## Hyperbolic Tilings

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Hyperbolic Tilings

Marek mentioned the hyperbolic tiling 3,9,9,9.

Here is a pic of it, I just made (by means of tyler):
9993-4col.gif
3,9,9,9 hyperbolic tiling - 2-coloring of both, the triangles and the nonagons
This one uses a 2-coloring for both the triangles and the nonagons, and thereby implicitely induces a 2-coloring of the vertices too. But this one is nicer and somehow looks easier to "understand" in its local structure than the version with identified ones each, which then surely would be uniform.

But still I have no idea for a broader theory on such exceptional hyperbolics: For this one does not seem to bend to any Dynkin diagram, nor the Theorem mentioned on my page on Coxeter domains - at least I don't see any so far.

Any further input to that quest?

--- rk
Klitzing
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### Re: Hyperbolic Tilings

Nice find. It's a "2/*3/3%". The twofold colouring is from a subgroup *3 3 3 3.

This polytope is a 2/*3/3%, which has more to do with a truncated cube than anything else.

Each of the polygons have full triangular symmetry, viz *3. The large symmetry is then * 3 3 3 3. Richard mentions a double-colouring. This equates to that one can use alternating colouring of the cells so that if a triangle were red, the adjacent 9-gons are green, and the opposite 9-gon is red. This opposite 9-gon is then against green triangles.

However, the fundemental region * 3 3 3 3 contains two vertices, and this is not where the unifrorm figure is transitive on. Instead, one notes that the digon between a pair of nine-gons has a rotational symmetry '2', and the overall symmetry is thus 2 * 3 3. This is not much different to the pyritohedral 3 * 2, that we can discuss the two together.

We begin by labeling the edges of the digon as /r, of the triangle as /m, and a third edge as /c, this appears in the end as a chord of the ninegon.

The ninegon then breaks down to a triangle 3/c with three attached trapezia. These are /m/c &# /r (or by Klitzing-notation: /m || /c). This is a trapezium formed by three consecutive sides of the ninegon, /r,/m,/r, and a base of 2.53208888076 being /c. This longer edge disappears to give a nine-gon, this is 3% is the core triangle, with missing edges, and the edges are actually 2/ (one edge) and *3/ (half an edge) in each of the six mirror-cells.
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wendy
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### Re: Hyperbolic Tilings

OK, now try to classify the AAAAB case

AAAAB

Edge types:
Code: Select all
`a b aa   aa A a : A xH -G,E2,B2b a aa   aa A b : B1 xB1 -Ab a aa   ba A a : C1 xE1 -B1,C1,Db a ba   aa A a : D xG -E1,F,C2a a ab   aa A b : E1 xC1 -Aa a ab   ba A a : F xF -B1,C1,Da a bb   aa A a : C2 xE2 -E1,F,C2a a aa   ab A b : G xD -Aa a aa   bb A a : E2 xC2 -B1,C1,Da a ba   ab A a : B2 xB2 -E1,F,C2a a aa   aa B a : H xA -H`

The vertex structure looks like this:

A/H
H/A
G/D E2/C2 B2/B2
E1/C1 F/F C2/E2
B1/B1 C1/E1 D/G

So there are theoretically 27 distinct configurations, although only some of them are possible; for example if you choose E2/C2 in third row, then you must have either E1/C1 or C2/E2 in fourth row.

The solutions are like this:

Solution 1: A/H-H/A-G/D-E1/C1-C1/E1 | A/H-H/A-E2/C2-C2/E2-D/G (AG,AE2DE1,C1,C2|HH) (AGAE2DE1,C1,C2|H)
In AAAB, one solution has two sub-solutions because A-type edge is not uniquely defined. This happens more often in AAAAB. Here, it can be surrounded as GAG and E1AE2 which are fundamentally distinct or as E1AG and GAE2 which are just mirror images. The "pure" solution 1a has a-gons divisible by four and b-gons even as their sides have to alternate between AG and AE2DE1 a-gons.
AG is a simple regular configuration with b-gon adjacent to every second edge of an a-gon while AE2DE1 is more complicated; b-gon is attached to every fourth edge and the a-gon two sides over (the D-edge) has two b-gons adjacent to it. C1 and C2 form third, chiral type of a-gon which is adjacent to b-gons only diagonally.
The mixed solution 1b retains the chiral C1 and C2, but replaces AG and AE2DE1 with AGAE2DE1 which needs to be divisible by 6. A hexagon with this configuration has two adjacent b-gons in meta position. b-gons can be odd here as they no longer alternate.

Solution 2: A/H-H/A-G/D-F/F-B1/B1 | A/H-H/A-B2/B2-F/F-D/G (AG,AB2FB1,DF|HH) (AGAB2FB1,DF|H) (AG,AB2FDFB1|HH), (AGAB2FDFB1|H)
This is an interesting solution as it has four separate cases. Not only do we have non-unique A-edges, but also F-edges! A-edges have a pure configuration (GAG and B1AB2) and a mixed configuration (B1AH and HAB2), and F-edges also have a pure configuration (DFD and B2FB1) and a mixed configuration (DFB1 and B2FD). And each of these can be specified independently, so we have:
2a: pure/pure solution. a-gons divisible by 4, b-gons even. You have AG a-gons from 1a. DF a-gons have no b-gons directly adjacent but the diagonal ones form different pattern than in C1 and C2. AB2FB1 is similar to AE2DE1, just with different configuration of diagonal b-gons.
2b: mixed/pure solution which retains DF but melds AG and AB2FB1 into AGAB2FB1, divisible by 6. b-gons can be odd.
2c: pure/mixed solution which retains AG but melds AB2FB1 and DF into AB2FDFB1, divisible by 6. b-gons must be even since they have alternating a-gons around them.
2d: mixed/mixed solution where all three types of a-gons meld into AGAB2FDFB1, divisible by 8. b-gons can be odd.

Solution 3: A/H-H/A-G/D-F/F-D/G (AG,DF|H)
The "most regular" solution with just simple AG and DF a-gons. This solution has nonchiral vertices.

Solution 4: A/H-H/A-G/D-C2/E2-C1/E1 | A/H-H/A-E2/C2-E1/C1-D/G (AG,AE2C1DC2E1|HH) (AGAE2C1DC2E1|H)
A-edges can be of pure type (GAG and E1AE2) or of mixed type (E1AG and GAE2).
4a: pure solution with simple AG a-gons and AE2C1DC2E1 divisible by 6. b-gons are even.
4b: mixed solution with AGAE2C1DC2E1 a-gons divisible by 8. b-gons can be odd.

Solution 5: A/H-H/A-E2/C2-E1/C1-B1/B1 | A/H-H/A-B2/B2-C2/E2-C1/E1 (AB2C2E1,AE2C1B1|H) (AB2C2E1AE2C1B1|HH)
A-edges can be of mixed type (E1AB2 and B1AE2) or of pure type (B1AB2 and E1AE2). However, here the mixed-type solution leads to two classes of a-gons and the pure-type solution to only one!
5a: mixed solution with a-gons divisible by 4. They are AB2C2E1 and its mirror image AE2C1B1. b-gons can be even.
5b: pure solution where both a-gon types merge into AB2C2E1AE2C1B1. b-gons must be even here as their edges alternate between B1AB2 and E1AE2 edges of the a-gons.

Solution 6: A/H-H/A-E2/C2-C2/E2-B1/B1 | A/H-H/A-B2/B2-E1/C1-C1/E1 (AB2E1,C1|H)/(AE2B1,C2|H) (AB2E1AE2B1,C1,C2|HH)
Here, once again, A-edges can be of mixed type (E1AB2 and B1AE2) or of pure type (B1AB2 and E1AE2).
6a: mixed solution with a-gons divisible by 3. This solution is chiral -- all vertices have the same chirality. You have either AB2E1 and C1 a-gons or AE2B1 and C2. The snub cube, snub dodecahedron and various snub {3,n} tilings all belong to this category.
6b: pure solution melding AB2E1 and AE2B1 into AB2E1AE2B1 divisible by 6. C1 and C2 from mixed solutions are still retained and now they are both present together. b-gons must be even here.

Solution 7: A/H-H/A-B2/B2-F/F-B1/B1 (AB2FB1|H)
Second solution with symmetrical vertices. a-gons are divisible by 4 and all have the form AB2FB1.

Only solutions 2d, 4b, 5b and 7 are "primitive" in the sense that all a-gons form only a single class. Perhaps 5a can be counted as well as all a-gons are isomorphic there except for chirality.

1a: 4|a, 2|b
1b: 6|a
2a: 4|a, 2|b
2b: 6|a
2c: 6|a, 2|b
2d: 8|a
3: 2|a
4a: 6|a, 2|b
4b: 8|a
5a: 4|a
5b: 8|a, 2|b
6a: 3|a
6b: 6|a, 2|b
7: 4|a

So this is the 14 solutions. Now for some examples:

n,3,3,3,3 - the snubs belong to 6a.

2,4,4,4,4 tiling:
This degenerate tiling is what you get when you take normal square tiling of Euclidean plane and mark one edge per vertex. 5 solutions work here, leading to 5 different patterns:

1a:
Code: Select all
`+---+===+---+---+|C1 |AE2|C2 |AE2||   |DE1|   |DE1|+---+---+---+===+|AE2║AG ║AE2|AG ||DE1║   ║DE1|   |+---+---+---+===+|C2 |AE2|C1 |AE2||   |DE1|   |DE1|+---+===+---+---+║AE2|AG |AE2║AG ║║DE1|   |DE1║   ║+---+===+---+---+`

2a:
Code: Select all
`+---+---+---+---+║AB2|DF |AB2║AG ║║FB1|   |FB1║   ║+---+---+---+---+|AB2║AG ║AB2|DF ||FB1║   ║FB1|   |+---+---+---+---+║AB2|DF |AB2║AG ║║FB1|   |FB1║   ║+---+---+---+---+|AB2║AG ║AB2|DF ||FB1║   ║FB1|   |+---+---+---+---+`

3:
Code: Select all
`+---+---+---+---+|DF |DF |DF |DF ||   |   |   |   |+---+---+---+---+║AG ║AG ║AG ║AG ║║   ║   ║   ║   ║+---+---+---+---+|DF |DF |DF |DF ||   |   |   |   |+---+---+---+---+║AG ║AG ║AG ║AG ║║   ║   ║   ║   ║+---+---+---+---+`

5a (since codes are too long, I mark the squares here just with L or R):
Code: Select all
`+---+===+---+---+|R  |R  |L  ║L  ||   |   |   ║   |+===+---+---+---+|R  |L  ║L  |R  ||   |   ║   |   |+---+---+---+===+|L  ║L  |R  |R  ||   ║   |   |   |+---+---+===+---+║L  |R  |R  |L  ║║   |   |   |   ║+---+===+---+---+`

7 (no markings are necessary as all squares look the same):
Code: Select all
`+---+---+---+---+|   ║   |   ║   ||   ║   |   ║   |+---+---+---+---+║   |   ║   |   ║║   |   ║   |   ║+---+---+---+---+|   ║   |   ║   ||   ║   |   ║   |+---+---+---+---+║   |   ║   |   ║║   |   ║   |   ║+---+---+---+---+`

3,4,4,4,4 is actually poorer - it lacks 1a and 2a solutions, keeping only 3, 5a and 7 because of odd b-gons. 4a,4,4,4,4 ({4,5} with one square per vertex of different color) has all five solutions but you need 6,4,4,4,4 to actually showcase all five "properly".

For a=6, there are 4 solutions that are always present (1b, 2b, 3 and 6a) and 3 more that only work for even b-gons (2c, 4a and 6b). 2,6,6,6,6 ({6,4} with one marked edge per vertex) and (4,6,6,6,6) can showcase all seven while 3,6,6,6,6 can only showcase the first four.

For a=8, there are 5 solutions that work for all b-gons (2d, 3, 4b, 5a and 7) and 3 solutions that only work for even b-gons (1a, 2a and 5b). 2,8,8,8,8 or 4,8,8,8,8 can showcase all 8, 3,8,8,8,8 can showcase the first five.

For a=9 (and also 15, 21 etc.), we get only the snub 6a solution again, like for a=3.
a=10 (and also 14, 22 etc.) has only the most regular solution 3.

a=12 has combination of a=4 and a=6 with 6 universal solutions (1b, 2b, 3, 5a, 6a and 7) and 5 even solutions (1a, 2a, 2c, 4a and 6b)

a=16 has the same 5+3 solutions as a=8.

a=18 has the same 4+3 solutions as a=6.

a=20 has the same 3+2 solutions as a=4.

a=24 has finally all 14 solutions: 8 universal (1b, 2b, 2d, 3, 4b, 5a, 6a and 7) and 6 even (1a, 2a, 2c, 4a, 5b and 6b).

a=5, 7, 11, 13, 17, 19 and 23 have no solutions. For a > 24, just divide by 24 and use the remainder.
Marek14
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### Re: Hyperbolic Tilings

Now let's have look at some other 5-valent tilings.

AAAAA is trivial, it corresponds to {n,5} tilings.

AAABB: this becomes interesting as I realized my method can be expanded a little bit to allow for more types than I originally thought.

Edge data:
Code: Select all
`b b ba   aa A a : A xF -D,B2b a ab   ba A b : B1 xB1 -Ab a bb   ba A a : C xD -B1,Ca a ab   bb A b : D xC -Aa a bb   bb A a : B2 xB2 -B1,Ca b aa   aa B a : E xE -Fa a aa   ab B b : F xA -E`

The vertex sequence is:

A/F
E/E
F/A
D/C B2/B2
B1/B1 C/D

Four possible sequences are translated into two symmetrical solutions and one chiral solution. These are related to solutions for AAAB.

Solution 1: A/F-E/E-F/A-D/C-B1/B1 | A/F-E/E-F/A-B2/B2-C/D
Let's try to analyze it a bit differently than before. We see that A, E and F all appear twice as a head (A/x) and twice as a tail (x/A). For A, we can see there's a pure form (B1AB2 and DAD) and a mixed form (B1AD and DAB2). Let's try the pure form first:

A/F-E/E-F/A'-D/C-B1/B1 | A'/F-E/E-F/A-B2/B2-C/D

Here I have the second type (DAD) a prime to separate it. The thing is: if A corresponds to F, then A' should naturally correspond to F', shouldn't it?

A/F-E/E-F'/A'-D/C-B1/B1 | A'/F'-E/E-F/A-B2/B2-C/D

And THIS introduces assymetry into E's: now we have pure E configuration (FEF and F'EF') and mixed E configuration (FEF' and F'EF), leading into two possibilities (if we mark the second E in every case):

Pure: A/F-E/E'-F'/A'-D/C-B1/B1 | A'/F'-E'/E-F/A-B2/B2-C/D
Mixed: A/F-E/E-F'/A'-D/C-B1/B1 | A'/F'-E'/E'-F/A-B2/B2-C/D

These now have ALL edge types unique and so they can be written out as:
1a - pure/pure: (AB2CB1,A'D|EF,E'F')
1b - pure/mixed: (AB2CB1,A'D|EF'E'F)

In 1b, b-gons are divisible by four and their F edges are alternately adjacent to the two distinct types of a-gons, while in 1a they only need to be even and each of them is exclusively adjacent to one type or the other.

How will this look for A-mixed solution?

A/F-E/E-F/A-D/C-B1/B1 | A'/F'-E/E-F'/A'-B2/B2-C/D

E-pure: FEF and F'E'F'
E-mixed: FEF' and F'E'F

1c - mixed/pure: A/F-E/E-F/A-D/C-B1/B1 | A'/F'-E'/E'-F'/A'-B2/B2-C/D (ADA'B2CB1|EF,E'F')
1d - mixed/mixed: A/F-E/E'-F/A-D/C-B1/B1 | A'/F'-E'/E-F'/A'-B2/B2-C/D (ADA'B2CB1|EF'E'F')

The difference is more subtle here -- there is only one type of a-gon, but it can be attached to a b-gon in two chiral orientations. 1c has chiral b-gons that are only attached to one orientation of a-gons while 1d has achiral b-gons that are alternately attached to both orientations.

This is the most regular solution.

Solution 3: A/F-E/E-F/A-B2/B2-B1/B1 (AB2B1|EF)
This is a solution with a-gons divisible by 3 that corresponds to 3,3,3,4,4 planar tiling.

So there are actually 6 solutions instead of 4 like I've originally thought. Let's tabulate them:

1a: 4|a, 2|b
1b: 4|a, 4|b
1c: 6|a, 2|b
1d: 6|a, 4|b
2: 2|a, 2|b
3: 3|a, 2|b

Examples:
a=3 (or 9) - for b=4 we get the laminate planar tiling. We get one solution (3) for every even b.

a=4 (or - we get two solutions (1a and 2) for every even b and one extra solution (1b) for b divisible by 4. Simplest examples are three distinct 2-colorings of {4,5}, two 4,4,4,6,6 tilings and three 4,4,4,8,8 tilings.

a=6 - we get three solutions (1c, 2 and 3) for every even b and one extra solution (1d) for b divisible by 4. Simplest example are four 6,6,6,4,4 tilings.

a=10 - we get only the most regular solution (2)

a=12 - we get four solutions (1a, 1c, 2 and 3) for every even b and two extra solutions (1b and 1d) for b divisible by 4. Simplest example are six 12,12,12,4,4 tilings.

a=5 (or 7, or 11) - no solutions.

For a > 12, take the remainder.
Marek14
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### Re: Hyperbolic Tilings

Let's see what my extension idea brings for next type of tilings, AABAB.

Edge data:

Code: Select all
`a b ab   aa A b : A1 xE1 -C,A2a b ab   ba A a : B xG -Da b aa   ab A b : C xF -C,A2a b aa   bb A a : A2 xE2 -Db a ba   ab A b : D xD -A1,Bb a aa   ba B a : E1 xA1 -E1,Fb a ba   aa B a : F xC -G,E2a a ab   ba B a : G XB -E1,Fa a bb   aa B a : E2 xA2 -G,E2`

Vertex sequence is:
A1/E1 B/G
E1/A1 F/C
C/F A2/E2
G/B E2/A2
D/D

So there are 16 possible vertex sequences that lead to a total of 4 distinct solutions and 6 solutions in total.

Solution 1: A1/E1-E1/A1-C/F-G/B-D/D | B/G-F/C-A2/E2-E2/A2-D/D
1a - D-pure (BDB and A2D'A1): A1/E1-E1/A1-C/F-G/B-D/D' | B/G-F/C-A2/E2-E2/A2-D'/D (A1CA2D',BD|E1,E2,FG)
1b - D-mixed (BDA1 and A2D'B): A1/E1-E1/A1-C/F-G/B-D/D | B/G-F/C-A2/E2-E2/A2-D'/D' (A1CA2D'BD|E1,E2,FG)

Solution 2: A1/E1-E1/A1-A2/E2-E2/A2-D/D (A1A2D|E1,E2)

Solution 3: A1/E1-F/C-A2/E2-G/B-D/D | B/G-E1/A1-C/F-E2/A2-D/D
3a - D-pure (BDB and A2D'A1): A1/E1-F/C-A2/E2-G/B-D/D' | B/G-E1/A1-C/F-E2/A2-D'/D (A1CA2D',BD|E1FE2G)
3b - D-mixed (BDA1 and A2D'B): A1/E1-F/C-A2/E2-G/B-D/D | B/G-E1/A1-C/F-E2/A2-D'/D' (A1CA2D'BD|E1FE2G)

Solution 4: B/G-F/C-C/F-G/B-D/D (BD,C|FG)

1a: 4|a, 2|b
1b: 6|a, 2|b
2: 3|a
3a: 4|a, 4|b
3b: 6|a, 4|b
4: 2|a, 2|b

Examples:

a=3: 1 solution (2) representing snub {n,n} tilings including planar 3,3,4,3,4

a=4: 2 solutions (1a and 4) for even b with one additional solution (3a) for b divisible by 4. Includes three 2-colorings of {4,5}, two 4,4,6,4,6 tilings and three 4,4,8,4,8 tilings.

a=6: 1 solution (2) for arbitrary b, 2 more solutions (1b and 4) for even b and one final solution (3b) for b divisible by 4. Includes a single 6,6,3,6,3 tiling and four 6,6,4,6,4 tilings.

a=12: 1 solution (2) for arbitrary b, 3 more solutions (1a, 1b and 4) for even b and 2 final solutions (3a and 3b) for b divisible by 4. Includes a single 12,12,3,12,3 tiling and six 12,12,4,12,4 tilings.
Marek14
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### Re: Hyperbolic Tilings

Type AAABC:

Edge data:
Code: Select all
`b c ba   aa A a : A xP -L,J2,G2,C2c b ca   aa A a : B xN -J1,K,H2,D2c a ab   ba A c : C1 xC1 -Ac a ab   ca A b : D1 xG1 -Bc a bb   ca A a : E1 xJ1 -C1,D1,E1,Fc a cb   ba A a : F xL -G1,H1,I,E2b a ac   ba A c : G1 xD1 -Ab a ac   ca A b : H1 xH1 -Bb a bc   ca A a : I xK -C1,D1,E1,Fb a cc   ba A a : E2 xJ2 -G1,H1,I,E2a a ac   bb A c : J1 xE1 -Aa a ac   cb A b : K xI -Ba a bc   cb A a : H2 xH2 -C1,D1,E1,Fa a cc   bb A a : D2 xG2 -G1,H1,I,E2a a ab   bc A c : L xF -Aa a ab   cc A b : J2 xE2 -Ba a bb   cc A a : G2 xD2 -C1,D1,E1,Fa a cb   bc A a : C2 xC2 -G1,H1,I,E2a c aa   aa B a : M xO -Na a aa   ac B c : N xB -Ma b aa   aa C a : O xM -Pa a aa   ab C b : P xA -O`

Vertex sequence:

A/P
O/M
N/B
J1/E1 K/I H2/H2 D2/G2
C1/C1 D1/G1 E1/J1 F/L

This type, unlike the previous ones, is intrinsically chiral, which means that we need to build a second vertex sequence that is a mirror image of the first. Previously, we've seen some solutions with symmetrical vertices; that can't happen here.

B/N
M/O
P/A
L/F J2/E2 G2/D2 C2/C2
G1/D1 H1/H1 I/K E2/J2

There's 16 possible sequences and 6 of those are solutions. This type is "coarse" enough to not admit any finer solutions: every valid sequence leads to one solution.

Solution 1: A/P-O/M-N/B-J1/E1-E1/J1 | B/N-M/O-P/A-J2/E2-E2/J2 (AJ2BJ1,E1,E2|MN|OP)
Solution 2: A/P-O/M-N/B-K/I-C1/C1 | B/N-M/O-P/A-C2/C2-I/K (AC2IC1,BK|MN|OP)
Solution 3: A/P-O/M-N/B-K/I-F/L | B/N-M/O-P/A-L/F-I/K (AL,BK,FI|MN|OP)
Solution 4: A/P-O/M-N/B-H2/H2-C1/C1 | B/N-M/O-P/A-C2/C2-H1/H1 (AC2H1BH2C1|MN|OP)
Solution 5: A/P-O/M-N/B-H2/H2-F/L | B/N-M/O-P/A-L/F-H1/H1 (AL,BH2FH1|MN|OP)
Solution 6: A/P-O/M-N/B-D2/G2-D1/G1 | B/N-M/O-P/A-G2/D2-G1/D1 (AG2D1BD2G1|MN|OP)

1: 4|a, 2|b, 2|c
2: 4|a, 2|b, 2|c
3: 2|a, 2|b, 2|c
4: 6|a, 2|b, 2|c
5: 4|a, 2|b, 2|c
6: 6|a, 2|b, 2|c

Solutions 2 and 5 are basically the same thing, only with b and c switched. This doesn't hold for "hexagonal" solutions 4 and 6 -- those are really intrinsically different.

Examples:

a=4: There are four solutions (1, 2, 3 and 5). This includes four three-colorings of {4,5} (two of them identical except for colors), four two-colorings of 4,4,4,4,6, four two-colorings of 4,4,4,6,6 (two of them identical except for colors) and four 4,4,4,6,8 tilings.
a=6: There are three solutions (3, 4 and 6). This includes three two-colorings of 6,6,6,4,4, three two-colorings of 6,6,6,6,4 and three 6,6,6,4,8 tilings.
a=12: There are six solutions (1, 2, 3, 4, 5 and 6). This includes six two-colorings of 12,12,12,4,4 (two of them identical except for colors) and six 12,12,12,4,6 tilings.
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABAC:

Edge data:
Code: Select all
`a c ab   aa A b : A1 xM1 -H,C2a c ab   ba A a : B xO -I,F2a b ac   aa A c : C1 xJ1 -E,A2a b ac   ca A a : D xL -F1,Ga c aa   ab A b : E xN -H,C2a c aa   bb A a : A2 xM2 -I,F2c a ba   ab A c : F1 xF1 -A1,Bc a ca   ab A b : G xI -C1,Da b aa   ac A c : H xK -E,A2a b aa   cc A a : C2 xJ2 -F1,Gb a ba   ac A c : I xG -A1,Bb a ca   ac A b : F2 xF2 -C1,Dc a aa   ca B a : J1 xC1 -J1,Kc a ca   aa B a : K xH -L,J2a a ac   ca B a : L xD -J1,Ka a cc   aa B a : J2 xC2 -L,J2b a aa   ba C a : M1 xA1 -M1,Nb a ba   aa C a : N xE -O,M2a a ab   ba C a : O xB -M1,Na a bb   aa C a : M2 xA2 -O,M2`

Vertex sequence:
A1/M1 B/O
M1/A1 N/E
H/K C2/J2
L/D J2/C2
F1/F1 G/I

Reverse vertex sequence:
C1/J1 D/L
J1/C1 K/H
E/N A2/M2
O/B M2/A2
I/G F2/F2

Out of 32 sequences, 8 are valid. This is a coarse type:

Solution 1: A1/M1-M1/A1-H/K-L/D-F1/F1 | D/L-K/H-A2/M2-M2/A2-F2/F2 (A1HA2F2DF1|KL|M1,M2)
Solution 2: A1/M1-M1/A1-H/K-L/D-G/I | D/L-K/H-A2/M2-M2/A2-I/G (A1HA2I,DG|KL|M1,M2)
Solution 3: A1/M1-M1/A1-C2/J2-J2/C2-F1/F1 | C1/J1-J1/C1-A2/M2-M2/A2-F2/F2 (A1C2F1|J2|M1)/(C1A2F2|J1|M2)
Solution 4: A1/M1-M1/A1-C2/J2-J2/C2-G/I | C1/J1-J1/C1-A2/M2-M2/A2-I/G (A1C2GC1A2I|J1,J2|M1,M2)
Solution 5: B/O-N/E-H/K-L/D-F1/F1 | D/L-K/H-E/N-O/B-F2/F2 (BF2DF1,EH|KL|NO)
Solution 6: B/O-N/E-H/K-L/D-G/I | D/L-K/H-E/N-O/B-I/G (BI,DG,EH|KL|NO)
Solution 7: B/O-N/E-C2/J2-J2/C2-F1/F1 | C1/J1-J1/C1-E/N-O/B-F2/F2 (BF2C1EC2F1|J1,J2|NO)
Solution 8: B/O-N/E-C2/J2-J2/C2-G/I | C1/J1-J1/C1-E/N-O/B-I/G (BI,C1EC2G|J1,J2|NO)

1: 6|a, 2|b
2: 4|a, 2|b
3: 3|a
4: 6|a
5: 4|a, 2|b, 2|c
6: 2|a, 2|b, 2|c
7: 6|a, 2|c
8: 4|a, 2|c

Solutions 1/7 and 2/8 form pairs and can be transformed into each other by switching b and c. Solution 3 is chiral.

Examples:

a=3: 1 solution exists (3), representing various snub {m,n} tilings. Includes a chiral 2-coloring of 3,3,4,3,4 and the 3,3,4,3,5 tiling.
a=4: 1 solution (2) for b even and c arbitrary, 1 solution (its twin for b arbitrary and c even, 2 extra solutions (5 and 6) exist for both b and c even. Includes a 2-coloring of 4,4,4,4,3, one 4,4,3,4,6 tiling, four 3-colorings of {4,5} (two of them identical except for colors), four 2-colorings of 4,4,4,4,6 and four 4,4,6,4,8 tilings.
a=6: 2 solutions (3 and 4) for arbitrary b and c, 1 solution (1) for b even and c arbitrary, 1 solution (its twin 7) for b arbitrary and c even and 1 solution (6) for b and c both even. Includes two 2-colorings of 6,6,3,6,3 (one of them chiral), two 6,6,3,6,5 tilings (one of them chiral), three 6,6,3,6,4 tilings (one of them chiral), five 2-colorings of 6,6,6,6,4 (one of them chiral, two of them identical except for colors) and five 6,6,4,6,8 tilings.
a=12: 2 solutions (3 and 4) for arbitrary b and c, 2 solutions (1 and 2) for b even and c arbitrary, 2 solutions (their twins 7 and for b arbitrary and c even and 2 solutions (5 and 6) for b and c both even. Includes two 2-colorings of 12,12,3,12,3 (one of them chiral), two 12,12,3,12,5 tilings (one of them chiral), four 12,12,3,12,4 tilings (one of them chiral), and eight 12,12,4,12,6 tilings.
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABBC:

Edge data:
Code: Select all
`b c bb   ba A a : A xL -E,C2b b bc   ca A a : B xI -C1,Dc a bb   bb A c : C1 xC1 -Ac a cb   bb A b : D xE -Bb a bb   bc A c : E xD -Ab a cb   bc A b : C2 xC2 -Bc b aa   aa B c : F1 xF1 -Hc b ca   aa B a : G xJ -Ia c aa   ab B b : H xK -J,F2a a ac   cb B b : I xB -F1,Ga b aa   ac B c : J xG -Ha b ca   ac B a : F2 xF2 -Ib b ba   aa C a : K xH -La a ab   bb C b : L xA -K`

Vertex sequence:
A/L
K/H
J/G F2/F2
I/B
C1/C1 D/E

Reverse vertex sequence:
B/I
F1/F1 G/J
H/K
L/A
E/D C2/C2

There are 4 possible sequences, all of them valid. This is a coarse type:

Solution 1: A/L-K/H-J/G-I/B-C1/C1 | B/I-G/J-H/K-L/A-C2/C2 (AC2BC1|GI,HJ|KL)
Solution 2: A/L-K/H-J/G-I/B-D/E | B/I-G/J-H/K-L/A-E/D (AE,BD|GI,HJ|KL)
Solution 3: A/L-K/H-F2/F2-I/B-C1/C1 | B/I-F1/F1-H/K-L/A-C2/C2 (AC2BC1|F1HF2I|KL)
Solution 4: A/L-K/H-F2/F2-I/B-D/E | B/I-F1/F1-H/K-L/A-E/D (AE,BD|F1HF2I|KL)

1: 4|a, 2|b, 2|c
2: 2|a, 2|b, 2|c
3: 4|a, 4|b, 2|c
4: 2|a, 4|b, 2|c

Solutions 1 and 4 are twins obtained by switching a and b.

Examples:
a=4, b=4: all four solutions exist, including 4 3-colorings of {4,5} (two of them identical except for colors) and 4 2-colorings of 4,4,4,4,6 (two of them identical except for colors)
a=4, b=6: two solutions exist (1 and 2), including two 2-colorings of 4,4,4,6,6 and two 4,4,6,6,8 tilings.
a=4, b=8: all four solutions exist, including 4 2-colorings of 4,4,4,8,8 and four 4,4,8,8,6 tilings.
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type ABABC:

Edge data:
Code: Select all
`b c ba   ab A b : A xP -G,E2,B2c b aa   bb A c : B1 xI1 -Ac b aa   cb A b : C1 xL1 -B1,C1,Dc b ca   ab A b : D xN -E1,F,C2a b ac   bb A c : E1 xJ1 -Aa b ac   cb A b : F xM -B1,C1,Da b cc   ab A b : C2 xL2 -E1,F,C2a b ab   bc A c : G xK -Aa b ab   cc A b : E2 xJ2 -B1,C1,Da b cb   ac A b : B2 xI2 -E1,F,C2a c ab   ba B a : H xO -N,L2,I2c a bb   aa B c : I1 xB1 -Hc a bb   ca B a : J1 xE1 -I1,J1,Kc a cb   ba B a : K xG -L1,M,J2b a bc   aa B c : L1 xC1 -Hb a bc   ca B a : M xF -I1,J1,Kb a cc   ba B a : J2 xE2 -L1,M,J2b a ba   ac B c : N xD -Hb a ba   cc B a : L2 xC2 -I1,J1,Kb a ca   bc B a : I2 xB2 -L1,M,J2a b ab   ba C a : O xH -Pb a ba   ab C b : P xA -O`

Vertex sequence:
A/P
O/H
N/D L2/C2 I2/B2
E1/J1 F/M C2/L2
I1/B1 J1/E1 K/G

Reverse vertex sequence:
B1/I1 C1/L1 D/N
H/O
P/A
G/K E2/J2 B2/I2
L1/C1 M/F J2/E2

There is 27 possible sequences, 4 of which are valid. This is a coarse type.

Solution 1: A/P-O/H-N/D-E1/J1-J1/E1 | D/N-H/O-P/A-E2/J2-J2/E2 (AE2DE1|HN,J1,J2|OP)
Solution 2: A/P-O/H-N/D-F/M-K/G | D/N-H/O-P/A-G/K-M/F (AG,DF|HN,KM|OP)
Solution 3: A/P-O/H-L2/C2-C2/L2-K/G | C1/L1-H/O-P/A-G/K-L1/C1 (AG,C1,C2|HL2KL1|OP)
Solution 4: A/P-O/H-I2/B2-F/M-I1/B1 | B1/I1-H/O-P/A-B2/I2-M/F (AB2FB1|HI2MI1|OP)

1: 4|a, 2|b, 2|c
2: 2|a, 2|b, 2|c
3: 2|a, 4|b, 2|c
4: 4|a, 4|b, 2|c

Solutions 1 and 3 are twins obtained by switching a and b.

Examples:
a=4, b=4: all four solutions exist, including 4 3-colorings of {4,5} (two of them identical except for colors) and 4 2-colorings of 4,4,4,4,6 (two of them identical except for colors)
a=4, b=6: two solutions exist (1 and 2), including two 2-colorings of 4,4,6,4,6 and two 4,6,4,6,8 tilings.
a=4, b=8: all four solutions exist, including 4 2-colorings of 4,4,8,4,8 and four 4,8,4,8,6 tilings.
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type ABBAC - trivial, only one solution exists for every even a and b; c is not constrained. Both a-gons and b-gons are simple alternations. Examples are a 2-coloring of 4,4,4,4,3 or the 4,6,6,4,3 tiling.

Type AABCD:

Edge data:
Code: Select all
`c d cb   ba A a : A xK -E,C2c b cd   da A a : B xG -C1,Dd a bc   cb A d : C1 xC1 -Ad a dc   cb A b : D xE -Bb a bc   cd A d : E xD -Ab a dc   cd A b : C2 xC2 -Bd c da   aa B a : F xI -Ga a ad   dc B c : G xB -Fa d aa   ab C b : H xJ -Ia b aa   ad C d : I xF -Hb c ba   aa D a : J xH -Ka a ab   bc D c : K xA -J`

Vertex sequence:
A/K
J/H
I/F
G/B
C1/C1 D/E

Reverse vertex sequence:
B/G
F/I
H/J
K/A
E/D C2/C2

There are 2 possible sequences, both of them valid. This is a coarse type.

Solution 1: A/K-J/H-I/F-G/B-C1/C1 | B/G-F/I-H/J-K/A-C2/C2 (AC2BC1|FG|HI|JK)
Solution 2: A/K-J/H-I/F-G/B-D/E | B/G-F/I-H/J-K/A-E/D (AE,BD|FG|HI|JK)

1: 4|a, 2|b, 2|c, 2|d
2: 2|a, 2|b, 2|c, 2|d

Examples:

Two 4-colorings of {4,5}
Two 4,4,6,8,10 tilings

Type ABACD:

Edge data:
Code: Select all
`c d ca   ab A b : A xL -E,C2d c da   ab A b : B xJ -C1,Da b ad   cc A d : C1 xF1 -Aa b ad   dc A c : D xH -Ba b ac   cd A d : E xG -Aa b ac   dd A c : C2 xF2 -Bd a cc   da B a : F1 xC1 -F1,Gd a dc   ca B a : G xE -H,F2c a cd   da B a : H xD -F1,Gc a dd   ca B a : F2 xC2 -H,F2a d ab   ba C a : I xK -Jb a ba   ad C d : J xB -Ia c ab   ba D a : K xI -Lb a ba   ac D c : L xA -K`

Vertex sequence:
A/L
K/I
J/B
C1/F1 D/H
F1/C1 G/E

Reverse vertex sequence:
B/J
I/K
L/A
E/G C2/F2
H/D F2/C2

There are 4 possible sequences, 2 of which are valid. This is a coarse type.

Solution 1: A/L-K/I-J/B-C1/F1-F1/C1 | B/J-I/K-L/A-C2/F2-F2/C2 (AC2BC1|F1,F2|IJ|KL)
Solution 2: A/L-K/I-J/B-D/H-G/E | B/J-I/K-L/A-E/G-H/D (AE,BD|GH|IJ|KL)

1: 4|a, 2|c, 2|d
2: 2|a, 2|b, 2|c, 2|d

Examples:

A 3-coloring of 4,4,4,4,3
The 4,3,4,6,8 tiling
Two 4-colorings of {4,5}
Two 4,6,4,8,10 tilings
A 4-coloring of 4,4,6,4,6
The 6,4,6,8,10 tiling

Type ABCDE - trivial, only one solution (a simple alternation) exists for every even a, b, c, d and e.
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

For 6-valents:

Type AAAAAA - trivial, {n,6} tilings including planar 3,3,3,3,3,3.

Type AAAAAB:

Edge data:
Code: Select all
`a a b a aa a A a a : A xL -K,I2,F2,B2a b a a aa a A b a : B1 xB1a b a a aa a A a b : C1 xF1 -B1,C1,D1,Ea b a a ba a A a a : D1 xI1 -F1,G1,H,D2a b a b aa a A a a : E xK -I1,J,G2,C2b a a a aa a A b a : F1 xC1b a a a aa a A a b : G1 xG1 -B1,C1,D1,Eb a a a ba a A a a : H xJ -F1,G1,H,D2b a a b aa a A a a : D2 xI2 -I1,J,G2,C2a a a a ab a A b a : I1 xD1a a a a ab a A a b : J xH -B1,C1,D1,Ea a a a bb a A a a : G2 xG2 -F1,G1,H,D2a a a b ab a A a a : C2 xF2 -I1,J,G2,C2a a a a aa b A b a : K xEa a a a aa b A a b : I2 xD2 -B1,C1,D1,Ea a a a ba b A a a : F2 xC2 -F1,G1,H,D2a a a b aa b A a a : B2 xB2 -I1,J,G2,C2a a a a aa a B a a : L xA -L`

Vertex sequence:
A/L
L/A
K/E I2/D2 F2/C2 B2/B2
I1/D1 J/H G2/G2 C2/F2
F1/C1 G1/G1 H/J D2/I2
B1/B1 C1/F1 D1/I1 E/K

There are 256 possible sequences, 30 of them valid. This is a fine type.
Solution 1: A/L-L/A-K/E-I1/D1-G1/G1-D1/I1 | A/L-L/A-I2/D2-G2/G2-D2/I2-E/K
1a - A-pure (KAK and I1A'I2): A'/L'-L/A-K/E-I1/D1-G1/G1-D1/I1 | A/L-L'/A'-I2/D2-G2/G2-D2/I2-E/K (AK,A'I2EI1,D1G1,D2G2|LL')
1b - A-mixed (I1AK and KA'I2): A/L-L/A-K/E-I1/D1-G1/G1-D1/I1 | A'/L'-L'/A'-I2/D2-G2/G2-D2/I2-E/K (AKAI2EI1,D1G1,D2G2|L,L')

Solution 2: A/L-L/A-K/E-I1/D1-H/J-D1/I1 | A/L-L/A-I2/D2-J/H-D2/I2-E/K
2a - A-pure (KAK and I1A'I2): A'/L'-L/A-K/E-I1/D1-H/J-D1/I1 | A/L-L'/A'-I2/D2-J/H-D2/I2-E/K (AK,A'I2EI1,D1HD2J|LL')
2b - A-mixed (I1AK and KA'I2): A/L-L/A-K/E-I1/D1-H/J-D1/I1 | A'/L'-L'/A'-I2/D2-J/H-D2/I2-E/K (AKA'I2EI1,D1HD2J|L,L')

Solution 3: A/L-L/A-K/E-J/H-F1/C1-C1/F1 | A/L-L/A-F2/C2-C2/F2-H/J-E/K
3a - A-pure (KAK and F1A'F2): A'/L'-L/A-K/E-J/H-F1/C1-C1/F1 | A/L-L'/A'-F2/C2-C2/F2-H/J-E/K (AK,A'F2HF1,C1,C2,EJ|LL')
3b - A-mixed (F1AK and KA'F2): A/L-L/A-K/E-J/H-F1/C1-C1/F1 | A'/L'-L'/A'-F2/C2-C2/F2-H/J-E/K (AKA'F2HF1,C1,C2,EJ|L,L')

Solution 4: A/L-L/A-K/E-J/H-G1/G1-B1/B1 | A/L-L/A-B2/B2-G2/G2-H/J-E/K
4a - A-pure (KAK and B1A'B2): A'/L'-L/A-K/E-J/H-G1/G1-B1/B1 | A/L-L'/A'-B2/B2-G2/G2-H/J-E/K (AK,A'B2G2HG1B1,EJ|LL')
4b - A-mixed (B1AK and KA'B2): A/L-L/A-K/E-J/H-G1/G1-B1/B1 | A'/L'-L'/A'-B2/B2-G2/G2-H/J-E/K (AKA'B2G2HJ1B1,EJ|L,L')

Solution 5: A/L-L/A-K/E-J/H-G1/G1-E/K | A/L-L/A-K/E-G2/G2-H/J-E/K
E-pure (JEJ and G1E'G2): A/L-L/A-K/E-J/H-G1/G1-E'/K' | A/L-L/A-K'/E'-G2/G2-H/J-E/K
5a - E-pure, A-pure (KAK and K'A'K'): A'/L'-L/A-K/E-J/H-G1/G1-E'/K' | A/L-L'/A'-K'/E'-G2/G2-H/J-E/K (AK,A'K',EJ,E'G2HE1|LL')
5b - E-pure, A-mixed (K'AK and KA'K'): A/L-L/A-K/E-J/H-G1/G1-E'/K' | A'/L'-L'/A'-K'/E'-G2/G2-H/J-E/K (AKA'K',EJ,E'G2HE1|L,L')
E-mixed (G1EJ and JE'G2): A/L-L/A-K/E-J/H-G1/G1-E/K | A/L-L/A-K'/E'-G2/G2-H/J-E'/K'
5c - E-mixed, A-pure (KAK and K'A'K'): A/L-L/A-K/E-J/H-G1/G1-E/K | A'/L'-L'/A'-K'/E'-G2/G2-H/J-E'/K' (AK,A'K',EJE'G2HE1|L,L')
5d - E-mixed, A-mixed (K'AK and KA'K'): A'/L'-L/A-K/E-J/H-G1/G1-E/K | A/L-L'/A'-K'/E'-G2/G2-H/J-E'/K' (AKA'K',EJE'G2HE1|LL')

Solution 6: A/L-L/A-K/E-J/H-H/J-B1/B1 | A/L-L/A-B2/B2-J/H-H/J-E/K
A-pure (KAK and B1A'B2): A'/L'-L/A-K/E-J/H-H/J-B1/B1 | A/L-L'/A'-B2/B2-J/H-H/J-E/K
6a - A-pure, J-pure (B2JB1 and EJ'E): A'/L'-L/A-K/E-J'/H'-H/J-B1/B1 | A/L-L'/A'-B2/B2-J/H-H'/J'-E/K (AK,A'B2JB1,EJ',HH'|LL')
6b - A-pure, J-mixed (EJB1 and B2J'E): A'/L'-L/A-K/E-J/H-H/J-B1/B1 | A/L-L'/A'-B2/B2-J'/H'-H'/J'-E/K (AK,A'B2J'EJB1,H,H'|LL')
A-mixed (B1AK and KA'B2): A/L-L/A-K/E-J/H-H/J-B1/B1 | A'/L'-L'/A'-B2/B2-J/H-H/J-E/K
6c - A-mixed, J-pure (B2JB1 and EJ'E): A/L-L/A-K/E-J'/H'-H/J-B1/B1 | A'/L'-L'/A'-B2/B2-J/H-H'/J'-E/K (AKA'B2JB1,EJ',HH'|L,L')
6d - A-mixed, J-mixed (EJB1 and B2J'E): A/L-L/A-K/E-J/H-H/J-B1/B1 | A'/L'-L'/A'-B2/B2-J'/H'-H'/J'-E/K (AKA'B2J'EJB1,H,H'|L,L')

Solution 7: A/L-L/A-K/E-J/H-H/J-E/K (AK,EJ,H|L)

Solution 8: A/L-L/A-K/E-J/H-D2/I2-D1/I1 | A/L-L/A-I2/D2-I1/D1-H/J-E/K
8a - A-pure (KAK and I1A'I2): A'/L'-L/A-K/E-J/H-D2/I2-D1/I1 | A/L-L'/A'-I2/D2-I1/D1-H/J-E/K (AK,A'I2D1HD2I1,EJ|LL')
8b - A-mixed (I1AK and KA'I2): A/L-L/A-K/E-J/H-D2/I2-D1/I1 | A'/L'-L'/A'-I2/D2-I1/D1-H/J-E/K (AKA'I2D1HD2I1,EJ|L,L')

Solution 9: A/L-L/A-K/E-G2/G2-F1/C1-C1/F1 | A/L-L/A-F2/C2-C2/F2-G1/G1-E/K
9a - A-pure (KAK and F1A'F2): A'/L'-L/A-K/E-G2/G2-F1/C1-C1/F1 | A/L-L'/A'-F2/C2-C2/F2-G1/G1-E/K (AK,A'F2G1EG2F1,C1,C2|LL')
9b - A-mixed (F1AK and KA'F2): A/L-L/A-K/E-G2/G2-F1/C1-C1/F1 | A'/L'-L'/A'-F2/C2-C2/F2-G1/G1-E/K (AKA'F2G1EG2F1,C1,C2|L,L')

Solution 10: A/L-L/A-K/E-G2/G2-G1/G1-B1/B1 | A/L-L/A-B2/B2-G2/G2-G1/G1-E/K
A-pure (KAK and B1A'B2): A'/L'-L/A-K/E-G2/G2-G1/G1-B1/B1 | A/L-L'/A'-B2/B2-G2/G2-G1/G1-E/K
10a - A-pure, G-regular: A'/L'-L/A-K/E-G2'/G2'-G1/G1-B1/B1 | A/L-L'/A'-B2/B2-G2/G2-G1'/G1'-E/K (AK,A'B2G2G1'EG2'G1B1|LL')
10b - A-pure, G-inverse: A'/L'-L/A-K/E-G2'/G2-G1'/G1-B1/B1 | A/L-L'/A'-B2/B2-G2/G2'-G1/G1'-E/K (AK,A'B2G2G1'EG2'G1B1|LL')
A-mixed (B1AK and KA'B2): A/L-L/A-K/E-G2/G2-G1/G1-B1/B1 | A'/L'-L'/A'-B2/B2-G2/G2-G1/G1-E/K
10c: A-mixed, G-regular: A/L-L/A-K/E-G2'/G2'-G1/G1-B1/B1 | A'/L'-L'/A'-B2/B2-G2/G2-G1'/G1'-E/K (AKA'B2G2G1'EG2'G1B1|L,L')
10d: A-mixed, G-inverse: A/L-L/A-K/E-G2'/G2-G1'/G1-B1/B1 | A'/L'-L'/A'-B2/B2-G2/G2'-G1/G1'-E/K (AKA'B2G2G1'EG2'G1B1|L,L')
The "G-regular" and "G-inverse" marks show whether G-edges are paired G1/G1 and G1'/G1' (regular) or G1/G1' (inverse). The pairs of solutions are almost identical except for switched chirality of certain vertices. I originally thought there might also be solutions with polygons AB2G2G1B1 and EG2G1, but those don't appear in the more detailed analysis; they would apparently require mixed G-pairs (for example having G1/G1' simultaneously with G2/G2); this would probably break uniformity.

Solution 11: A/L-L/A-K/E-G2/G2-G1/G1-E/K (AK,EG2G1|L)

Solution 12: A/L-L/A-K/E-G2/G2-H/J-B1/B1 | A/L-L/A-B2/B2-J/H-G1/G1-E/K
12a - A-pure (KAK and B1A'B2): A'/L'-L/A-K/E-G2/G2-H/J-B1/B1 | A/L-L'/A'-B2/B2-J/H-G1/G1-E/K (AK,A'B2JB1,EG2HG1|LL')
12b - A-mixed (B1AK and KA'B2): A/L-L/A-K/E-G2/G2-H/J-B1/B1 | A'/L'-L'/A'-B2/B2-J/H-G1/G1-E/K (AKA'B2JB1,EG2HG1|L,L')

Solution 13: A/L-L/A-K/E-G2/G2-D2/I2-D1/I1 | A/L-L/A-I2/D2-I1/D1-G1/G1-E/K
13a - A-pure (KAK and I1A'I2): A'/L'-L/A-K/E-G2/G2-D2/I2-D1/I1 | A/L-L'/A'-I2/D2-I1/D1-G1/G1-E/K (AK,A'I2D1G1EG2D2I1|LL')
13b - A-mixed (I1AK and KA'I2): A/L-L/A-K/E-G2/G2-D2/I2-D1/I1 | A'/L'-L'/A'-I2/D2-I1/D1-G1/G1-E/K (AKA'I2D1G1EG2D2I1|L,L')

Solution 14: A/L-L/A-K/E-C2/F2-G1/G1-C1/F1 | A/L-L/A-F2/C2-G2/G2-F1/C1-E/K
14a - A-pure (KAK and F1A'F2): A'/L'-L/A-K/E-C2/F2-G1/G1-C1/F1 | A/L-L'/A'-F2/C2-G2/G2-F1/C1-E/K (AK,A'F2G1C1EC2G2F1|LL')
14b - A-mixed (F1AK and KA'F2): A/L-L/A-K/E-C2/F2-G1/G1-C1/F1 | A'/L'-L'/A'-F2/C2-G2/G2-F1/C1-E/K (AKA'F2G1C1EC2G2F1|L,L')

Solution 15: A/L-L/A-K/E-C2/F2-H/J-C1/F1 | A/L-L/A-F2/C2-J/H-F1/C1-E/K
15a - A-pure (KAK and F1A'F2): A'/L'-L/A-K/E-C2/F2-H/J-C1/F1 | A/L-L'/A'-F2/C2-J/H-F1/C1-E/K (AK,A'F2HF1,C1EC2J|LL')
15b - A-mixed (F1AK and KA'F2): A/L-L/A-K/E-C2/F2-H/J-C1/F1 | A'/L'-L'/A'-F2/C2-J/H-F1/C1-E/K (AKA'F2HF1,C1EC2J|L,L')

Solution 16: A/L-L/A-I2/D2-I1/D1-F1/C1-C1/F1 | A/L-L/A-F2/C2-C2/F2-D2/I2-D1/I1
16a - A-pure (I1AI2 and F1A'F2): A'/L'-L/A-I2/D2-I1/D1-F1/C1-C1/F1 | A/L-L'/A'-F2/C2-C2/F2-D2/I2-D1/I1 (AI2D1F1A'F2D2I1,C1,C2|LL')
16b - A-mixed (F1AI2 and I1A'F2): A/L-L/A-I2/D2-I1/D1-F1/C1-C1/F1 | A'/L'-L'/A'-F2/C2-C2/F2-D2/I2-D1/I1 (AI2D1F1,A'F2D2I1,C1,C2|L,L')

Solution 17: A/L-L/A-I2/D2-I1/D1-G1/G1-B1/B1 | A/L-L/A-B2/B2-G2/G2-D2/I2-D1/I1
17a - A-pure (I1AI2 and B1A'B2): A'/L'-L/A-I2/D2-I1/D1-G1/G1-B1/B1 | A/L-L'/A'-B2/B2-G2/G2-D2/I2-D1/I1 (AI2D1G1B1A'B2G2D2I1|LL')
17b - A-mixed (B1AI2 and I1A'B2): A/L-L/A-I2/D2-I1/D1-G1/G1-B1/B1 | A'/L'-L'/A'-B2/B2-G2/G2-D2/I2-D1/I1 (AI2D1G1B1,A'B2G2D2I1|L,L')

Solution 18: A/L-L/A-I2/D2-I1/D1-H/J-B1/B1 | A/L-L/A-B2/B2-J/H-D2/I2-D1/I1
18a - A-pure (I1AI2 and B1A'B2): A'/L'-L/A-I2/D2-I1/D1-H/J-B1/B1 | A/L-L'/A'-B2/B2-J/H-D2/I2-D1/I1 (AI2D1HD2I1,A'B2JB1|LL')
18b - A-mixed (B1AI2 and I1A'B2): A/L-L/A-I2/D2-I1/D1-H/J-B1/B1 | A'/L'-L'/A'-B2/B2-J/H-D2/I2-D1/I1 (AI2D1HD2I1A'B2JB1|L,L')

Solution 19: A/L-L/A-I2/D2-I1/D1-D2/I2-D1/I1 (AI2D1D2I1|L)

Solution 20: A/L-L/A-I2/D2-J/H-D2/I2-B1/B1 | A/L-L/A-B2/B2-I1/D1-H/J-D1/I1
20a - A-pure (I1AI2 and B1A'B2): A'/L'-L/A-I2/D2-J/H-D2/I2-B1/B1 | A/L-L'/A'-B2/B2-I1/D1-H/J-D1/I1 (AI2B1A'B2I1,HD2JD1|LL')
20b - A-mixed (B1AI2 and I1A'B2): A/L-L/A-I2/D2-J/H-D2/I2-B1/B1 | A'/L'-L'/A'-B2/B2-I1/D1-H/J-D1/I1 (AI2B1,A'B2I1,D1HD2J|L,L')

Solution 21: A/L-L/A-I2/D2-G2/G2-D2/I2-B1/B1 | A/L-L/A-B2/B2-I1/D1-G1/G1-D1/I1
21a - A-pure (I1AI2 and B1A'B2): A'/L'-L/A-I2/D2-G2/G2-D2/I2-B1/B1 | A/L-L'/A'-B2/B2-I1/D1-G1/G1-D1/I1 (AI2B1A'B2I1,D1G1,D2G2|LL')
21b - A-mixed (B1AI2 and I1A'B2): A/L-L/A-I2/D2-G2/G2-D2/I2-B1/B1 | A'/L'-L'/A'-B2/B2-I1/D1-G1/G1-D1/I1 (AI2B1,D2G2|L)/(A'B2I1,D1G1|L')

Solution 22: A/L-L/A-I2/D2-C2/F2-D2/I2-C1/F1 | A/L-L/A-F2/C2-I1/D1-F1/C1-D1/I1
22a - A-pure (I1AI2 and F1A'F2): A'/L'-L/A-I2/D2-C2/F2-D2/I2-C1/F1 | A/L-L'/A'-F2/C2-I1/D1-F1/C1-D1/I1 (AI2C1D1F1A'F2D2C2I1|LL')
22b - A-mixed (F1AI2 and I1A'F2): A/L-L/A-I2/D2-C2/F2-D2/I2-C1/F1 | A'/L'-L'/A'-F2/C2-I1/D1-F1/C1-D1/I1 (AI2C1D1F1,A'F2D2C2I1|L,L')

Solution 23: A/L-L/A-F2/C2-J/H-F1/C1-B1/B1 | A/L-L/A-B2/B2-C2/F2-H/J-C1/F1
23a - A-pure (F1AF2 and B1A'B2): A'/L'-L/A-F2/C2-J/H-F1/C1-B1/B1 | A/L-L'/A'-B2/B2-C2/F2-H/J-C1/F1 (AF2HF1,A'B2C2JC1B1|LL')
23b - A-mixed (B1AF2 and F1A'B2): A/L-L/A-F2/C2-J/H-F1/C1-B1/B1 | A'/L'-L'/A'-B2/B2-C2/F2-H/J-C1/F1 (AF2HF1A'B2C2JC1B1|L,L')

Solution 24: A/L-L/A-F2/C2-G2/G2-F1/C1-B1/B1 | A/L-L/A-B2/B2-C2/F2-G1/G1-C1/F1
24a - A-pure (F1AF2 and B1A'B2): A'/L'-L/A-F2/C2-G2/G2-F1/C1-B1/B1 | A/L-L'/A'-B2/B2-C2/F2-G1/G1-C1/F1 (AF2G1C1B1A'B2C2G2F1|LL')
24b - A-mixed (B1AF2 and F1A'B2): A/L-L/A-F2/C2-G2/G2-F1/C1-B1/B1 | A'/L'-L'/A'-B2/B2-C2/F2-G1/G1-C1/F1 (AF2G1C1B1,A'B2C2G2F1|L)

Solution 25: A/L-L/A-F2/C2-C2/F2-F1/C1-C1/F1 (AF2F1,C1,C2|L)

Solution 26: A/L-L/A-F2/C2-C2/F2-G1/G1-B1/B1 | A/L-L/A-B2/B2-G2/G2-F1/C1-C1/F1
26a - A-pure (F1AF2 and B1A'B2): A'/L'-L/A-F2/C2-C2/F2-G1/G1-B1/B1 | A/L-L'/A'-B2/B2-G2/G2-F1/C1-C1/F1 (AF2G1B1A'B2G2F1,C1,C2|LL')
26b - A-mixed (B1AF2 and F1A'B2): A/L-L/A-F2/C2-C2/F2-G1/G1-B1/B1 | A'/L'-L'/A'-B2/B2-G2/G2-F1/C1-C1/F1 (AF2G1B1,C2|L)/(A'B2G2F1,C1|L')

Solution 27: A/L-L/A-F2/C2-C2/F2-H/J-B1/B1 | A/L-L/A-B2/B2-J/H-F1/C1-C1/F1
27a - A-pure (F1AF2 and B1A'B2): A'/L'-L/A-F2/C2-C2/F2-H/J-B1/B1 | A/L-L'/A'-B2/B2-J/H-F1/C1-C1/F1 (AF2HF1,A'B2JB1,C1,C2|LL')
27b - A-mixed (B1AF2 and F1A'B2): A/L-L/A-F2/C2-C2/F2-H/J-B1/B1 | A'/L'-L'/A'-B2/B2-J/H-F1/C1-C1/F1 (AF2HF1A'B2JB1,C1,C2|L,L')

Solution 28: A/L-L/A-B2/B2-J/H-G1/G1-B1/B1 | A/L-L/A-B2/B2-G2/G2-H/J-B1/B1
28a: A-pure, B-regular: A'/L'-L/A-B2/B2-J/H-G1/G1-B1'/B1' | A/L-L'/A'-B2'/B2'-G2/G2-H/J-B1/B1 (AB2JB1,A'B2'G2HG1B1'|LL')
28b: A-pure, B-inverse: A'/L'-L/A-B2/B2'-J/H-G1/G1-B1/B1' | A/L-L'/A'-B2'/B2-G2/G2-H/J-B1'/B1 (AB2G2HG1B1,A'B2'JB1'|LL')
28c: A-mixed, B-regular: A/L-L/A-B2/B2-J/H-G1/G1-B1'/B1' | A'/L'-L'/A'-B2'/B2'-G2/G2-H/J-B1/B1 (AB2JB1A'B2'G2HG1B1'|L,L')
28d: A-mixed, B-inverse: A/L-L/A-B2/B2'-J/H-G1/G1-B1/B1' | A'/L'-L'/A'-B2'/B2-G2/G2-H/J-B1'/B1 (AB2G2HG1B1A'B2'JB1'|L,L')
28a and 28b (as well as 28c and 28d) differ in chiralities of some vertices.

Solution 29: A/L-L/A-B2/B2-J/H-H/J-B1/B1 (AB2JB1,H|L)

Solution 30: A/L-L/A-B2/B2-G2/G2-G1/G1-B1/B1 (AB2G2G1B1|L)

1a: 4|a, 2|b
1b: 6|a
2a: 4|a, 2|b
2b: 12|a
3a: 4|a, 2|b
3b: 6|a
4a: 6|a, 2|b
4b: 8|a
5a: 4|a, 2|b
5b: 4|a
5c: 6|a
5d: 12|a, 2|b
6a: 4|a, 2|b
6b: 6|a, 2|b
6c: 6|a
6d: 8|a
7: 2|a
8a: 6|a, 2|b
8b: 8|a
9a: 6|a, 2|b
9b: 8|a
10a: 8|a, 2|b
10b: 8|a, 2|b
10c: 10|a
10d: 10|a
11: 6|a
12a: 4|a, 2|b
12b: 12|a
13a: 8|a, 2|b
13b: 10|a
14a: 8|a, 2|b
14b: 10|a
15a: 4|a, 2|b
15b: 12|a
16a: 8|a, 2|b
16b: 4|a
17a: 10|a, 2|b
17b: 5|a
18a: 12|a, 2|b
18b: 10|a
19: 5|a
20a: 12|a, 2|b
20b: 12|b
21a: 6|a, 2|b
21b: 6|a
22a: 10|a, 2|b
22b: 5|a
23a: 12|a, 2|b
23b: 10|a
24a: 10|a, 2|b
24b: 5|a
25: 3|a
26a: 8|a, 2|b
26b: 4|a
27a: 4|a, 2|b
27b: 8|a
28a: 12|a, 2|b
28b: 12|a, 2|b
28c: 10|a
28d: 10|a
29: 4|a
30: 5|a

So there seem to be 62 solutions in all, with possibility of some new shenanigans in groups 10 and 28.
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Way too fast to follow, Marek

What was the reason for your disallowance, which you mentioned in [a,a,a,a,b]?
So there are theoretically 27 distinct configurations, although only some of them are possible; for example if you choose E2/C2 in second row, then you must have either E1/C1 or C2/E2 in third row.

At least I cannot spot such a restriction right from the mere larger vertex configurations.
Might that come in a posteriori only, by virtue of the thereafter to be considered edge cirquits of the tiles?

--- rk
Klitzing
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Posts: 1512
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: Hyperbolic Tilings

I'm still trying to figure out how Marek14's code work. At the moment, I am relying on Klitzing's piccies to get info.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger

wendy
Pentonian

Posts: 1918
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

### Re: Hyperbolic Tilings

wendy wrote:I'm still trying to figure out how Marek14's code work. At the moment, I am relying on Klitzing's piccies to get info.

His idea in the great run is not too hard.

Your input at first is just a vertex configuration be means of several tiles per vertex, e.g. [a,a,a,b] just describes
Code: Select all
`a | a--+--a | b`

Next he derives all possible edge configurations (as seen from some capitalized tile). E.g. he draws
Code: Select all
`a a bb A a`

(and labls that by B2) which rather should display more clearly the configuration
Code: Select all
`a | a | b--+---+--b | A | a`

Then he considers larger vertex configurations by sticking such edge configurations around a vertex. Thus e.g. he could get something like
Code: Select all
`    a | a        --+--    a | B | A | b--+---+---+--a | A | A | a    --+--        a | b    `

(This is the one he decodes as E|A-D|C-B1|B1-A|E.)
Then, admitted, he remains a bit unclear on how to group such patches (so for [a,a,a,b] it is understood) and derives therefrom sides sequences around each of the tiles. Potentially those thereby fall into multiple ones, i.e. getting differently sequenced tiles a1 and a2, both having just the same sides count.
Here he just provides the repetition pattern, and thus derives thus some restriction on the divisibility of the sides counts of those tiles.
Thus the final outcome for [a,a,a,b] can be given according to
• Case 1 : tile a1 has alteting sides A-D-..., a2 is regular with sides C-..., and b is regular as well with sides E-... Accordingly the shapes of a1 and a2 have to be even sided. No restriction to b.
• Case 2 : tile a1 as above, tile a2 having sides pattern A-B2-C-B2-..., thus the (common) shape of a1 and a2 has to be divisable by 4. As b again has sides E-..., but alternates in adjacent a1 and a2, it has to be evensided here.
• Case 3 : only a single tile of type a, having sides pattern A-D-A-B2-C-B2-... Thus it is divisable by 6. b again is the regular E-... with no further restriction.
• Case 4 : here a has sides pattern A-B2-B1-... periodically. Hence it is divisable by 3. b again is regular (E-...) with no restriction.

But again, as already stated above, there are some gaps to be filled in within some details - already in his outline of [a,a,a,a,b].

--- rk
Klitzing
Pentonian

Posts: 1512
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: Hyperbolic Tilings

Klitzing wrote:But again, as already stated above, there are some gaps to be filled in within some details - already in his outline of [a,a,a,a,b].

--- rk

When I posted [a,a,a,a,b], it was still a bit unclear what to do when one edge type is present multiple times in the whole configuration. Only today I realized that I can use apostrophes to make all edges unique and that this will also allow for finer resolution. So I ran through all remaining 5-valent vertices and tried this approach on [a,a,a,a,a,b]. It's still a bit unclear in categories 10 and 28 and I wonder if additional solutions could be found there -- but the results I got were truly surprising!

As for the disallowance you asked about (i.e. how to infer valid vertex sequences from the set of all possible sequences):

Let's consider [a,a,a,a,b] sequence:
A/H
H/A
G/D E2/C2 B2/B2
E1/C1 F/F C2/E2
B1/B1 C1/E1 D/G

Now, let's select an invalid sequence:

A/H-H/A-E2/C2-F/F-D/G

We'll join it with the mirror image:

A/H-H/A-E2/C2-F/F-D/G | A/H-H/A-G/D-F/F-C1/E1

And now you can see the problem: You can't write the proper polygons.
If you start with AE2... then you can't continue. There's no entry of form x/E2 in the normal or mirror sequence; E2 is an orphan that only appears once. The valid sequences are those which don't run into this problem. In this case, the E2/C2 entry must be balanced by C2/E2 as fourth entry... or by E1/C1 which will then give C2/E2 in the mirror sequence.

A/H-H/A-E2/C2-C2/E2-D/G | A-H-H/A-G/D-E1/C1-C1/E1
A/H-H/A-E2/C2-E1/C1-D/G | A-H-H/A-G/D-C2/E2-C1/E1
Marek14
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Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

If I rewrite the [a,a,a,b] in the light of my today's notation, the solutions would be:

Solution 1: A/E-E/A-D/C-B1/B1 | A/E-E/A-B2/B2-C/D

Solution 3: A/E-E/A-B2/B2-B1/B1 (AB2B1|E)

From this, it can be seen exactly why b must be even in 1a. We can also see that in 1b, b-gons fall in two chiral groups.

Rewriting [a,a,a,a,b] in this way, we get

Solution 1: A/H-H/A-G/D-E1/C1-C1/E1 | A/H-H/A-E2/C2-C2/E2-D/G
1a: A'/H'-H/A-G/D-E1/C1-C1/E1 | A/H-H'/A'-E2/C2-C2/E2-D/G (AG,A'E2DE1,C1,C2|HH')
1b: A/H-H/A-G/D-E1/C1-C1/E1 | A'/H'-H'/A'-E2/C2-C2/E2-D/G (AGA'E2DE1,C1,C2|H,H')

Solution 2: A/H-H/A-G/D-F/F-B1/B1 | A/H-H/A-B2/B2-F/F-D/G
1a: A'/H'-H/A-G/D-F'/F-B1/B1 | A/H-H'/A'-B2/B2-F/F'-D/G (AG,A'B2FB1,DF'|HH')
1b: A'/H'-H/A-G/D-F/F-B1/B1 | A/H-H'/A'-B2/B2-F'/F'-D/G (AG,A'B2F'DFB1|HH')
1c: A/H-H/A-G/D-F'/F-B1/B1 | A'/H'-H'/A'-B2/B2-F/F'-D/G (AGA'B2FB1,DF'|H,H')
1d: A/H-H/A-G/D-F/F-B1/B1 | A'/H'-H'/A'-B2/B2-F'/F'-D/G (AGA'B2F'DFB1|H,H')

Solution 3: A/H-H/A-G/D-F/F-D/G (AG,DF|H)

Solution 4: A/H-H/A-G/D-C2/E2-C1/E1 | A/H-H/A-E2/C2-E1/C1-D/G
4a: A'/H'-H/A-G/D-C2/E2-C1/E1 | A/H-H'/A'-E2/C2-E1/C1-D/G (AG,A'E2C1DC2E1|HH')
4b: A/H-H/A-G/D-C2/E2-C1/E1 | A'/H'-H'/A'-E2/C2-E1/C1-D/G (AGA'E2C1DC2E1|H,H')

Solution 5: A/H-H/A-E2/C2-E1/C1-B1/B1 | A/H-H/A-B2/B2-C2/E2-C1/E1
5a: A'/H'-H/A-E2/C2-E1/C1-B1/B1 | A/H-H'/A'-B2/B2-C2/E2-C1/E1 (AE2C1B1A'B2C2E1|H,H')
5b: A/H-H/A-E2/C2-E1/C1-B1/B1 | A'/H'-H'/A'-B2/B2-C2/E2-C1/E1 (AE2C1B1,A'B2C2E1|H,H')

Solution 6: A/H-H/A-E2/C2-C2/E2-B1/B1 | A/H-H/A-B2/B2-E1/C1-C1/E1
6a: A'/H'-H/A-E2/C2-C2/E2-B1/B1 | A/H-H'/A'-B2/B2-E1/C1-C1/E1 (AE2B1A'B2E1,C1,C2|HH')
6b: A/H-H/A-E2/C2-C2/E2-B1/B1 | A'/H'-H'/A'-B2/B2-E1/C1-C1/E1 (AE2B1,C2|H)/(A'B2E1,C1|H')

Solution 7: A/H-H/A-B2/B2-F/F-B1/B1 (AB2FB1|H)

I probably switched around some a and b subsolutions to always have them in pure-mixed order. The pure ones require even-sided b while the mixed ones tend to have two chiral b-gons.
Marek14
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Posts: 1137
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### Re: Hyperbolic Tilings

Marek14 wrote:
Klitzing wrote:But again, as already stated above, there are some gaps to be filled in within some details - already in his outline of [a,a,a,a,b].

--- rk

[...]
As for the disallowance you asked about (i.e. how to infer valid vertex sequences from the set of all possible sequences):
[...]
Now, let's select an invalid sequence:
[...]
We'll join it with the mirror image:
[...]
And now you can see the problem: You can't write the proper polygons.
[...]

Aha, that was exactly what I missed, thanx.
Your mysterious grouping was only with the mirror image (at most). Then (at least) the sequences around the tiles should be closable.
Other / larger groupings might provide tilings as well. But those then would no longer be uniform. - Got it.
--- rk
Klitzing
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Posts: 1512
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Location: Heidenheim, Germany

### Re: Hyperbolic Tilings

Type AAAABB:

Edge data:

Code: Select all
`a b b b aa a A a a : A xI -G,E2,B2b b a a aa a A b b : B1 xB1 -Ab b a a ba a A a b : C1 xE1 -B1,C1,Db b a b ba a A a a : D xG -E1,F,C2b a a a ab a A b b : E1 xC1 -Ab a a a bb a A a b : F xF -B1,C1,Db a a b bb a A a a : C2 xE2 -E1,F,C2a a a a ab b A b b : G xD -Aa a a a bb b A a b : E2 xC2 -B1,C1,Da a a b bb b A a a : B2 xB2 -E1,F,C2a a b a aa a B a a : H xH -Ia a a a aa b B b a : I xA -H`

Vertex sequence:
A/I
H/H
I/A
G/D E2/C2 B2/B2
E1/C1 F/F C2/E2
B1/B1 C1/E1 D/G

27 possible sequences, leading to 7 basic solutions:

Solution 1: A/I-H/H-I/A-G/D-E1/C1-C1/E1 | A/I-H/H-I/A-E2/C2-C2/E2-D/G
1a: A'/I'-H'/H-I/A-G/D-E1/C1-C1/E1 | A/I-H/H'-I'/A'-E2/C2-C2/E2-D/G (AG,A'E2DE1,C1,C2|HI,H'I')
1b: A'/I'-H/H-I/A-G/D-E1/C1-C1/E1 | A/I-H'/H'-I'/A'-E2/C2-C2/E2-D/G (AG,A'E2DE1,C1,C2|HIH'I')
1c: A/I-H/H-I/A-G/D-E1/C1-C1/E1 | A'/I'-H'/H'-I'/A'-E2/C2-C2/E2-D/G (AGA'E2DE1,C1,C2|HI,H'I')
1d: A/I-H'/H-I/A-G/D-E1/C1-C1/E1 | A'/I'-H/H'-I'/A'-E2/C2-C2/E2-D/G (AGA'E2DE1,C1,C2|HIH'I')

Solution 2: A/I-H/H-I/A-G/D-F/F-B1/B1 | A/I-H/H-I/A-B2/B2-F/F-D/G
2a: A'/I'-H'/H-I/A-G/D-F'/F-B1/B1 | A/I-H/H'-I'/A'-B2/B2-F/F'-D/G (AG,A'B2FB1,DF'|HI,H'I')
2b: A'/I'-H/H-I/A-G/D-F'/F-B1/B1 | A/I-H'/H'-I'/A'-B2/B2-F/F'-D/G (AG,A'B2FB1,DF'|HIH'I')
2c: A'/I'-H'/H-I/A-G/D-F/F-B1/B1 | A/I-H/H'-I'/A'-B2/B2-F'/F'-D/G (AG,A'B2F'DFB1|HI,H'I')
2d: A'/I'-H/H-I/A-G/D-F/F-B1/B1 | A/I-H'/H'-I'/A'-B2/B2-F'/F'-D/G (AG,A'B2F'DFB1|HIH'I')
2e: A/I-H/H-I/A-G/D-F'/F-B1/B1 | A'/I'-H'/H'-I'/A'-B2/B2-F/F'-D/G (AGA'B2FB1,DF'|HI,H'I')
2f: A/I-H'/H-I/A-G/D-F'/F-B1/B1 | A'/I'-H/H'-I'/A'-B2/B2-F/F'-D/G (AGA'B2FB1,DF'|HIH'I')
2g: A/I-H/H-I/A-G/D-F/F-B1/B1 | A'/I'-H'/H'-I'/A'-B2/B2-F'/F'-D/G (AGA'B2F'DFB1|HI,H'I')
2h: A/I-H'/H-I/A-G/D-F/F-B1/B1 | A'/I'-H/H'-I'/A'-B2/B2-F'/F'-D/G (AGA'B2F'DFB1|HIH'I')

Solution 3: A/I-H/H-I/A-G/D-F/F-D/G (AG,DF|HI)

Solution 4: A/I-H/H-I/A-G/D-C2/E2-C1/E1 | A/I-H/H-I/A-E2/C2-E1/C1-D/G
4a: A'/I'-H'/H-I/A-G/D-C2/E2-C1/E1 | A/I-H/H'-I'/A'-E2/C2-E1/C1-D/G (AG,A'E2C1DC2E1|HI,H'I')
4b: A'/I'-H/H-I/A-G/D-C2/E2-C1/E1 | A/I-H'/H'-I'/A'-E2/C2-E1/C1-D/G (AG,A'E2C1DC2E1|HIH'I')
4c: A/I-H/H-I/A-G/D-C2/E2-C1/E1 | A'/I'-H'/H'-I'/A'-E2/C2-E1/C1-D/G (AGA'E2C1DC2E1|HI,H'I')
4d: A/I-H'/H-I/A-G/D-C2/E2-C1/E1 | A'/I'-H/H'-I'/A'-E2/C2-E1/C1-D/G (AGA'E2C1DC2E1|HIH'I')

Solution 5: A/I-H/H-I/A-E2/C2-E1/C1-B1/B1 | A/I-H/H-I/A-B2/B2-C2/E2-C1/E1
5a: A'/I'-H'/H-I/A-E2/C2-E1/C1-B1/B1 | A/I-H/H'-I'/A'-B2/B2-C2/E2-C1/E1 (AE2C1B1A'B2C2E1|HI,H'I')
5b: A'/I'-H/H-I/A-E2/C2-E1/C1-B1/B1 | A/I-H'/H'-I'/A'-B2/B2-C2/E2-C1/E1 (AE2C1B1A'B2C2E1|HIH'I')
5c: A/I-H/H-I/A-E2/C2-E1/C1-B1/B1 | A'/I'-H'/H'-I'/A'-B2/B2-C2/E2-C1/E1 (AE2C1B1,A'B2C2E1|HI,H'I')
5d: A/I-H'/H-I/A-E2/C2-E1/C1-B1/B1 | A'/I'-H/H'-I'/A'-B2/B2-C2/E2-C1/E1 (AE2C1B1,A'B2C2E1|HIH'I')

Solution 6: A/I-H/H-I/A-E2/C2-C2/E2-B1/B1 | A/I-H/H-I/A-B2/B2-E1/C1-C1/E1
6a: A'/I'-H'/H-I/A-E2/C2-C2/E2-B1/B1 | A/I-H/H'-I'/A'-B2/B2-E1/C1-C1/E1 (AE2B1A'B2E1,C1,C2|HI,H'I')
6b: A'/I'-H/H-I/A-E2/C2-C2/E2-B1/B1 | A/I-H'/H'-I'/A'-B2/B2-E1/C1-C1/E1 (AE2B1A'B2E1,C1,C2|HIH'I')
6c: A/I-H/H-I/A-E2/C2-C2/E2-B1/B1 | A'/I'-H'/H'-I'/A'-B2/B2-E1/C1-C1/E1 (AE2B1,C2|HI)/(A'B2E1,C1|H'I')
6d: A/I-H'/H-I/A-E2/C2-C2/E2-B1/B1 | A'/I'-H/H'-I'/A'-B2/B2-E1/C1-C1/E1 (AE2B1,A'B2E1,C1,C2|HIH'I')

Solution 7: A/I-H/H-I/A-B2/B2-F/F-B1/B1 (AB2FB1|HI)

3: 2|a, 2|b

6c: 3|a, 2|b
6d: 3|a, 4|b

1a, 2a, 5c, 7: 4|a, 2|b
1b, 2b, 5d: 4|a, 4|b

1c, 2c, 2e, 4a, 6a: 6|a, 2|b
1d, 2d, 2f, 4b, 6b: 6|a, 4|b

2g, 4c, 5a: 8|a, 2|b
2h, 4d, 5b: 8|a, 4|b
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AAABAB:

Edge data:
Code: Select all
`b a b a aa a A b a : A1 xG1 -E,A2b a b a ba a A a a : B xI -F,C2a b a a bb a A b a : C1 xC1 -A1,Ba b a b ab a A a b : D xF -C1,Da a b a aa b A b a : E xH -E,A2a a b a ba b A a a : A2 xG2 -F,C2b a a a ba b A b a : F xD -A1,Bb a a b aa b A a b : C2 xC2 -C1,Da b a a aa a B a b : G1 xA1 -G1,Ha b a b aa a B a a : H xE -I,G2a a a a ab a B a b : I xB -G1,Ha a a b ab a B a a : G2 xA2 -I,G2`

Vertex sequence:
A1/G1 B/I
G1/A1 H/E
E/H A2/G2
I/B G2/A2
F/D C2/C2
C1/C1 D/F

64 possible sequences, leading to 14 basic solutions:

Solution 1: A1/G1-G1/A1-E/H-I/B-F/D-C1/C1 | B/I-H/E-A2/G2-G2/A2-C2/C2-D/F (A1EA2C2DC1,BF|G1,G2,HI)

Solution 2: A1/G1-G1/A1-E/H-I/B-F/D-D/F | B/I-H/E-A2/G2-G2/A2-F/D-D/F
2a: A1/G1-G1/A1-E/H-I/B-F'/D'-D/F | B/I-H/E-A2/G2-G2/A2-F/D-D'/F' (A1EA2F,BF',DD'|G1,G2,HI)
2b: A1/G1-G1/A1-E/H-I/B-F/D-D/F | B/I-H/E-A2/G2-G2/A2-F'/D'-D'/F' (A1EA2F'BF,D,D'|G1,G2,HI)

Solution 3: A1/G1-G1/A1-E/H-I/B-C2/C2-C1/C1 | B/I-H/E-A2/G2-G2/A2-C2/C2-C1/C1
3a: A1/G1-G1/A1-E/H-I/B-C2'/C2'-C1'/C1' | B/I-H/E-A2/G2-G2/A2-C2/C2-C1/C1 (A1EA2C2C1BC2'C1'|G1,G2,HI)
3b: A1/G1-G1/A1-E/H-I/B-C2'/C2-C1/C1' | B/I-H/E-A2/G2-G2/A2-C2/C2'-C1'/C1 (A1EA2C2C1BC2'C1'|G1,G2,HI)

Solution 4: A1/G1-G1/A1-E/H-I/B-C2/C2-D/F | B/I-H/E-A2/G2-G2/A2-F/D-C1/C1 (A1EA2F,BC2DC1|G1,G2,HI)

Solution 5: A1/G1-G1/A1-A2/G2-G2/A2-F/D-C1/C1 | A1/G1-G1/A1-A2/G2-G2/A2-C2/C2-D/F
5a: A1/G1-G1'/A1'-A2'/G2'-G2/A2-F/D-C1/C1 | A1'/G1'-G1/A1-A2/G2-G2'/A2'-C2/C2-D/F (A1A2FA1'A2'C2DC1|G1G1',G2G2')
5a: A1/G1-G1/A1-A2/G2-G2/A2-F/D-C1/C1 | A1'/G1'-G1'/A1'-A2'/G2'-G2'/A2'-C2/C2-D/F (A1A2FA1'A2'C2DC1|G1,G1',G2,G2')

Solution 6: A1/G1-G1/A1-A2/G2-G2/A2-F/D-D/F (A1A2F,D|G1,G2)

Solution 7: A1/G1-G1/A1-A2/G2-G2/A2-C2/C2-C1/C1 (A1A2C2C1|G1,G2)

Solution 8: A1/G1-H/E-A2/G2-I/B-F/D-C1/C1 | B/I-G1/A1-E/H-G2/A2-C2/C2-D/F (A1EA2C2DC1,BF|G1HG2I)

Solution 9: A1/G1-H/E-A2/G2-I/B-F/D-D/F | B/I-G1/A1-E/H-G2/A2-F/D-D/F
9a: A1/G1-H/E-A2/G2-I/B-F'/D'-D/F | B/I-G1/A1-E/H-G2/A2-F/D-D'/F' (A1EA2F,BF',DD'|G1HG2I)
9a: A1/G1-H/E-A2/G2-I/B-F/D-D/F | B/I-G1/A1-E/H-G2/A2-F'/D'-D'/F' (A1EA2F'BF,D,D'|G1HG2I)

Solution 10: A1/G1-H/E-A2/G2-I/B-C2/C2-C1/C1 | B/I-G1/A1-E/H-G2/A2-C2/C2-C1/C1
10a: A1/G1-H/E-A2/G2-I/B-C2'/C2'-C1'/C1' | B/I-G1/A1-E/H-G2/A2-C2/C2-C1/C1 (A1EA2C2C1BC2'C1'|G1HG2I)
10b: A1/G1-H/E-A2/G2-I/B-C2'/C2-C1/C1' | B/I-G1/A1-E/H-G2/A2-C2/C2'-C1'/C1 (A1EA2C2C1BC2'C1'|G1HG2I)

Solution 11: A1/G1-H/E-A2/G2-I/B-C2/C2-D/F | B/I-G1/A1-E/H-G2/A2-F/D-C1/C1 (A1EA2F,BC2DC1|G1HG2I)

Solution 12: B/I-H/E-E/H-I/B-F/D-C1/C1 | B/I-H/E-E/H-I/B-C2/C2-D/F
12a: B'/I'-H'/E'-E/H-I/B-F/D-C1/C1 | B/I-H/E-E'/H'-I'/B'-C2/C2-D/F (BF,B'C2DC1,EE'|HI,H'I')
12b: B'/I'-H/E-E/H-I/B-F/D-C1/C1 | B/I-H'/E'-E'/H'-I'/B'-C2/C2-D/F (BF,B'C2DC1,E,E'|HIH'I')
12c: B/I-H/E-E/H-I/B-F/D-C1/C1 | B'/I'-H'/E'-E'/H'-I'/B'-C2/C2-D/F (BFB'C2DC1,E,E'|HI,H'I')
12d: B/I-H'/E'-E/H-I/B-F/D-C1/C1 | B'/I'-H/E-E'/H'-I'/B'-C2/C2-D/F (BFB'C2DC1,EE'|HIH'I')

Solution 13: B/I-H/E-E/H-I/B-F/D-D/F (BF,D,E|HI)

Solution 14: B/I-H/E-E/H-I/B-C2/C2-C1/C1 (BC2C1,E|HI)

2|a, 2|b - 13

3|a - 6
3|a, 2|b - 14

4|a - 7
4|a, 2|b - 2a, 4, 12a
4|a, 4|b - 9a, 11, 12b

6|a, 2|b - 1, 2b, 12c
6|a, 4|b - 8, 9b, 12d

8|a - 5b
8|a, 2|b - 3a, 3b, 5a
8|a, 4|b - 10a, 10b
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Bonus: five pseudotilings 4,4,4,2,4,2

2a (1 = A1EA2F, 2 = BF', 3 = DD'):
Code: Select all
`+===+---+===+---+===+---+| 1 ║ 2 ║ 1 | 3 | 1 ║ 2 ║|   ║   ║   |   |   ║   ║+===+---+===+---+===+---+║ 1 | 3 | 1 ║ 2 ║ 1 | 3 |║   |   |   ║   ║   |   |+===+---+===+---+===+---+| 1 ║ 2 ║ 1 | 3 | 1 ║ 2 ║|   ║   ║   |   |   ║   ║+===+---+===+---+===+---+║ 1 | 3 | 1 ║ 2 ║ 1 | 3 |║   |   |   ║   ║   |   |+===+---+===+---+===+---+| 1 ║ 2 ║ 1 | 3 | 1 ║ 2 ║|   ║   ║   |   |   ║   ║+===+---+===+---+===+---+║ 1 | 3 | 1 ║ 2 ║ 1 | 3 |║   |   |   ║   ║   |   |+===+---+===+---+===+---+`

4 (1 = A1EA2F, 2 = BC2DC1):
Code: Select all
`+===+---+===+---+===+---+║ 1 | 2 ║ 1 | 2 ║ 1 | 2 ║║   |   ║   |   ║   |   ║+===+---+===+---+===+---+| 1 ║ 2 | 1 ║ 2 | 1 ║ 2 ||   ║   |   ║   |   ║   |+===+---+===+---+===+---+║ 1 | 2 ║ 1 | 2 ║ 1 | 2 ║║   |   ║   |   ║   |   ║+===+---+===+---+===+---+| 1 ║ 2 | 1 ║ 2 | 1 ║ 2 ||   ║   |   ║   |   ║   |+===+---+===+---+===+---+║ 1 | 2 ║ 1 | 2 ║ 1 | 2 ║║   |   ║   |   ║   |   ║+===+---+===+---+===+---+| 1 ║ 2 | 1 ║ 2 | 1 ║ 2 ||   ║   |   ║   |   ║   |+===+---+===+---+===+---+`

7 (all squares are identical):
Code: Select all
`+---+===+---+===+---+===+║   |   ║   |   ║   |   ║║   |   ║   |   ║   |   ║+===+---+===+---+===+---+|   ║   |   ║   |   ║   ||   ║   |   ║   |   ║   |+---+===+---+===+---+===+║   |   ║   |   ║   |   |║   |   ║   |   ║   |   |+===+---+===+---+===+---+|   ║   |   ║   |   ║   ||   ║   |   ║   |   ║   |+---+===+---+===+---+===+║   |   ║   |   ║   |   |║   |   ║   |   ║   |   |+===+---+===+---+===+---+|   ║   |   ║   |   ║   ||   ║   |   ║   |   ║   |+---+===+---+===+---+===+`

12a (1 = BF, 2 = B'C2DC1, 3 = EE'):
Code: Select all
`+===+---+===+---+===+---+| 1 | 2 ║ 3 ║ 2 | 1 | 2 ║|   |   ║   ║   |   |   ║+===+---+===+---+===+---+║ 3 ║ 2 | 1 | 2 ║ 3 ║ 2 |║   ║   |   |   ║   ║   |+===+---+===+---+===+---+| 1 | 2 ║ 3 ║ 2 | 1 | 2 ║|   |   ║   ║   |   |   ║+===+---+===+---+===+---+║ 3 ║ 2 | 1 | 2 ║ 3 ║ 2 |║   ║   |   |   ║   ║   |+===+---+===+---+===+---+| 1 | 2 ║ 3 ║ 2 | 1 | 2 ║|   |   ║   ║   |   |   ║+===+---+===+---+===+---+║ 3 ║ 2 | 1 | 2 ║ 3 ║ 2 |║   ║   |   |   ║   ║   |+===+---+===+---+===+---+`

13 (1 = BD, 2 = D, 3 = E):
Code: Select all
`+===+---+===+---+===+---+║ 3 ║ 1 ║ 3 ║ 1 ║ 3 ║ 1 ║║   ║   ║   ║   ║   ║   ║+===+---+===+---+===+---+| 1 | 2 | 1 | 2 | 1 | 2 ||   |   |   |   |   |   |+===+---+===+---+===+---+║ 3 ║ 1 ║ 3 ║ 1 ║ 3 ║ 1 ║║   ║   ║   ║   ║   ║   ║+===+---+===+---+===+---+| 1 | 2 | 1 | 2 | 1 | 2 ||   |   |   |   |   |   |+===+---+===+---+===+---+║ 3 ║ 1 ║ 3 ║ 1 ║ 3 ║ 1 ║║   ║   ║   ║   ║   ║   ║+===+---+===+---+===+---+| 1 | 2 | 1 | 2 | 1 | 2 ||   |   |   |   |   |   |+===+---+===+---+===+---+`
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABAAB is trivial and has only one solution -- doubly wound AAB.

Type AAABBB:

Edge data:

Code: Select all
`b b b b ba a A a a : A xH -D,B2b b a a bb a A b b : B1 xB1 -Ab b a b bb a A a b : C xD -B1,Cb a a a bb b A b b : D xC -Ab a a b bb b A a b : B2 xB2 -B1,Ca b b a aa a B b a : E1 xE1 -G,E2a b b b aa a B a a : F xG -Ha a b a aa b B b a : G xF -G,E2a a b b aa b B a a : E2 xE2 -Ha a a a ab b B b b : H xA -E1,F`

Vertex sequence:
A/H
E1/E1 F/G
G/F E2/E2
H/A
D/C B2/B2
B1/B1 C/D

16 possible sequences, leading to 10 basic solutions. These are typically related to combinations of solutions of AAAB:

Solution 1: A/H-E1/E1-G/F-H/A-D/C-B1/B1 | A/H-F/G-E2/E2-H/A-B2/B2-C/D

Solution 2: A/H-E1/E1-G/F-H/A-D/C-C/D | A/H-F/G-E2/E2-H/A-D/C-C/D

Solution 3: A/H-E1/E1-G/F-H/A-B2/B2-B1/B1 | A/H-F/G-E2/E2-H/A-B2/B2-B1/B1
3a: A'/H'-E1/E1-G/F-H/A-B2/B2-B1/B1 | A/H-F/G-E2/E2-H'/A'-B2'/B2'-B1'/B1' (AB2B1A'B2'B1'|E1GE2H',FH)
3b: A'/H'-E1/E1-G/F-H/A-B2/B2'-B1'/B1 | A/H-F/G-E2/E2-H'/A'-B2'/B2-B1/B1' (AB2B1A'B2'B1'|E1GE2H',FH)
3c: A/H-E1/E1-G/F-H/A-B2/B2-B1/B1 | A'/H'-F/G-E2/E2-H'/A'-B2'/B2'-B1'/B1' (AB2B1,A'B2'B1'|E1GE2H'FH)
3d: A/H-E1/E1-G/F-H/A-B2/B2'-B1'/B1 | A'/H'-F/G-E2/E2-H'/A'-B2'/B2-B1/B1' (AB2B1,A'B2'B1'|E1GE2H'FH)

Solution 4: A/H-E1/E1-G/F-H/A-B2/B2-C/D | A/H-F/G-E2/E2-H/A-D/C-B1/B1
Can you spot the difference between these and solutions 1a and 1b?

Solution 5: A/H-E1/E1-E2/E2-H/A-D/C-B1/B1 | A/H-E1/E1-E2/E2-H/A-B2/B2-C/D
These solutions are twins of solutions from group 3 obtained by switching a and b.

Solution 7: A/H-E1/E1-E2/E2-H/A-B2/B2-B1/B1 (AB2B1,E1E2H)

Solution 8: A/H-F/G-G/F-H/A-D/C-B1/B1 | A/H-F/G-G/F-H/A-B2/B2-C/D
These solutions are twins of solutions from group 2 obtained by switching a and b.

Solution 10: A/H-F/G-G/F-H/A-B2/B2-B1/B1 (AB2B1|FH,G)
This solution is a twin of solution 6 obtained by switching a and b.

2|a, 2|b: 9
2|a, 3|b: 6
2|a, 4|b: 2a
2|a, 6|b: 2c

3|a, 2|b: 10
3|a, 3|b: 7
3|a, 6|b: 3c, 3d

4|a, 2|b: 8a
4|a, 4|b: 1a, 2b, 4a, 8b
4|a, 6|b: 2d, 5a, 5b

6|a, 2|b: 8c
6|a, 3|b: 5c, 5d
6|a, 4|b: 3a, 3b, 8d
6|a, 6|b: 1b, 4b
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABABB:

Edge data:
Code: Select all
`a b b a ab a A b b : A1 xP1 -H1,G2,I,C2a b b a bb a A a b : B1 xU1 -J1,Ka b b b ab a A b a : C1 xX1 -L,E2,H2,A2a b b b ab a A a b : D x Z -M,J2b a b a ab a A b b : E1 xQ1 -H1,G2,I,C2b a b a bb a A a b : F xV -J1,Kb a b b ab a A b a : G1 xW2 -L,E2,H2,A2b a b b ab a A a b : B2 xU2 -M,J2a b b a aa b A b b : H1 xR1 -H1,G2,I,C2a b b a ba b A a b : G2 xW1 -J1,Ka b b b aa b A b a : I xY -L,E2,H2,A2a b b b aa b A a b : C2 xX2 -M,J2b b a b aa b A b b : J1 xJ1 -A1,B1,C1,Db b a b ba b A b a : K xM -E1,F,G1,B2a a b a ab b A b b : L xS -H1,G2,I,C2a a b a bb b A a b : E2 xQ2 -J1,Ka a b b ab b A b a : H2 xR2 -L,E2,H2,A2a a b b ab b A a b : A2 xP2 -M,J2a b a b ab b A b b : M xK -A1,B1,C1,Da b a b bb b A b a : J2 xJ2 -E1,F,G1,B2b a b a aa a B a b : N1 xN1 -X1,W2,Y,R2b a b a ba a B a a : O xT -Z,U2,X2,P2b b a a ba a B b a : P1 xA1 -N1,Ob b a a ba a B a b : Q1 xE1 -P1,Q1,R1,Sb b a b aa a B b a : R1 xH1 -T,N2b b a b ba a B a a : S xL -U1/V/W1/Q2a a b a ab a B a b : T xO -X1,W2,Y,R2a a b a bb a B a a : N2 xN2 -Z,U2,X2,P2b a a a bb a B b a : U1 xB1 -N1,Ob a a a bb a B a b : V xF -P1,Q1,R1,Sb a a b ab a B b a : W1 xG2 -T,N2b a a b bb a B a a : Q2 xE2 -U1/V/W1/Q2a b a a ba b B b a : X1 xC1 -N1,Oa b a a ba b B a b : W2 xG1 -P1,Q1,R1,Sa b a b aa b B b a : Y xI -T,N2a b a b ba b B a a : R2 xH2 -U1/V/W1/Q2b a a a ba b B b a : Z xD -N1,Ob a a a ba b B a b : U2 xB2 -P1,Q1,R1,Sb a a b aa b B b a : X2 xC2 -T,N2b a a b ba b B a a : P2 xA2 -U1/V/W1/Q2`

Vertex sequence:
A1/P1 B1/U1 C1/X1 D/Z
N1/N1 O/T
X1/C1 W2/G1 Y/I R2/H2
L/S E2/Q2 H2/R2 A2/P2
U1/B1 V/F W1/G2 Q2/E2
J1/J1 K/M

Reverse vertex sequence:
E1/Q1 G1/W2 F/V B2/U2
P1/A1 R1/H1 Q1/E1 S/L
H1/R1 I/Y G2/W1 C2/X2
T/O N2/N2
Z/D X2/C2 U2/B2 P2/A2
M/K J2/J2

1024 possible sequences, leading to 40 solutions:

Solution 1: 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)
Solution 2: 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)
Solution 3: 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)
Solution 4: 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)
Solution 5 (twin of solution 2): 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)
Solution 6: 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)
Solution 7 (twin of solution 4): 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)
Solution 8: 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)
Solution 9: 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)
Solution 10: 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)
Solution 11 (chiral): 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)
Solution 12: 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)
Solution 13: 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)
Solution 14: 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)
Solution 15: 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)
Solution 16: 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)
Solution 17 (twin of solution 9): 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)
Solution 18 (twin of solution 13): 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)
Solution 19 (chiral, twin of solution 11): 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)
Solution 20 (twin of solution 15): 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)
Solution 21 (twin of solution 10): 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)
Solution 22 (twin of solution 14): 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)
Solution 23 (twin of solution 12): 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)
Solution 24 (twin of solution 16): 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)
Solution 25: 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)
Solution 26: 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)
Solution 27: 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)
Solution 28: 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)
Solution 29: 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)
Solution 30: 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)
Solution 31 (twin of solution 29): 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)
Solution 32: 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)
Solution 33 (twin of solution 26): 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)
Solution 34: 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)
Solution 35 (twin of solution 28): 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)
Solution 36: 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)
Solution 37 (twin of solution 32): 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)
Solution 38: 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)
Solution 39 (twin of solution 30): 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)
Solution 40 (twin of solution 38): 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)

2|a, 2|b: 36
2|a, 3|b: 11
2|a, 4|b: 13, 28, 40
2|a, 6|b: 9, 15, 32

3|a, 2|b: 19
3|a, 3|b: 1
3|a, 4|b: 23
3|a, 6|b: 5

4|a, 2|b: 18, 35, 38
4|a, 3|b: 12
4|a, 4|b: 8, 14, 22, 27, 30, 34, 39
4|a, 6|b: 4, 10, 16, 26, 31

6|a, 2|b: 17, 20, 37
6|a, 3|b: 2
6|a, 4|b: 7, 21, 24, 29, 33
6|a, 6|b: 3, 6, 25
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type ABABAB is trivial, with only one alternating solution without restrictions for a or b.

Type AAAABC:

Edge data:
Code: Select all
`a b c b aa a A a a : A xAa -W,U2,R2,N2,I2,C2a c b c aa a A a a : B xY -U1,V,S2,O2,J2,D2b c a a aa a A c b : C1 xC1 -Ab c a a aa a A b c : D1 xI1 -Bb c a a ba a A a c : E1 xN1 -C1,D1,E1,F1,G1,Hb c a a ca a A a b : F1 xR1 -I1,J1,K1,L1,M,G2b c a b ca a A a a : G1 xU1 -N1,O1,P1,Q,L2,F2b c a c ba a A a a : H xW -R1,S1,T,P2,K2,E2c b a a aa a A c b : I1 xD1 -Ac b a a aa a A b c : J1/J1 -Bc b a a ba a A a c : K1 xO1 -C1,D1,E1,F1,G1,Hc b a a ca a A a b : L1 xS1 -I1,J1,K1,L1,M,G2c b a b ca a A a a : M xV -N1,O1,P1,Q,L2,F2c b a c ba a A a a : G2 xU2 -R1,S1,T,P2,K2,E2c a a a ab a A c b : N1 xE1 -Ac a a a ab a A b c : O1 xK1 -Bc a a a bb a A a c : P1 xP1 -C1,D1,E1,F1,G1,Hc a a a cb a A a b : Q xT -I1,J1,K1,L1,M,G2c a a b cb a A a a : L2 xS2 -N1,O1,P1,Q,L2,F2c a a c bb a A a a : F2 xR2 -R1,S1,T,P2,K2,E2b a a a ac a A c b : R1 xF1 -Ab a a a ac a A b c : S1 xL1 -Bb a a a bc a A a c : T xQ -C1,D1,E1,F1,G1,Hb a a a cc a A a b : P2 xP2 -I1,J1,K1,L1,M,G2b a a b cc a A a a : K2 xO2 -N1,O1,P1,Q,L2,F2b a a c bc a A a a : E2 xN2 -R1,S1,T,P2,K2,E2a a a a ac b A c b : U1 xG1 -Aa a a a ac b A b c : V xM -Ba a a a bc b A a c : S2 xL2 -C1,D1,E1,F1,G1,Ha a a a cc b A a b : O2 xK2 -I1,J1,K1,L1,M,G2a a a b cc b A a a : J2 xJ2 -N1,O1,P1,Q,L2,F2a a a c bc b A a a : D2 xI2 -R1,S1,T,P2,K2,E2a a a a ab c A c b : W xH -Aa a a a ab c A b c : U2 xG2 -Ba a a a bb c A a c : R2 xF2 -C1,D1,E1,F1,G1,Ha a a a cb c A a b : N2 xE2 -I1,J1,K1,L1,M,G2a a a b cb c A a a : I2 xD2 -N1,O1,P1,Q,L2,F2a a a c bb c A a a : C2 xC2 -R1,S1,T,P2,K2,E2a a c a aa a B a a : X xZ -Ya a a a aa c B c a : Y xB -Xa a b a aa a C a a : Z xX -Aaa a a a aa b C b a : Aa xA -Z`

(Since there's more than 26 distinct edge types, the 27th is marked Aa.)

Vertex sequence:
A/Aa
Z/X
Y/B
U1/G1 V/M S2/L2 O2/K2 J2/J2 D2/I2
N1/E1 O1/K1 P1/P1 Q/T L2/S2 F2/R2
C1/C1 D1/I1 E1/N1 F1/R1 G1/U1 H/W

Reverse vertex sequence:
B/Y
X/Z
Aa/A
W/H U2/G2 R2/F2 N2/E2 I2/D2 C2/C2
R1/F1 S1/L1 T/Q P2/P2 K2/O2 E2/N2
I1/D1 J1/J1 K1/O1 L1/S1 M/V G2/U2

216 possible sequences, leading to 20 solutions:

Solution 1: A/Aa-Z/X-Y/B-U1/G1-P1/P1-G1/U1 | B/Y-X/Z-Aa/A-U2/G2-P2/P2-G2/U2 (AU2BU1,G1P1,G2P2|XY|ZAa)
Solution 2: A/Aa-Z/X-Y/B-U1/G1-Q/T-G1/U1 | B/Y-X/Z-Aa/A-U2/G2-T/Q-G2/U2 (AU2BU1,G1QG2T|XY|ZAa)
Solution 3: A/Aa-Z/X-Y/B-V/M-N1/E1-E1/N1 | B/Y-X/Z-Aa/A-N2/E2-E2/N2-M/V (AN2MN1,BV,E1,E2|XY|ZAa)
Solution 4: A/Aa-Z/X-Y/B-V/M-P1/P1-C1/C1 | B/Y-X/Z-Aa/A-C2/C2-P2/P2-M/V (AC2P2MP1C1,BV|XY|ZAa)
Solution 5: A/Aa-Z/X-Y/B-V/M-P1/P1-H/W | B/Y-X/Z-Aa/A-W/H-P2/P2-M/V (AW,BV,HP2MP1|XY|ZAa)
Solution 6: A/Aa-Z/X-Y/B-V/M-Q/T-C1/C1 | B/Y-X/Z-Aa/A-C2/C2-T/Q-M/V (AC2TC1,BV,MQ|XY|ZAa)
Solution 7: A/Aa-Z/X-Y/B-V/M-Q/T-H/W | B/Y-X/Z-Aa/A-W/H-T/Q-M/V (AW,BV,HT,MQ|XY|ZAa)
Solution 8: A/Aa-Z/X-Y/B-V/M-F2/R2-F1/R1 | B/Y-X/Z-Aa/A-R2/F2-R1/F1-M/V (AR2F1MF2R1,BV|XY|ZAa)
Solution 9: A/Aa-Z/X-Y/B-S2/L2-L2/S2-C1/C1 | B/Y-X/Z-Aa/A-C2/C2-S1/L1-L1/S1 (AC2S1BS2C1,L1,L2|XY|ZAa)
Solution 10: A/Aa-Z/X-Y/B-S2/L2-L2/S2-H/W | B/Y-X/Z-Aa/A-W/H-S1/L1-L1/S1 (AW,BS2HS1,L1,L2|XY|ZAa)
Solution 11: A/Aa-Z/X-Y/B-O2/K2-O1/K1-C1/C1 | B/Y-X/Z-Aa/A-C2/C2-K2/O2-K1/O1 (AC2K2O1BO2K1C1|XY|ZAa)
Solution 12 (twin of solution : A/Aa-Z/X-Y/B-O2/K2-O1/K1-H/W | B/Y-X/Z-Aa/A-W/H-K2/O2-K1/O1 (AW,BO2K1HK2O1|XY|ZAa)
Solution 13 (twin of solution 9): A/Aa-Z/X-Y/B-J2/J2-N1/E1-E1/N1 | B/Y-X/Z-Aa/A-N2/E2-E2/N2-J1/J1 (AN2J1BJ2N1,E1,E2|XY|ZAa)
Solution 14: A/Aa-Z/X-Y/B-J2/J2-P1/P1-C1/C1 | B/Y-X/Z-Aa/A-C2/C2-P2/P2-J1/J1 (AC2P2J1BJ2P1C1|XY|ZAa)
Solution 15 (twin of solution 4): A/Aa-Z/X-Y/B-J2/J2-P1/P1-H/W | B/Y-X/Z-Aa/A-W/H-P2/P2-J1/J1 (AW,BJ2P1HP2J1|XY|ZAa)
Solution 16: A/Aa-Z/X-Y/B-J2/J2-Q/T-C1/C1 | B/Y-X/Z-Aa/A-C2/C2-T/Q-J1/J1 (AC2TC1,BJ2QJ1|XY|ZAa)
Solution 17 (twin of solution 6): A/Aa-Z/X-Y/B-J2/J2-Q/T-H/W | B/Y-X/Z-Aa/A-W/H-T/Q-J1/J1 (AW,BJ2QJ1,HT|XY|ZAa)
Solution 18 (twin of solution 11): A/Aa-Z/X-Y/B-J2/J2-F2/R2-F1/R1 | B/Y-X/Z-Aa/A-R2/F2-R1/F1-J1/J1 (AR2F1J1BJ2F2R1|XY|ZAa)
Solution 19: A/Aa-Z/X-Y/B-D2/I2-P1/P1-D1/I1 | B/Y-X/Z-Aa/A-I2/D2-P2/P2-I1/D1 (AI2P1D1BD2P2I1|XY|ZAa)
Solution 20: A/Aa-Z/X-Y/B-D2/I2-Q/T-D1/I1 | B/Y-X/Z-Aa/A-I2/D2-T/Q-I1/D1 (AI2QI1,BD2TD1|XY|ZAa)

2|a, 2|b, 2|c: 7

4|a, 2|b, 2|c: 1, 2, 3, 5, 6, 10, 16, 17, 20

6|a, 2|b, 2|c: 4, 8, 9, 12, 13, 15

8|a, 2|b, 2|c: 11, 14, 18, 19
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AAABAC:

Edge data:
Code: Select all
`b a c a aa a A b a : A1 xT1 -O,C2b a c a ba a A a a : B xV -P,M2,I2,E2c a b a aa a A c a : C1 xQ1 -L,A2c a b a ca a A a a : D xS -M1,N,J2,F2a c a a bb a A c a : E1 xE1 -A1,Ba c a a cb a A b a : F1 xI1 -C1,Da c a b ab a A a c : G1 xM1 -E1,F1,G1,Ha c a c ab a A a b : H xP -I1,J1,K,G2a b a a bc a A c a : I1 xF1 -A1,Ba b a a cc a A b a : J1 xJ1 -C1,Da b a b ac a A a c : K xN -E1,F1,G1,Ha b a c ac a A a b : G2 xM2 -I1,J1,K,G2a a c a aa b A b a : L xU -O,C2a a c a ba b A a a : A2 xT2 -P,M2,I2,E2c a a a ba b A c a : M1 xG1 -A1,Bc a a a ca b A b a : N xK -C1,Dc a a b aa b A a c : J2 xJ2 -E1,F1,G1,Hc a a c aa b A a b : F2 xI2 -I1,J1,K,G2a a b a aa c A c a : O xR -L,A2a a b a ca c A a a : C2 xQ2 -M1,N,J2,F2b a a a ba c A c a : P xH -A1,Bb a a a ca c A b a : M2 xG2 -C1,Db a a b aa c A a c : I2 xF2 -E1,F1,G1,Hb a a c aa c A a b : E2 xE2 -I1,J1,K,G2a c a a aa a B a c : Q1 xC1 -Q1,Ra c a c aa a B a a : R xO -S,Q2a a a a ac a B a c : S xD -Q1,Ra a a c ac a B a a : Q2 xC2 -S,Q2a b a a aa a C a b : T1 xA1 -T1,Ua b a b aa a C a a : U xL -V,T2a a a a ab a C a b : V xB -T1,Ua a a b ab a C a a : T2 xA2 -V,T2`

Vertex sequence:
A1/T1 B/V
T1/A1 U/L
O/R C2/Q2
S/D Q2/C2
M1/G1 N/K J2/J2 F2/I2
E1/E1 F1/I1 G1/M1 H/P

Reverse vertex sequence:
C1/Q1 D/S
Q1/C1 R/O
L/U A2/T2
V/B T2/A2
P/H M2/G2 I2/F2 E2/E2
I1/F1 J1/J1 K/N G2/M2

256 possible sequences, leading to 24 solutions:

Solution 1: A1/T1-T1/A1-O/R-S/D-M1/G1-G1/M1 | D/S-R/O-A2/T2-T2/A2-M2/G2-G2/M2 (A1OA2M2DM1,G1,G2|RS|T1,T2)
Solution 2: A1/T1-T1/A1-O/R-S/D-N/K-E1/E1 | D/S-R/O-A2/T2-T2/A2-E2/E2-K/N (A1OA2E2KE1,DN|RS|T1,T2)
Solution 3: A1/T1-T1/A1-O/R-S/D-N/K-H/P | D/S-R/O-A2/T2-T2/A2-P/H-K/N (A1OA2P,DN,HK|RS|T1,T2)
Solution 4: A1/T1-T1/A1-O/R-S/D-J2/J2-E1/E1 | D/S-R/O-A2/T2-T2/A2-E2/E2-J1/J1 (A1OA2E2J1DJ2E1|RS|T1,T2)
Solution 5: A1/T1-T1/A1-O/R-S/D-J2/J2-H/P | D/S-R/O-A2/T2-T2/A2-P/H-J1/J1 (A1OA2P,DJ2HJ1|RS|T1,T2)
Solution 6: A1/T1-T1/A1-O/R-S/D-F2/I2-F1/I1 | D/S-R/O-A2/T2-T2/A2-I2/F2-I1/F1 (A1OA2I2F1DF2I1|RS|T1,T2)
Solution 7 (chiral): A1/T1-T1/A1-C2/Q2-Q2/C2-M1/G1-G1/M1 | C1/Q1-Q1/C1-A2/T2-T2/A2-M2/G2-G2/M2 (A1C2M1,G1|Q2|T1)/(C1A2M2,G2|Q1|T2)
Solution 8: A1/T1-T1/A1-C2/Q2-Q2/C2-N/K-E1/E1 | C1/Q1-Q1/C1-A2/T2-T2/A2-E2/E2-K/N (A1C2NC1A2E2KE1|Q1,Q2|T1,T2)
Solution 9: A1/T1-T1/A1-C2/Q2-Q2/C2-N/K-H/P | C1/Q1-Q1/C1-A2/T2-T2/A2-P/H-K/N (A1C2NC1A2P,HK|Q1,Q2|T1,T2)
Solution 10 (chiral): A1/T1-T1/A1-C2/Q2-Q2/C2-J2/J2-E1/E1 | C1/Q1-Q1/C1-A2/T2-T2/A2-E2/E2-J1/J1 (A1C2J2E1|Q2|T1)/(C1A2E2J1|Q1|T2)
Solution 11: A1/T1-T1/A1-C2/Q2-Q2/C2-J2/J2-H/P | C1/Q1-Q1/C1-A2/T2-T2/A2-P/H-J1/J1 (A1C2J2HJ1C1A2P|Q1,Q2|T1,T2)
Solution 12: A1/T1-T1/A1-C2/Q2-Q2/C2-F2/I2-F1/I1 | C1/Q1-Q1/C1-A2/T2-T2/A2-I2/F2-I1/F1 (A1C2F2I1,C1A2I2F1|Q1,Q2|T1,T2)
Solution 13: B/V-U/L-O/R-S/D-M1/G1-G1/M1 | D/S-R/O-L/U-V/B-M2/G2-G2/M2 (BM2DM1,G1,G2,LO|RS|UV)
Solution 14: B/V-U/L-O/R-S/D-N/K-E1/E1 | D/S-R/O-L/U-V/B-E2/E2-K/N (BE2KE1,DN,LO|RS|UV)
Solution 15: B/V-U/L-O/R-S/D-N/K-H/P | D/S-R/O-L/U-V/B-P/H-K/N (BP,DN,HK,LO|RS|UV)
Solution 16: B/V-U/L-O/R-S/D-J2/J2-E1/E1 | D/S-R/O-L/U-V/B-E2/E2-J1/J1 (BE2J1DJ2E1,LO|RS|UV)
Solution 17 (twin of solution 14): B/V-U/L-O/R-S/D-J2/J2-H/P | D/S-R/O-L/U-V/B-P/H-J1/J1 (BP,DJ2HJ1,LO|RS|UV)
Solution 18: B/V-U/L-O/R-S/D-F2/I2-F1/I1 | D/S-R/O-L/U-V/B-I2/F2-I1/F1 (BI2F1DF2I1,LO|RS|UV)
Solution 19 (twin of solution 1): B/V-U/L-C2/Q2-Q2/C2-M1/G1-G1/M1 | C1/Q1-Q1/C1-L/U-V/B-M2/G2-G2/M2 (BM2C1LC2M1,G1,G2|Q1,Q2|UV)
Solution 20 (twin of solution 5): B/V-U/L-C2/Q2-Q2/C2-N/K-E1/E1 | C1/Q1-Q1/C1-L/U-V/B-E2/E2-K/N (BE2KE1,C1LC2N|Q1,Q2|UV)
Solution 21 (twin of solution 3): B/V-U/L-C2/Q2-Q2/C2-N/K-H/P | C1/Q1-Q1/C1-L/U-V/B-P/H-K/N (BP,C1LC2N,HK|Q1,Q2|UV)
Solution 22 (twin of solution 4): B/V-U/L-C2/Q2-Q2/C2-J2/J2-E1/E1 | C1/Q1-Q1/C1-L/U-V/B-E2/E2-J1/J1 (BE2J1C1LC2J2E1|Q1,Q2|UV)
Solution 23 (twin of solution 2): B/V-U/L-C2/Q2-Q2/C2-J2/J2-H/P | C1/Q1-Q1/C1-L/U-V/B-P/H-J1/J1 (BP,C1LC2J2HJ1|Q1,Q2|UV)
Solution 24 (twin of solution 6): B/V-U/L-C2/Q2-Q2/C2-F2/I2-F1/I1 | C1/Q1-Q1/C1-L/U-V/B-I2/F2-I1/F1 (BI2F1C1LC2F2I1|Q1,Q2|UV)

2|a, 2|b, 2|c: 15

3|a: 7

4|a: 10, 12
4|a, 2|c: 20, 21
4|a, 2|b: 3, 5
4|a, 2|b, 2|c: 13, 14, 17

6|a: 9
6|a, 2|c: 19, 23
6|a, 2|b: 1, 2
6|a, 2|b, 2|c: 16, 18

8|a: 7, 11
8|a, 2|c: 22, 24
8|a, 2|b: 4, 6
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABAAC:

Edge data:
Code: Select all
`a a c a ab a A a b : A xG -E,C2a a b a ac a A a c : B xF -C1,Da c a b aa b A c a : C1 xC1 -Aa c a c aa b A b a : D xE -Ba b a b aa c A c a : E xD -Aa b a c aa c A b a : C2 xC2 -Bc a a a ca a B a a : F xB -Fb a a a ba a C a a : G xA -G`

Vertex sequence:
A/G
G/A
E/D C2/C2
B/F
F/B
C1/C1 D/E

4 possible sequences, leading to 3 basic solutions:
Solution 1: A/G-G/A-E/D-B/F-F/B-C1/C1 | A/G-G/A-C2/C2-B/F-F/B-D/E
1a: A'/G'-G/A-E/D-B'/F'-F/B-C1/C1 | A/G-G'/A'-C2/C2-B/F-F'/B'-D/E (AE,A'C2BC1,B'D|FF'|GG')
1b: A'/G'-G/A-E/D-B/F-F/B-C1/C1 | A/G-G'/A'-C2/C2-B'/F'-F'/B'-D/E (AE,A'C2B'DBC1|F,F'|GG')
1c: A/G-G/A-E/D-B'/F'-F/B-C1/C1 | A'/G'-G'/A'-C2/C2-B/F-F'/B'-D/E (AEA'C2BC1,B'D|FF'|G,G')
1d: A/G-G/A-E/D-B/F-F/B-C1/C1 | A'/G'-G'/A'-C2/C2-B'/F'-F'/B'-D/E (AEA'C2B'DBC1|F,F'|G,G')
Solutions 1b and 1c are twins.

Solution 2: A/G-G/A-E/D-B/F-F/B-D/E (AE,BD|F|G)

Solution 3: A/G-G/A-C2/C2-B/F-F/B-C1/C1 (AC2BC1|F|G)

2|a: 2

4|a: 3
4|a, 2|b, 2|c: 1a

6|a, 2|c: 1b
6|a, 2|b: 1c

8|a: 1d
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AAABBC:

Edge data:
Code: Select all
`b b c b ba a A a a : A xS -L,J2,G2,C2c b b b ca a A a a : B xP -J1,K,H2,D2b c a a bb a A c b : C1 xC1 -Ab c a a cb a A b b : D1 xG1 -Bb c a b bb a A a c : E1 xJ1 -C1,D1,E1,Fb c a c bb a A a b : F xL -G1,H1,I,E2b b a a bc a A c b : G1 xD1 -Ab b a a cc a A b b : H1 xH1 -Bb b a b bc a A a c : I xK -C1,D1,E1,Fb b a c bc a A a b : E2 xJ2 -G1,H1,I,E2c a a a bb b A c b : J1 xE1 -Ac a a a cb b A b b : K xI -Bc a a b bb b A a c : H2 xH2 -C1,D1,E1,Fc a a c bb b A a b : D2 xG2 -G1,H1,I,E2b a a a bb c A c b : L xF -Ab a a a cb c A b b : J2 xE2 -Bb a a b bb c A a c : G2 xD2 -C1,D1,E1,Fb a a c bb c A a b : C2 xC2 -G1,H1,I,E2a c b a aa a B c a : M1 xM1 -Oa c b c aa a B a a : N xQ -Pa a c a aa b B b a : O xR -Q,M2a a a a ac b B b c : P xB -M1,Na a b a aa c B c a : Q xN -Oa a b c aa c B a a : M2 xM2 -Pa b b b aa a C a a : R xO -Sa a a a ab b C b b : S xA -R`

Vertex sequence:
A/S
R/O
Q/N M2/M2
P/B
J1/E1 K/I H2/H2 D2/G2
C1/C1 D1/G1 E1/J1 F/L

Reverse vertex sequence:
B/P
M1/M1 N/Q
O/R
S/A
L/F J2/E2 G2/D2 C2/C2
G1/D1 H1/H1 I/K E2/J2

32 possible sequences, leading to 12 solutions:

Solution 1: A/S-R/O-Q/N-P/B-J1/E1-E1/J1 | B/P-N/Q-O/R-S/A-J2/E2-E2/J2 (AJ2BJ1,E1,E2|NP,OQ|RS)
Solution 2: A/S-R/O-Q/N-P/B-K/I-C1/C1 | B/P-N/Q-O/R-S/A-C2/C2-I/K (AC2IC1,BK|NP,OQ|RS)
Solution 3: A/S-R/O-Q/N-P/B-K/I-F/L | B/P-N/Q-O/R-S/A-L/F-I/K (AL,BK,FI|NP,OQ|RS)
Solution 4: A/S-R/O-Q/N-P/B-H2/H2-C1/C1 | B/P-N/Q-O/R-S/A-C2/C2-H1/H1 (AC2H1BH2C1|NP,OQ|RS)
Solution 5: A/S-R/O-Q/N-P/B-H2/H2-F/L | B/P-N/Q-O/R-S/A-L/F-H1/H1 (AL,BH2FH1|NP,OQ|RS)
Solution 6: A/S-R/O-Q/N-P/B-D2/G2-D1/G1 | B/P-N/Q-O/R-S/A-G2/D2-G1/D1 (AG2D1BD2G1|NP,OQ|RS)
Solution 7: A/S-R/O-M2/M2-P/B-J1/E1-E1/J1 | B/P-M1/M1-O/R-S/A-J2/E2-E2/J2 (AJ2BJ1,E1,E2|M1OM2P|RS)
Solution 8: A/S-R/O-M2/M2-P/B-K/I-C1/C1 | B/P-M1/M1-O/R-S/A-C2/C2-I/K (AC2IC1,BK|M1OM2P|RS)
Solution 9: A/S-R/O-M2/M2-P/B-K/I-F/L | B/P-M1/M1-O/R-S/A-L/F-I/K (AL,BK,FI|M1OM2P|RS)
Solution 10: A/S-R/O-M2/M2-P/B-H2/H2-C1/C1 | B/P-M1/M1-O/R-S/A-C2/C2-H1/H1 (AC2H1BH2C1|M1OM2P|RS)
Solution 11: A/S-R/O-M2/M2-P/B-H2/H2-F/L | B/P-M1/M1-O/R-S/A-L/F-H1/H1 (AL,BH2FH1|M1OM2P|RS)
Solution 12: A/S-R/O-M2/M2-P/B-D2/G2-D1/G1 | B/P-M1/M1-O/R-S/A-G2/D2-G1/D1 (AG2D1BD2G1|M1OM2P|RS)

2|a, 2|b, 2|c: 3
2|a, 4|b, 2|c: 9

4|a, 2|b, 2|c: 1, 2, 5
4|a, 4|b, 2|c: 7, 8, 11

6|a, 2|b, 2|c: 4, 6
6|a, 4|b, 2|c: 10, 12
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AAABCB:

Edge data:
Code: Select all
`b c b c ba a A a a : A xF -D,B2c b a a bb a A b c : B1 xB1 -Ac b a b cb a A a b : C xD -B1,Cb a a a bc b A b c : D xC -Ab a a b cc b A a b : B2 xB2 -B1,Ca b c b aa a B a a : E xG -Fa a a a ab c B c b : F xA -Ea a b a aa b C b a : G xE -G`

Vertex sequence:
A/F
E/G
G/E
F/A
D/C B2/B2
B1/B1 C/D

4 possible sequences, leading to 3 basic solutions:
Solution 1: A/F-E/G-G/E-F/A-D/C-B1/B1 | A/F-E/G-G/E-F/A-B2/B2-C/D

Solution 3: A/F-E/G-G/E-F/A-B2/B2-B1/B1 (AB2B1|EF|G)

2|a, 2|b: 2

3|a, 2|b: 3

4|a, 2|b, 2|c: 1a
4|a, 4|b: 1b

6|a, 2|b: 1c
6|a, 4|b, 2|c: 1d
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABBAC:

Edge data:
Code: Select all
`b a c a ab a A b b : A1 xP1 -H,C2b a c a bb a A a b : B xR -I,F2a b b b ac a A c a : C1 xM1 -E,A2a b b b ac a A a c : D xO -F1,Ga a c a ab b A b b : E xQ -H,C2a a c a bb b A a b : A2 xP2 -I,F2a c a b bb b A c a : F1 xF1 -A1,Ba c a c ab b A b b : G xI -C1,Da b b b aa c A c a : H xN -E,A2a b b b aa c A a c : C2 xM2 -F1,Gb b a b ba c A c a : I xG -A1,Bb b a c aa c A b b : F2 xF2 -C1,Dc a b a aa a B a c : J1 xJ1 -M1,Nc a b a ca a B a a : K xL -O,M2a a b a ac a B a c : L xK -M1,Na a b a cc a B a a : J2 xJ2 -O,M2a c a a ca b B b a : M1 xC1 -J1,Ka c a c aa b B b a : N xH -L,J2c a a a ca b B b a : O xD -J1,Kc a a c aa b B b a : M2 xC2 -L,J2b b a a ba a C a b : P1 xA1 -P1,Qb b a b ba a C a a : Q xE -R,P2b a a a bb a C a b : R xB -P1,Qb a a b bb a C a a : P2 xA2 -R,P2`

Vertex sequence:
A1/P1 B/R
P1/A1 Q/E
H/N C2/M2
L/K J2/J2
O/D M2/C2
F1/F1 G/I

Reverse vertex sequence:
C1/M1 D/O
J1/J1 K/L
M1/C1 N/H
E/Q A2/P2
R/B P2/A2
I/G F2/F2

64 possible sequences, leading to 16 solutions:

Solution 1: A1/P1-P1/A1-H/N-L/K-O/D-F1/F1 | D/O-K/L-N/H-A2/P2-P2/A2-F2/F2 (A1HA2F2DF1|KO,LN|P1,P2)
Solution 2: A1/P1-P1/A1-H/N-L/K-O/D-G/I | D/O-K/L-N/H-A2/P2-P2/A2-I/G (A1HA2I,DG|KO,LN|P1,P2)
Solution 3: A1/P1-P1/A1-H/N-J2/J2-O/D-F1/F1 | D/O-J1/J1-N/H-A2/P2-P2/A2-F2/F2 (A1HA2F2DF1|J1NJ2O|P1,P2)
Solution 4: A1/P1-P1/A1-H/N-J2/J2-O/D-G/I | D/O-J1/J1-N/H-A2/P2-P2/A2-I/G (A1HA2I,DG|J1NJ2O|P1,P2)
Solution 5: A1/P1-P1/A1-C2/M2-L/K-M2/C2-F1/F1 | C1/M1-K/L-M1/C1-A2/P2-P2/A2-F2/F2 (A1C2F1,C1A2F2|KM2LM1|P1,P2)
Solution 6: A1/P1-P1/A1-C2/M2-L/K-M2/C2-G/I | C1/M1-K/L-M1/C1-A2/P2-P2/A2-I/G (A1C2GC1A2I|KM2LM1|P1,P2)
Solution 7 (chiral): A1/P1-P1/A1-C2/M2-J2/J2-M2/C2-F1/F1 | C1/M1-J1/J1-M1/C1-A2/P2-P2/A2-F2/F2 (A1C2F1|J2M2|P1)/(C1A2F2|J1M1|P2)
Solution 8: A1/P1-P1/A1-C2/M2-J2/J2-M2/C2-G/I | C1/M1-J1/J1-M1/C1-A2/P2-P2/A2-I/G (A1C2GC1A2I|J1M1,J2M2|P1,P2)
Solution 9: B/R-Q/E-H/N-L/K-O/D-F1/F1 | D/O-K/L-N/H-E/Q-R/B-F2/F2 (BF2DF1,EH|KO,LN|QR)
Solution 10: B/R-Q/E-H/N-L/K-O/D-G/I | D/O-K/L-N/H-E/Q-R/B-I/G (BI,DG,EH|KO,LN|QR)
Solution 11: B/R-Q/E-H/N-J2/J2-O/D-F1/F1 | D/O-J1/J1-N/H-E/Q-R/B-F2/F2 (BF2DF1,EH|J1NJ2O|QR)
Solution 12: B/R-Q/E-H/N-J2/J2-O/D-G/I | D/O-J1/J1-N/H-E/Q-R/B-I/G (BI,DG,EH|J1NJ2O|QR)
Solution 13: B/R-Q/E-C2/M2-L/K-M2/C2-F1/F1 | C1/M1-K/L-M1/C1-E/Q-R/B-F2/F2 (BF2C1EC2F1|KM2LM1|QR)
Solution 14: B/R-Q/E-C2/M2-L/K-M2/C2-G/I | C1/M1-K/L-M1/C1-E/Q-R/B-I/G (BI,C1EC2G|KM2LM1|QR)
Solution 15: B/R-Q/E-C2/M2-J2/J2-M2/C2-F1/F1 | C1/M1-J1/J1-M1/C1-E/Q-R/B-F2/F2 (BF2C1EC2F1|J1M1,J2M2|QR)
Solution 16: B/R-Q/E-C2/M2-J2/J2-M2/C2-G/I | C1/M1-J1/J1-M1/C1-E/Q-R/B-I/G (BI,C1EC2G|J1M1,J2M2|QR)

2|a, 2|b, 2|c: 10
2|a, 4|b, 2|c: 12

3|a, 2|b: 7
3|a, 4|b: 5

4|a, 2|b: 2
4|a, 2|b, 2|c: 9, 16
4|a, 4|b: 4
4|a, 4|b, 2|c: 11, 14

6|a, 2|b: 1, 8
6|a, 2|b, 2|c: 15
6|a, 4|b: 3, 6
6|a, 4|b, 2|c: 13
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABACB:

Edge data:
Code: Select all
`a c b a ab a A c b : A1 xL1 -Fa c b a cb a A a b : B1 xO1 -G1,Ha c b c ab a A a b : C xQ -I,G2c a b a ab a A c b : D1 xM1 -Fc a b a cb a A a b : E xP -G1,Hc a b c ab a A a b : B2 xO2 -I,G2a b c b aa b A b a : F xS -J,D2,A2c b a b aa b A b c : G1 xG1 -A1,B1,Cc b a b ca b A b a : H xI -D1,E,B2a b a b ac b A b c : I xH -A1,B1,Ca b a b cc b A b a : G2 xG2 -D1,E,B2a a b a ab c A c b : J xN -Fa a b a cb c A a b : D2 xM2 -G1,Ha a b c ab c A a b : A2 xL2 -I,G2b a c a ba a B a a : K xR -Q,O2,L2b c a a ba a B c a : L1 xA1 -Kb c a a ba a B a c : M1 xD1 -L1,M1,Nb c a c ba a B a a : N xJ -O1,P,M2b a a a bc a B c a : O1 xB1 -Kb a a a bc a B a c : P xE -L1,M1,Nb a a c bc a B a a : M2 xD2 -O1,P,M2b a a a ba c B c a : Q xC -Kb a a a ba c B a c : O2 xB2 -L1,M1,Nb a a c ba c B a a : L2 xA2 -O1,P,M2a a b a ab a C a b : R xK -Sa b a b aa b C b a : S xF -R`

Vertex sequence:
A1/L1 B1/O1 C/Q
K/R
S/F
J/N D2/M2 A2/L2
O1/B1 P/E M2/D2
G1/G1 H/I

Reverse vertex sequence:
D1/M1 E/P B2/O2
L1/A1 M1/D1 N/J
F/S
R/K
Q/C O2/B2 L2/A2
I/H G2/G2

54 possible sequences, leading to 8 solutions:

Solution 1: A1/L1-K/R-S/F-A2/L2-P/E-G1/G1 | E/P-L1/A1-F/S-R/K-L2/A2-G2/G2 (A1FA2G2EG1|KL2PL1|RS)
Solution 2: A1/L1-K/R-S/F-A2/L2-P/E-H/I | E/P-L1/A1-F/S-R/K-L2/A2-I/H (A1FA2I,EH|KL2PL1|RS)
Solution 3: B1/O1-K/R-S/F-J/N-O1/B1-G1/G1 | B2/O2-N/J-F/S-R/K-O2/B2-G2/G2 (B1G1,B2G2,FJ|KO2NO1|RS)
Solution 4: B1/O1-K/R-S/F-J/N-O1/B1-H/I | B2/O2-N/J-F/S-R/K-O2/B2-I/H (B1HB2I,FJ|KO2NO1|RS)
Solution 5: C/Q-K/R-S/F-J/N-P/E-G1/G1 | E/P-N/J-F/S-R/K-Q/C-G2/G2 (CG2EG1,FJ|KQ,NP|RS)
Solution 6: C/Q-K/R-S/F-J/N-P/E-H/I | E/P-N/J-F/S-R/K-Q/C-I/H (CI,EH,FJ|KQ,NP|RS)
Solution 7: C/Q-K/R-S/F-D2/M2-M2/D2-G1/G1 | D1/M1-M1/D1-F/S-R/K-Q/C-G2/G2 (CG2D1FD2G1|KQ,M1,M2|RS)
Solution 8: C/Q-K/R-S/F-D2/M2-M2/D2-H/I | D1/M1-M1/D1-F/S-R/K-Q/C-I/H (CI,D1FD2H|KQ,M1,M2|RS)

2|a, 2|b, 2|c: 6
2|a, 4|b, 2|c: 3

4|a, 2|b, 2|c: 5, 8
4|a, 4|b, 2|c: 2, 4

6a, 2|b, 2|c: 7
6|a, 4|b, 2|c: 1
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Hyperbolic Tilings

Type AABABC:

Edge data:
Code: Select all
`a b c b ab a A a b : A xS -J,G2b a b a ac a A b c : B1 xK1 -E1,D2,Fb a b a bc a A a c : C xO -G1,Hb a b c ac a A b a : D1 xP2 -I,B2,E2a c b a aa b A b c : E1 xL1 -E1,D2,Fa c b a ba b A a c : D2 xP1 -G1,Ha c b c aa b A b a : F xQ -I,B2,E2b c a b aa b A c b : G1 xG1 -Ab c a c ba b A b a : H xJ -B1,C,D1a a b a ac b A b c : I xM -E1,D2,Fa a b a bc b A a c : B2 xK2 -G1,Ha a b c ac b A b a : E2 xL2 -I,B2,E2a b a b ab c A c b : J xH -Aa b a c bb c A b a : G2 xG2 -B1,C,D1c b a a ca a B a b : K1 xB1 -K1,L1,Mc b a b aa a B c a : L1 xE1 -Nc b a b ca a B a a : M xI -O,P1,K2a a c a ab a B a b : N xR -P2,Q,L2c a a a cb a B a b : O xC -K1,L1,Mc a a b ab a B c a : P1 xD2 -Nc a a b cb a B a a : K2 xB2 -O,P1,K2a b a a ca c B a b : P2 xD1 -K1,L1,Ma b a b aa c B c a : Q xF -Na b a b ca c B a a : L2 xE2 -O,P1,K2b a b a ba a C a a : R xN -Sb a a a ba b C b a : S xA -R`

Vertex sequence:
A/S
R/N
P2/D1 Q/F L2/E2
I/M B2/K2 E2/L2
O/C P1/D2 K2/B2
G1/G1 H/J

Reverse vertex sequence:
B1/K1 D1/P2 C/O
L1/E1 K1/B1 M/I
E1/L1 F/Q D2/P1
N/R
S/A
J/H G2/G2

54 possible sequences, leading to 8 solutions:

Solution 1: A/S-R/N-P2/D1-I/M-P1/D2-G1/G1 / D1/P2-M/I-D2/P1-N/R-S/A-G2/G2 (AG2D1ID2G1|MP1NP2|RS)
Solution 2: A/S-R/N-P2/D1-I/M-P1/D2-H/J / D1/P2-M/I-D2/P1-N/R-S/A-J/H (AJ,D1ID2H|MP1NP2|RS)
Solution 3: A/S-R/N-Q/F-I/M-O/C-G1/G1 / C/O-M/I-F/Q-N/R-S/A-G2/G2 (AG2CG1,FI|MO,NQ|RS)
Solution 4: A/S-R/N-Q/F-I/M-O/C-H/J / C/O-M/I-F/Q-N/R-S/A-J/H (AJ,CH,FI|MO,NQ|RS)
Solution 5: A/S-R/N-Q/F-B2/K2-K2/B2-G1/G1 / B1/K1-K1/B1-F/Q-N/R-S/A-G2/G2 (AG2B1FB2G1|K1,K2,NQ|RS)
Solution 6: A/S-R/N-Q/F-B2/K2-K2/B2-H/J / B1/K1-K1/B1-F/Q-N/R-S/A-J/H (AJ,B1FB2H|K1,K2,NQ|RS)
Solution 7: A/S-R/N-L2/E2-E2/L2-O/C-G1/G1 / C/O-L1/E1-E1/L1-N/R-S/A-G2/G2 (AG2CG1,E1,E2|L1NL2O|RS)
Solution 8: A/S-R/N-L2/E2-E2/L2-O/C-H/J / C/O-L1/E1-E1/L1-N/R-S/A-J/H (AJ,CH,E1,E2|L1NL2O|RS)

2|a, 2|b, 2|c: 4
2|a, 4|b, 2|c: 8

4|a, 2|b, 2|c: 3, 6
4|a, 4|b, 2|c: 2, 7

6|a, 4|b, 2|c: 1, 5
Marek14
Pentonian

Posts: 1137
Joined: Sat Jul 16, 2005 6:40 pm

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