Tilings

Higher-dimensional geometry (previously "Polyshapes").

Tilings

Postby Marek14 » Sun Feb 15, 2015 12:15 pm

I dusted off (re-created) my old research in uniform tilings. Works for spherical, Euclidean and hyperbolic ones.

Here's a sample for 3- and 4-polygon vertices:

Code: Select all
AAA

  a
a A a : A xA -A

A/A-A/A-A/A (A)

---

AAB

  b
a A a : A xC -B

  a
b A b : B xB -A

  a
a B a : C xA -C

A/C-C/A-B/B (AB|C)

---

ABC

  c
b A b : A xF -B

  b
c A c : B xD -A

  c
a B a : C xE -D

  a
c B c : D xB -C

  b
a C a : E xC -F

  a
b C b : F xA -E

A/F-E/C-D/B | B/D-C/E-F/A (AB|CD|EF)

----

AAAA

a a a
a A a : A xA -A

A/A-A/A-A/A-A/A (A)

----

AAAB

a b a
a A a : A xE -D,B2

b a a
a A b : B1 xB1 -A

b a b
a A a : C xD -B1,C

a a a
b A b : D xC -A

a a b
b A a : B2 xB2 -B1,C

a a a
a B a : E xA -E

A/E-E/A-D/C-B1/B1 | A/E-E/A-B2/B2-C/D (AD,AB2CB1|EE)(ADAB2CB1|E,E)
A/E-E/A-D/C-C/D (AD,C|E)
A/E-E/A-B2/B2-B1/B1 (AB2B1|E)

----

AABB

b b b
a A a : A xD -B

b a b
b A b : B xB -A

a b a
a B a : C xC -D

a a a
b B b : D xA -C

A/D-C/C-D/A-B/B (AB|CD)

----

ABAB

a b a
b A b : A xB -A

b a b
a B a : B xA -B

A/B-B/A-A/B-B/A (A|B)

----

AABC

b c b
a A a : A xI -E,C1

c b c
a A a : B xG -C1,D

c a b
b A c : C1 xC1 -A

c a c
b A b : D xE -B

b a b
c A c : E xD -A

b a c
c A b : C2 xC2 -B

a c a
a B a : F xH -G

a a a
c B c : G xB -F

a b a
a C a : H xF -I

a a a
b C b : I xA -H

A/I-H/F-G/B-C1/C1 | B/G-F/H-I/A-C2/C2 (AC2BC1|FG|HI)
A/I-H/F-G/B-D/E | B/G-F/H-I/A-E/D (AE,BD|FG|HI)

----

ABAC

a c a
b A b : A xD -B

a b a
c A c : B xC -A

c a c
a B a : C xB -C

b a b
a C a : D xA -D

A/D-D/A-B/C-C/B (AB|C|D)

----

ABCD

c d c
b A b : A xH -B

c b c
d A d : B xD -A

d c d
a B a : C xE -D

d a d
c B c : D xB -C

a b a
d C d : E xC -F

a d a
b C b : F xG -E

b c b
a D a : G xF -H

b a b
c D c : H xA -G

A/H-G/F-E/C-D/B | B/D-C/E-F/G-H/A (AB|CD|EF|GH)


If we use one as an example of my algorithm:

AABC - general configuration of the tiling. This case is a-gon, a-gon, b-gon and c-gon around a vertex, in this order. If a,b and c are not all distinct, this can be used for colored tilings where some cell types are colored with different colors, preserving uniformity. Some polygons can be 2-gons -- this is considered as colored or otherwise highlighted edges.

b c b
a A a : A xI -E,C1

c b c
a A a : B xG -C1,D

c a b
b A c : C1 xC1 -A

c a c
b A b : D xE -B

b a b
c A c : E xD -A

b a c
c A b : C2 xC2 -B

a c a
a B a : F xH -G

a a a
c B c : G xB -F

a b a
a C a : H xF -I

a a a
b C b : I xA -H

These are edge configurations. The capital letter is mother polygon, then you see the sequence of polygons around it, from preceding edge to the one we see to the following one. Each configuration then receives a marker (a letter, with 1 or 2 for chiral configurations). Turning the configuration upside down will give a corresponding configuration (shown after x). After the dash, you see which edges can follow this one.

A/I-H/F-G/B-C1/C1 | B/G-F/H-I/A-C2/C2 (AC2BC1|FG|HI)
A/I-H/F-G/B-D/E | B/G-F/H-I/A-E/D (AE,BD|FG|HI)

This is the heart of the algorithm. It shows a sequence of edges around a vertex (the sequence after | is the mirror image in cases where the vertex is chiral). In this case, the basic sequence itself is chiral, but even achiral sequences like AAAB can have chiral vertices. In parenthesis, you see polygons that form that particular tiling, with | separating their groups.

So in this case, AABC, there are two solutions: One with AC2BC1 a-gons, which means that their edges are adjacent to c, a, b, and a (and a must be divisible by 4), the second uses AE a-gons (with edges adjacent to c and a, alternating) and BD a-gons (with edges adjacent to b and a, alternating). b-gons and c-gons have always the same form.

You can imagine this tiling as 4,4,4A,4B, a kind of tri-coloring of {4,4}.
Marek14
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Re: Tilings

Postby wendy » Mon Feb 16, 2015 11:24 am

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.
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the dream we dream together is reality.
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Re: Tilings

Postby Marek14 » Mon Feb 16, 2015 4:27 pm

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)
Marek14
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