Thanks!

]]>Nicely done!

]]>[…]Excellent blog here! Also your site so much up fast![…]…

]]>Actually, this could be used to good realistic effcet. Sediment will tend to deposit in concavities and wind will tend to clean and abrade convexities giving them different colors. This is much the way that vegetation and sand distribution can be affected by slope. Also, if this could be applied to heightfields or masks, you could use this to flatten out the bottoms of concavities and convexities and possibly to give more of a streaky and abraded or bare rock texture to convexities. I may have to get one of those evil Windows machines just for your app.

]]>I’ll explain why lines of steepest descent interest me. They are evevrywhere tangent to the gradient of , the lowest non-trivial eigenfunction of the Neumann Laplacian. For your height 2 , base 1 isosceles triangle, it seems the “cold spot” has about -0.09 , at the pointy vertex. Maybe we already know that the two other vertices at the base are the only two “hot spots” with being about +0.03 there, and maybe +0.025 on the base, half-way between the two “hot spot” vertices. Say we do a mirror image of the 3d graph of about the base, thus extending ; could it be that the extended has a saddle point at x=0, y=0 ? It seems that to the left of the axis of symmetry of the height 2, base 1 triangle, the steepest ascent curves should slope downwards and to the left, and to the right of the axis of symmetry, the steepest ascent curves starting at the pointy vertex should slope downwards and to the right.

For nodal lines, I meant for the eigenfunctions. They show where is average-valued. For the isosceles case, we already know where they lie, by the of Laugesen and Siudeja.

It’s easy enough for me to generate some figures showing the nodal line for a few triangles.

As for your steepest descent comment:

do you mean we should plot the nodal line in the same fixed triangle as time increases (in the heat equation), and then look at the direction of steepest descent? This is relatively easy to do.

or do you mean we should plot the nodal line for triangles with angles which are close, and somehow study the nodal lines in there? This I don’t readily see how to do, since the domain (and hence the location of the nodal lines) will change from triangle to triangle.

Let me know if this is still of interest, and I can throw up some graphics.

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