On 04/08/2016 20:46, Roger Bagula wrote:
Steward Robert Hinsley,
Your tile has puppies...
The construction used is more restrictive than it needs to be.
It has 5 tiles for each of the 36 different IFSs that allow us to treat
the flowsnake as c6-symmetric for purposes of derivation by this
technique, keeping five elements of the outer ring fixed.
If we treat the flowsnake as c3-symmetric, there are 180 different
allowable IFSs that are not also allowable for the c6-symmetric case,
and 2 Pacman tiles for each of those, again keeping five elements of the
outer ring fixed.
This works, so 720 tiles when including both 7 element flowsnakes as
base tiles, unless there's some degeneracy that I've overlooked. All
have 3 copies in the unit cell.
If we treat the flowsnake as c2-symmetric, there are 1260 different IFSs
that are not also allowable for the c6-symmetric case, but only 1 Pacman
tile for each of those, again keeping five elements of the outer ring fixed.
This works, so 2520 IFS when including both 7 element flowsnakes as base
tiles. All have 2 copies in the unit cell. But the orientation of the
pair of elements at the same position is indepedent of the other
elements, and doesn't matter, as their combined shape is a flowsnake,
and the attractor is 6-fold degenerate. That reduces to number of such
tiles to 360.
Or 1080 further Pacmen tiles in total.
The next step is to investigate the permutation of the fixed cells.
(After many hours of computer time to generate 1080 tiles and their unit cells.)
In the c2-symmetric case this just generates the same set of tiles, as
the allowable permutations are swapping each pair of opposite tiles,
which is equivalent to using the IFS with that pair in the reverse
orientation.
In the c3-symmetric case the allowable permutations are swapping the two allowable target permutations, and permuting the other 3 positions. Permutations 246 -> 462 and 246-> 624 generate the same set of tiles as
246 -> 246, for the same reason as above, so they can be eliminated.
That leaves the swaps 24 -> 42, 26 -> 62 and 46 -> 64, and independently
35 -> 53, to be considered.
In the c6-symmetric case there are 120 allowable permutations. So if
that works that's over 30000 more tiles.
The same construction can be used with the Koch Snowflake, except that
this is d6-symmetric so the effect of throwing reflections in to the mix
has to be taken in account - I believe this means that you can end up
with tiles that have 12 copies in then unit cell - the obvious candidate
would be a Koch demi-teardrop.
Then we can take the symmetric 13 element dissections of the flowsnakes
or the Koch snowflake, and treating pairs of elements from the inner and
outer rings as units produce similar tiles with larger mouths.
We can also take the 19 element 2nd order flowsnakes, and produce
similar figures by treating groups of 3 elements as units.
The sub-tile holes remind me very much of the McWorter
L-system that pokes fractal holes in tilings...
Dieter Steemann did an L-system study of Flowsnake derivatives
based on L-system "closed paths" that was interesting last year
or the year before.
Thanks for keeping up your tiling work.
Roger Bagula
--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)