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 5fold 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 ?

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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/aquasiperiodictilingwithpentagrams/
Changing the size agin to (1500, 600), and being patient while the pattern bakes in the oven for awhile, reveals a gap that’s kiteshaped (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.
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.
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.