Assuming you have 0,0 and 1,0 as other 2 vertices 0.83, 0.3 gives obtuse triangle using Pythagorean theorem.

]]>hmm, it plots fine for me.

]]>Your 40-60-80 triangle is a bit obtuse, when plotted. It seems aa and bb should be different. In fact we should probably switch from 40-60-80 to something with rational vertices.

]]>Just FYI, I realized that in general we do not have a finite number of sheets. One can take, for example, the 45-67.5-67.5 isoceles triangle. Reflecting about the two congruent edges gives an octagon, but if on reflects long edge-short edge-long edge, the two images of the short edge are at right angles to each other. Repeating this process produces a square tiling of the plane. If you assume that the short edge has length 1, you can get an infinite number of square tilings of the plane translated by 1 in a diagonal direction — that is, the preimage of the corners of the original triangle contains Z[\sqrt{2}]^2. Since this set is dense, we can’t have a finite number of sheets.

]]>thanks are due to you- you’ve been incredibly patient and careful, and I appreciate your looking over the results. I fixed the print statement for the vertices. Who knows, maybe introduced some other error. I’ll stop for now.

]]>You deserve a break. You are doing a great job with the numerical side of the project.

]]>fixed, thanks. I’m going to take a break from coding now, since I’m making silly errors.

]]>I could not reply to your latest post. With old data even the third eigenfunction was good for hot-spots conjecture. I am not sure what vertices you are using for 40-60-80 triangle, but the old ones where giving me an obtuse triangle. You have right isosceles vertex for 40-60-80 in the new file (aa, bb values).

]]>You’re right!

Thanks for your patience- this has been a helpful discussion for me. This error, and the fact that I also got the second eigenvalue on the 40-60-80, suggested I had a systematic bug in the code I wrote today. I found it, fixed it, and have replaced the data:

http://www.math.sfu.ca/~nigam/polymath-figures/dump-data.odt

The conclusions remain the same.

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