[math-fun] flowsnake bathroom tile
[ Take 3 --- with apologies to Rich! ] After GoogleDocs rehashed their interface again --- which it admittedly needed --- I can't upload there any more. So I've attached a GIF graphic, which I hope will sneak under the 40Kb limit set by the math-fun server. An efficient algorithm (combinatorial, rather than generic polygon-filling) to fill the original snowflake curve required some thought on my part, although in the end it isn't very difficult. The natural tiles are rhombs, rather than RWG's hexagons [hexflo.png]. However, I haven't yet figured out a neat way to use them directly, rather than split into two triangles. The two regions are simply connected, unsurprisingly though not immediately obviously --- take the ends of a finite segment of curve and pull? Rather grotty Maple code & larger pics available on request. Fred Lunnon
The pictures didn't come through. On Mon, Aug 8, 2011 at 6:52 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
[ Take 3 --- with apologies to Rich! ]
After GoogleDocs rehashed their interface again --- which it admittedly needed --- I can't upload there any more. So I've attached a GIF graphic, which I hope will sneak under the 40Kb limit set by the math-fun server.
An efficient algorithm (combinatorial, rather than generic polygon-filling) to fill the original snowflake curve required some thought on my part, although in the end it isn't very difficult.
The natural tiles are rhombs, rather than RWG's hexagons [hexflo.png]. However, I haven't yet figured out a neat way to use them directly, rather than split into two triangles.
The two regions are simply connected, unsurprisingly though not immediately obviously --- take the ends of a finite segment of curve and pull?
Rather grotty Maple code & larger pics available on request.
Fred Lunnon
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Dan said (apparently, though not to math-fun?)
BUT there is a lot of choice in how 6 copies of each rosette are placed around itself, so this method can give a plethora of different curves in the limit. (Probably continuum many, given a countable number of discrete choices.)
As it stands, I don't think this is quite correct. Modulo Euclidean isometry, there appear to be only the two generation rules illustrated by Bill at <http://gosper.org/flopnoflop.png> . Notice that both series of curves are rotated making curve start and finish on a horizontal line; but the (nascent) island coastlines are mirror images and at distinct angles. Therefore they can't be combined in the course of a single generation: if the smaller coastlines fit together, their curve endpoints do not meet. I think it may be possible to vary the rule as a function of generation n, yielding 2^(n-1) visually distinct curves; however I have yet to concoct an actual algorithm. By the way, notice also that although the limit coastline has 6-fold rotational symmetry, the discrete approximants (ignoring curve endpoints) have only 3-fold. This is easy to see from the filled plot I attempted to post earlier, which has apparently been ambushed at the pass ... Fred Lunnon
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Fred lunnon -
Mike Stay