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Escher and Elliptic Curves
melquiades writes "Mathematician Hendrik Lenstra was struck by the blank spot in M. C. Escher's Print Gallery . Why is the spot blank there, he wondered, and what should go in it? Although Escher, who had only a high-school mathematics background, drew the picture by brilliant and methodical intuition, the mathematical machinery underlying the image turned out to be elliptic curves (which come up in factorization, cryptography, and the proof of Fermat's Last Theorem). Lenstra and his colleagues were able to generate several breathtaking possible completions for the missing space. Read the story at the ever-registration-required NYT."
Elliptic Curves:
curves of the form y^2 = Ax^3 + Bx^2 + Cx + D
pick values for A B C and D, the locus in 2 space (the cartesian plane, or R2) is the type of curve Escher was using.
In analysis, which is where all of the headline making math using Elliptic Curves, A B C and D (as well as x and y) can be complex numbers.
At this point things get complicated. I'm not going to fill up 1000 words explaining Riemann surfaces, algebraic functions, etc.
There are a lot of good pages out there.
In Capitalist America, bank robs you!
On page 717 in Godel, Escher, Bach, Hofstadter explains the "central blemish" as follows...
"Though the blemish seems like a defect, perhaps the defect lies in our expectations, for in fact Escher could not have completed that portion of the pircture without being inconsistent with the rules by which he was drawing the picture. The center of the whorl is -- and must be -- incomplete. Escher could have made it arbitrarily small, but he could not have gotten rid of it."
What Lenstra was able to do was to figure out the structure of the picture. From there, he was able to generate a suitable center so that none of the relationships between the four various pieces are disrupted.
This is the reason why this is some pretty neat work.