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M. Čtrnáct provided a full research, which uniform tilings of several given vertex configuration types would be possible in general. This research thereby is completely independent of any whatsoever expanded usage of Dynkin symbols and also of the given theorem provided in the context of Coxeter domains. In fact, it exceeds the reach of both realms.
He starts his research not directly with the mere vertex configuration, but rather with edge configurations. He uses capital letters for
some given tile, and lower case letters for the directly or diametrally adjacent ones. In the considered quest we get the following 6 edge
configurations. The code provided beside each configuration refers
by : to its label,
by ⇅ to its opposite configuration (the configuration having the edge-opposite tile capitalised,
in fact a 180° rotation around the respective edge center),
by ↶ to the set of possible edge configurations at the next edge of the main tile towards its left,
by ↷ to the set of possible edge configurations at the next edge of the main tile towards its right.
(These and several more similar research results originally were posted here.)
The local edge configurations then are
a | b | a --+---+-- a | A | a :A_{ } ⇅E_{ } ↶D,B_{1} ↷D,B_{2} b | a | a --+---+-- a | A | b :B_{1} ⇅B_{1} ↶C,B_{2} ↷A b | a | b --+---+-- a | A | a :C_{ } ⇅D_{ } ↶C,B_{2} ↷C,B_{1} a | a | a --+---+-- b | A | b :D_{ } ⇅C_{ } ↶A_{ } ↷A a | a | b --+---+-- b | A | a :B_{2} ⇅B_{2} ↶A_{ } ↷C,B_{1} a | a | a --+---+-- a | B | a :E_{ } ⇅A_{ } ↶E_{ } ↷E
The first 5 configurations describe the possible edges of an A-gon. B-gons do have just a single edge type, because they have to be surrounded completely by a-gons.
The next step then is to build (larger) vertex configurations: now not by means of mere incident tiles, rather by means of these just deduced incident edge configurations. By definition one has a b-gon incident to each vertex. And the edge configuration type here is E (with the edge-opposite configuration A). Thus we have to start with E|A, will have to consider the sequences ↷, and finally will end in A|E. The possible sequences here are:
E|A ↷ _{ }D|C_{ } ↷ _{ }C|D_{ } ↷ A|E E|A ↷ _{ }D|C_{ } ↷ B_{1}|B_{1} ↷ A|E E|A ↷ B_{2}|B_{2} ↷ _{ }C|D_{ } ↷ A|E E|A ↷ B_{2}|B_{2} ↷ B_{1}|B_{1} ↷ A|E
Or in terms of tile configurations these codings correspond respectively to the types:
a | a --+-- a | B | A | b --+---+---+-- a | A | A | a --+-- b | a E|A ↷ _{ }D|C_{ } ↷ _{ }C|D_{ } ↷ A|E a | a --+-- a | B | A | b --+---+---+-- a | A | A | a --+-- a | b E|A ↷ _{ }D|C_{ } ↷ B_{1}|B_{1} ↷ A|E a | a --+-- a | B | A | a --+---+---+-- a | A | A | b --+-- b | a E|A ↷ B_{2}|B_{2} ↷ _{ }C|D_{ } ↷ A|E a | a --+-- a | B | A | a --+---+---+-- a | A | A | b --+-- a | b E|A ↷ B_{2}|B_{2} ↷ B_{1}|B_{1} ↷ A|E
Here the first and last larger vertex configuration are 2 different symmetrical ones, while the others just form a single enantiomorph pair. Thus those 2 middle ones ought to be grouped.
Finally one considers the possible sequences of edge configurations around the tiles themselves. Here the (above grouped) larger vertex configurations already have to be considered independently. It shall be noted that the consecutive edge configurations around any tile always are given in parts by the respective substrings |x ↷ y|.
The pair of lowest numbers, which allows for all 4 cases, i.e. bows to all divisibility constraints simultanuously, thus would be a = {12} and b = {4}. That choice therefore is depicted above each.
[3,9,9,9] e.g. on the other hand would be unique, it can exist only as type 4. Moreover it is a quite evident example for not matching the requirements of the above mentioned theorem, because here all tiles are odd.
Conversely, putting together all these cases in an alternative way, we just get the restriction, that a needs to be divisable either by 2 or by 3 at least. E.g. a can neither be {5}, {7}, {11} etc. when required to become an [a,a,a,b] tiling.
Right according the lines of the previous section this quest can be solved too. Starting with the here possible edge configurations:
a | b | a a >+---+< a a | A | a :A_{ } ⇅H_{ } ↶B_{1},E_{1},G ↷G,E_{2},B_{2} b | a | a a >+---+< a a | A | b :B_{1} ⇅B_{1} ↶C_{1},F,E_{2} ↷A b | a | a a >+---+< b a | A | a :C_{1} ⇅E_{1} ↶C_{1},F,E_{2} ↷B_{1},C_{1},D b | a | b a >+---+< a a | A | a :D_{ } ⇅G_{ } ↶C_{1},F,E_{2} ↷E_{1},F,C_{2} a | a | a b >+---+< a a | A | b :E_{1} ⇅C_{1} ↶D,C_{2},B_{2} ↷A a | a | a b >+---+< b a | A | a :F_{ } ⇅F_{ } ↶D,C_{2},B_{2} ↷B_{1},C_{1},D a | a | b b >+---+< a a | A | a :C_{2} ⇅E_{2} ↶D,C_{2},B_{2} ↷E_{1},F,C_{2} a | a | a a >+---+< a b | A | b :G_{ } ⇅D_{ } ↶A_{ }_{ } ↷A a | a | a a >+---+< b b | A | a :E_{2} ⇅C_{2} ↶A_{ }_{ } ↷B_{1},C_{1},D a | a | b a >+---+< a b | A | a :B_{2} ⇅B_{2} ↶A_{ }_{ } ↷E_{1},F,C_{2} a | a | a a >+---+< a a | B | a :H_{ } ⇅A_{ } ↶H_{ }_{ } ↷H
Next one builds (larger) vertex configurations, as deduced by these incident edge configurations. The 27 possible sequences, grouped by mirror images, here are:
H|A ↷ _{ }G|D_{ } ↷ E_{1}|C_{1} ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ C_{2}|E_{2} ↷ _{ }D|G_{ } ↷ A|H † H|A ↷ _{ }G|D_{ } ↷ E_{1}|C_{1} ↷ C_{1}|E_{1} ↷ A|H H|A ↷ E_{2}|C_{2} ↷ C_{2}|E_{2} ↷ _{ }D|G_{ } ↷ A|H H|A ↷ _{ }G|D_{ } ↷ E_{1}|C_{1} ↷ _{ }D|G_{ } ↷ A|H H|A ↷ _{ }G|D_{ } ↷ C_{2}|E_{2} ↷ _{ }D|G_{ } ↷ A|H † H|A ↷ _{ }G|D_{ } ↷ _{ }F|F_{ } ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ _{ }F|F_{ } ↷ _{ }D|G_{ } ↷ A|H H|A ↷ _{ }G|D_{ } ↷ _{ }F|F_{ } ↷ C_{1}|E_{1} ↷ A|H H|A ↷ E_{2}|C_{2} ↷ _{ }F|F_{ } ↷ _{ }D|G_{ } ↷ A|H † H|A ↷ _{ }G|D_{ } ↷ _{ }F|F_{ } ↷ _{ }D|G_{ } ↷ A|H H|A ↷ _{ }G|D_{ } ↷ C_{2}|E_{2} ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ E_{1}|C_{1} ↷ _{ }D|G_{ } ↷ A|H † H|A ↷ _{ }G|D_{ } ↷ C_{2}|E_{2} ↷ C_{1}|E_{1} ↷ A|H H|A ↷ E_{2}|C_{2} ↷ E_{1}|C_{1} ↷ _{ }D|G_{ } ↷ A|H H|A ↷ E_{2}|C_{2} ↷ E_{1}|C_{1} ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ C_{2}|E_{2} ↷ C_{1}|E_{1} ↷ A|H H|A ↷ E_{2}|C_{2} ↷ E_{1}|C_{1} ↷ C_{1}|E_{1} ↷ A|H H|A ↷ E_{2}|C_{2} ↷ C_{2}|E_{2} ↷ C_{1}|E_{1} ↷ A|H † H|A ↷ E_{2}|C_{2} ↷ _{ }F|F_{ } ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ _{ }F|F_{ } ↷ C_{1}|E_{1} ↷ A|H † H|A ↷ E_{2}|C_{2} ↷ _{ }F|F_{ } ↷ C_{1}|E_{1} ↷ A|H † H|A ↷ E_{2}|C_{2} ↷ C_{2}|E_{2} ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ E_{1}|C_{1} ↷ C_{1}|E_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ E_{1}|C_{1} ↷ B_{1}|B_{1} ↷ A|H H|A ↷ B_{2}|B_{2} ↷ C_{2}|E_{2} ↷ B_{1}|B_{1} ↷ A|H † H|A ↷ B_{2}|B_{2} ↷ _{ }F|F_{ } ↷ B_{1}|B_{1} ↷ A|H
Now we try to build up edge sequences from those contained |x ↷ y| for the tiles a and b. Generally we restrict to uniform tilings only, therefore either a single such vertex configuration or at most the union with its mirror image can be used. Sequences which do not close (corresponding groups are marked † above) will have to be omitted in the following listing. This then results in the following possible cases:
The least case, which shows up all of the 14 possibilities, therefore would be [24^{4},4]. Obviously numbers much too high to display the tiling differences within according Poincaré disc displays each. Instead we provide in addition to the above grouping of several examples of tilings per case also a regrouping of different tiling cases per divisibility.
divisibility | e.g. | cases | |
3|a | 1|b | a=3, b=7 | 12 |
4|a | 1|b | a=4, b=3 | 7 , 10 , 14 |
4|a | 2|b | a=4, b=6 | 1 , 3 , 7 , 10 , 14 |
6|a | 1|b | a=6, b=3 | 2 , 4 , 7 , 12 |
6|a | 2|b | a=6, b=4 | 2 , 4 , 5 , 7 , 8 , 12 , 13 |
8|a | 1|b | a=8, b=3 | 6 , 7 , 9 , 14 |
8|a | 2|b | a=8, b=4 | 1 , 3 , 6 , 7 , 9 , 10 , 11 , 14 |
© – All the pics on [a,a,a,a,b] generously have been privately provided by M. Čtrnáct. Used colorings only aid for distinction; none the less all cases are uniform. Esp. yellow and green tiles resp. light and dark blue ones (and in case 10 also red and orange ones) often just distinguish enantiomorph pairings, although they would be symmetry equivalent.
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