Statistical physicists were right in their definitions. Mathematicians should take over computer graphics and fix the problem we ourselves created. Anything less would be dishonourable and dereliction of duty. You can’t expect these clowns to do anything right.
Mathematicians had erred in our notion of smoothness and we had been struggling with understanding the concepts of roughness and smoothness. We simply should pay careful attention to statistical physics models for these notions because things only look smooth in images looked at from a distance, which is why piecewise linear approximations are fine. Besovspace theory has global nothions of smoothness but we need better concepts of smoothness. The GaussBonnet theorem for surfaces tells us a bit about curvature integrals: the Euler characteristic of a genus g surface is 22g which must match the normalized integral of the Gauss curvature over the surface. This means that discretization of the surface piecewise linearly will do nothing to the topology and so piecewise linear approximation is possible for mathematically defined smoothness that is totally local having to do with certain derivatives existing locally. But from the statistical view of smoothness, this sort of local smoothness is not important; rather, what is smooth is dependent on a large system of particles having some properties in the large. Hydrodynamics is the most obvious science where this intuition is clearest. The human notion of smoothness is defined in terms of smooth flows and turbulent or chaotic behavior. In computer graphics three dimensional modeling the deficiency of the mathematicians’ definition is the clearest because human eye can see smoothness and chaos not by the mathematicians’ definition but the statistical physicists.
Thus we are compelled to peek in the attached.
Attachments area
Preview attachment yanglee2.pdf
yanglee2.pdf

2:18 PM (0 minutes ago)



Baxter. Now the mathematician Benoit Mandelbrot had made an attack on the classical notion of smoothness with introduction of fractals but I think the deeper issue of what is the right definition of smoothness is still an open issue. I am myself partisan to considering piecewise linear approximations of smooth compact riemannian twodimensional manifolds isometrically embedded in the flat Euclidean threedimensional spaces to be equally smooth because in computer graphics smoothness is a macro concept.
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