On May 20, 2014, at 1:05 PM, Warren D Smith <warren.wds@gmail.com> wrote:
I find reading this paper painful but think it probably is good underneath. The painfulness comes from the large length compared with the small amount of "here's a satisfying result for you". Specifically, it seems to me, that you must have shown, that you can write down a certain formula, for an analytic function of two variables, in closed form (using Schwarz Christoffel) such that complex plane "sections" of this function (1 complex dimensional sections of 2 complex dimensional thing) have level sets describing quasiperiodic Penrose-tiling-like sets. [Meanwhile other sections of other such functions lead to (boring) doubly-periodic sets and correspond to elliptic functions.]
Right? So then my question would be -- since that seems to be the whole point your paper is (or should be) driving at -- why didn't you actually do it?
Write down a freaking formula and get a Penrosian set! If you were really ambitious you could then actually compute some pictures… You never got as far as figure 1?
The point of this paper was not to make pretty pictures, but to come up with a unique realization of these geometrical objects that cuts across different symmetries, “local isomorphism classes”, etc. In a subject dominated by gadgetry — worms, cartwheels — it’s hard to see what properties are generic or even how to articulate those generic properties. The idea behind this paper developed after hearing Bill Thurston describe his geometrization approach to classifying manifolds. So instead of always messing around with gadgets — turning chickens into kites and darts — why not use geometry to select the best “representative” of the various classes, and make these the subject of study and classification? I completed the classification that was started in that paper. There are 35 classes, including 2 with 8-fold, 4 with 10-fold, and 14 with 12-fold symmetry (the others all have crystallographic symmetry). The classes are distinguished not just by their symmetry but also by their transformation properties — all that business involving worms and cartwheels ...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun