Is there an optimum near s=0.417 ?

Using s=0.417 instead of s=0.42 in the code of the last post I got a quasiperiodic structure with fewer and smaller gaps than the one shown in “Projection method for 5-fold rotational symmetry“. Further decreasing s results in somehow chaotic structures. Note that the gaps seem to have all the same shape. Do we have a tiling with a finite set of tiles ? Is this in some sense a best result ?

0417

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3 Responses to Is there an optimum near s=0.417 ?

  1. Eric says:

    A possible complication: changing the first line of code from size(600,600) to size(700,600) reveals diamond shaped gaps (thick rhombs), just like the gaps that showed up here:https://geometricolor.wordpress.com/2012/08/19/a-quasiperiodic-tiling-with-pentagrams/

    Changing the size agin to (1500, 600), and being patient while the pattern bakes in the oven for awhile, reveals a gap that’s kite-shaped (next to two diamonds).

    Whoops, I was about to post this comment, but then I changed the value of s to 4.178, and that seems to fix the problem. Maybe the value of s just has to be refined a bit more.

    • Peter Stampfli says:

      Thanks Eric, for your great results. Unfortunately, I have no idea how to find the best value for s other than refining as you did. And I have no idea what the optimum looks like and what it means.
      I am now working on the dualization method, which will produce tilings without any gaps. But this method is extremely difficult to work out and to present. Be patient.

      • Eric says:

        That’s great news. I’m very much looking forward to seeing the dualization method, particularly given your earlier blog entries showing the dualization method in action.

        For the above comment, I see I have a typo – I meant to write s=0.4178, and in fact, s=0.41785 works even better. But it is interesting that 4.178 does produce great results! Before noticing my typo, I tried values of s between 0.8 and 0.81 and saw quite similar results to 4.178 — for example, try 0.805.

        Other results: For 9 fold symmetry, values of s around 0.333 are interesting.

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