I am sorry I can not understand it. It seems the difference between the left and middle data sets is the extrinsic structure, rather than intrinsic structure. Many thanks.

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Have you ever considered writing a popular book on some topics in modern math, say Seiberg-Witten theory or whatever your speciality in math used to be? I’m thinking of something along the lines of Ash and Gross’ book Fearless Symmetry, where they bravely attempt to give an account of galois cohomology. You’ll be surprised by how many people would love to read something like that (as is also evidenced by the fact that Ash and Gross have gone on to write two other books of a similar nature).

]]>Makes sense. Thanks for following up!

-J

]]>1) If you moved the pen to a spot that’s earlier on the original path, it would end up returning to the place where the original path started, then re-trace that path. But it wouldn’t cross the path, i.e. there wouldn’t be a visible intersection – it would just look like a longer curve.

2) The pen never reaches an equilibrium point because the vectors get shorter and shorter, so the curve only meets the equilibrium point in the limit. So two different curves can only meet at an equilibrium point and in the limit. (Though again they don’t cross – they just meet at their ends. I realize this is a technicality, but I’m sticking to it.)

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