M | G1/W2-S/L-G2/W1-T/O-Z/D-M/K (DM,G1LG2K|OZ,SW1TW2)
D/Z-O/T-Y/I-L/S-V/F-J1/J1 | F/V-S/L-I/Y-T/O-Z/D-J2/J2 (DJ2FJ1,IL|OZ,SV,TY)
D/Z-O/T-Y/I-L/S-V/F-K/M | F/V-S/L-I/Y-T/O-Z/D-M/K (DM,FK,IL|OZ,SV,TY)
D/Z-O/T-Y/I-E2/Q2-Q2/E2-J1/J1 | E1/Q1-Q1/E1-I/Y-T/O-Z/D-J2/J2 (DJ2E1IE2J1|OZ,Q1,TY,Q2)
D/Z-O/T-Y/I-E2/Q2-Q2/E2-K/M | E1/Q1-Q1/E1-I/Y-T/O-Z/D-M/K (DM,E1IE2K|OZ,Q1,TY,Q2)
D/Z-O/T-R2/H2-H2/R2-V/F-J1/J1 | F/V-R1/H1-H1/R1-T/O-Z/D-J2/J2 (DJ2FJ1,H1,H2|OZ,R1TR2V)
D/Z-O/T-R2/H2-H2/R2-V/F-K/M | F/V-R1/H1-H1/R1-T/O-Z/D-M/K (DM,FK,H1,H2|OZ,R1TR2V)
H1: (BaabB)
H2: (BbaaB)
B1J1: (AbabBabbA)
DM: (AbabBBabA)
FK: (AbbaBabbA)
B2J2: (AbbaBbabA)
IL: (BaabBbaaB)
A1G2J1: (AbabBaabBabbA)
C1E2J1: (AbabBbaaBabbA)
E1C2J2: (AbbaBaabBbabA)
G1A2J2: (AbbaBbaaBbabA)
A1IA2M: (AbabBaabBbaaBbabA)
B1KB2M: (AbabBabbAbbaBbabA)
C1LC2M: (AbabBbaaBaabBbabA)
DJ2FJ1: (AbabBbabAbbaBabbA)
E1IE2K: (AbbaBaabBbaaBabbA)
G1LG2K: (AbbaBbaaBaabBabbA)
A1G2KG1A2M: (AbabBaabBabbAbbaBbaaBbabA)
A1IA2J2FJ1: (AbabBaabBbaaBbabAbbaBabbA)
C1LC2J2FJ1: (AbabBbaaBaabBbabAbbaBabbA)
C1E2KE1C2M: (AbabBbaaBabbAbbaBaabBbaaB)
DJ2E1IE2J1: (AbabBbabAbbaBaabBbaaBabbA)
DJ2G1LG2J1: (AbabBbabAbbaBbaaBaabBabbA)
Q1: (AabbA)
Q2: (AbbaA)
N1X1: (AabaBaabA)
OZ: (AabaBabaA)
SV: (AabbAbbaA)
TY: (AbaaBaabA)
N2X2: (AbaaBabaA)
N1W2P1: (AabaBaabAabbA)
N1R2U1: (AabaBaabAbbaA)
R1N2U2: (AabbAbaaBabaA)
N2P2W1: (AbaaBabaAbbaA)
N1YN2Z: (AabaBaabAbaaBabaA)
OU2SU1: (AabaBabaAabbAbbaA)
OX2TX1: (AabaBabaAbaaBaabA)
OP2VP1: (AabaBabaAbbaAabbA)
R1TR2V: (AabbAbaaBaabAbbaA)
SW1TW2: (AabbAbbaAbaaBaabA)
N1W2SW1N2Z: (AabaBaabAabbAbbaAbaaBabaA)
N1YN2U2SU1: (AabaBaabAbaaBabaAabbAbbaA)
N1YN2P2VP1: (AabaBaabAbaaBabaAbbaAabbA)
N1R2VR1N2Z: (AabaBaabAbbaAabbAbaaBabaA)
OU2R1TR2U1: (AabaBabaAabbAbaaBaabAbbaA)
OP2W1TW2P1: (AabaBabaAbbaAbaaBaabAabbA)
The code at the end is description of each individual polygon solution that appears among the solutions. Among the 40 solutions (though some are duplicates obtained by switching A and B polygons), you can find:
D/Z-O/T-Y/I-L/S-V/F-K/M | F/V-S/L-I/Y-T/O-Z/D-M/K (DM,FK,IL|OZ,SV,TY) - the semiregular solution. This basically treats the tiling as a variant of ABCDEF -- each of A and B can be uniformly colored with three colors.
A1/P1-N1/N1-W2/G1-A2/P2-W1/G2-J1/J1 | G1/W2-P1/A1-G2/W1-N2/N2-P2/A2-J2/J2 (A1G2J1,G1A2J2|N1W2P1,N2P2W1) - the triangular solution. This provides a way to color {3,6} with two colors with this assymetric color configuration at a vertex.
If you wish to find solution for (4,4,3,4,3,3), you have B1/U1-N1/N1-R2/H2-H2/R2-U1/B1-J1/J1 | B2/U2-R1/H1-H1/R1-N2/N2-U2/B2-J2/J2 (B1J1,H2|N1R2U1)/(B2J2,H1|R1N2U2) which is a kind of snub existing in two chiral versions -- but you also get more complicated solution B1/U1-N1/N1-R2/H2-H2/R2-U1/B1-K/M | B2/U2-R1/H1-H1/R1-N2/N2-U2/B2-M/K (B1KB2M,H1,H2|N1R2U1,R1N2U2) which mixes both vertex chiralities together.
A 2-coloring of {4,6} under these conditions leads to 14 solutions altogether -- some of them are symmetrical, some form mirrored pairs:
A1/P1-O/T-Y/I-A2/P2-V/F-K/M | F/V-P1/A1-I/Y-T/O-P2/A2-M/K (A1IA2M,FK|OP2VP1,TY)
B1/U1-O/T-Y/I-L/S-U1/B1-J1/J1 | B2/U2-S/L-I/Y-T/O-U2/B2-J2/J2 (B1J1,B2J2,IL|OU2SU1,TY)
B1/U1-O/T-Y/I-L/S-U1/B1-K/M | B2/U2-S/L-I/Y-T/O-U2/B2-M/K (B1KB2M,IL|OU2SU1,TY)
C1/X1-N1/N1-X1/C1-L/S-V/F-K/M | F/V-S/L-C2/X2-N2/N2-X2/C2-M/K (C1LC2M,FK|N1X1,SV,N2X2)
C1/X1-O/T-X1/C1-L/S-V/F-K/M | F/V-S/L-C2/X2-T/O-X2/C2-M/K (C1LC2M,FK|OX2TX1,SV)
D/Z-N1/N1-Y/I-L/S-V/F-J1/J1 | F/V-S/L-I/Y-N2/N2-Z/D-J2/J2 (DJ2FJ1,IL|N1YN2Z,SV)
D/Z-N1/N1-Y/I-L/S-V/F-K/M | F/V-S/L-I/Y-N2/N2-Z/D-M/K (DM,FK,IL|N1YN2Z,SV)
D/Z-N1/N1-Y/I-E2/Q2-Q2/E2-K/M | E1/Q1-Q1/E1-I/Y-N2/N2-Z/D-M/K (DM,E1IE2K|N1YN2Z,Q1,Q2)
D/Z-O/T-W2/G1-L/S-W1/G2-K/M | G1/W2-S/L-G2/W1-T/O-Z/D-M/K (DM,G1LG2K|OZ,SW1TW2)
D/Z-O/T-Y/I-L/S-V/F-J1/J1 | F/V-S/L-I/Y-T/O-Z/D-J2/J2 (DJ2FJ1,IL|OZ,SV,TY)
D/Z-O/T-Y/I-L/S-V/F-K/M | F/V-S/L-I/Y-T/O-Z/D-M/K (DM,FK,IL|OZ,SV,TY)
D/Z-O/T-Y/I-E2/Q2-Q2/E2-K/M | E1/Q1-Q1/E1-I/Y-T/O-Z/D-M/K (DM,E1IE2K|OZ,Q1,TY,Q2)
D/Z-O/T-R2/H2-H2/R2-V/F-J1/J1 | F/V-R1/H1-H1/R1-T/O-Z/D-J2/J2 (DJ2FJ1,H1,H2|OZ,R1TR2V)
D/Z-O/T-R2/H2-H2/R2-V/F-K/M | F/V-R1/H1-H1/R1-T/O-Z/D-M/K (DM,FK,H1,H2|OZ,R1TR2V)