Comments on: Polymath7 research threads 2: the Hot Spots Conjecture

By: Bartlomiej Siudeja

Bartlomiej Siudeja — Tue, 26 Jun 2012 02:29:59 +0000

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

By: Nilima Nigam

Nilima Nigam — Tue, 26 Jun 2012 01:57:49 +0000

hmm, it plots fine for me.

By: Bartlomiej Siudeja

Bartlomiej Siudeja — Tue, 26 Jun 2012 00:50:58 +0000

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.

By: Craig H

Craig H — Mon, 25 Jun 2012 19:39:20 +0000

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.

By: Terence Tao

Terence Tao — Sun, 24 Jun 2012 19:23:15 +0000

I’ve just rolled the thread over again, as this one is hitting the 100-comment mark: http://polymathprojects.org/2012/06/24/polymath7-research-threads-3-the-hot-spots-conjecture/

By: Nilima Nigam

Nilima Nigam — Sat, 23 Jun 2012 06:51:27 +0000

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.

By: Bartlomiej Siudeja

Bartlomiej Siudeja — Sat, 23 Jun 2012 06:38:02 +0000

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

By: Nilima Nigam

Nilima Nigam — Sat, 23 Jun 2012 06:36:23 +0000

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

By: Bartlomiej Siudeja

Bartlomiej Siudeja — Sat, 23 Jun 2012 06:14:04 +0000

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).

By: Nilima Nigam

Nilima Nigam — Sat, 23 Jun 2012 06:06:13 +0000

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.