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."

Comments

Great collection of Escher images

[tolleyl]

This is a page of Escher images that are posted with permission of the copyright holder. It's one of the best collections on the web. http://www.cs.unc.edu/~davemc/Pic/Escher/

For those who don't know Escher

[Anonymous Coward]

World of Escher

For the curious:

[colmore]

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.

According to Hofstadter...

[mwhansen]

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.

Re:I can't believe it!

[jafac]

1. I've studied Escher, and I'm utterly convinced that he knew exactly what elliptical curves were. He may not have understood it in a mathematically analytical sense, more of as a intuitive sense.

2. His work was primarily in lithography. You don't worry too much about the fine precision of "dull tools" like pencils and pens. Traditional lithography is done on a large limestone slab, with a grease pencil, yes, but you can sharpen the pencil and achieve very fine lines, because it's very soft - and ultimately, you're more limited by the grain of the paper in your resolution than anything else.
(next, the grease pencil acts as a resist, and the stone is chemically etched, and then ink applied. The raised, or non-etched bits of the stone surface press ink into the paper, the depressed bits do not.)
Escher also worked a lot in woodcut and engraving - those techniques are fairly obvious, and in woodcut, at least, you are pretty limited in resolution, as far as the grain of the wood goes.
In any case, drawing out the center, as it goes, is not impossible - because EVERY object you draw has infinitely small detail on it. Part of the technique of a good artist is knowing when to suggest detail and when to actually render it, and at what point, actually rendering it will yeild an effect that is not desirable. Had Escher chosen to render this portion of the drawing, it would have been a simple matter of rendering the details down to a certain point, and thereafter, simply suggesting it - knowing that, nobody's going to be examining the central part of the drawing with a microscope. The human eye only sees so much.
It's more likely that he concluded that the human eye of the viewer would have been drawn to this central point, and the problem would have been that attention would be needlessly focussed on the details there, instead of the outer portions of the drawing.

3. Escher was Dutch. I know we've all seen enough racial profiling in the past year, but the stereotype holds true - you'll be hard pressed to find a lazy, or even "laid back" (to use the politically correct term) Dutchman. Enough with the generalizing - just a brief study of the individual's life, and you'll know that he was a very intense, hard working man, and a very prolific artist. Looking at some of his studies and sketches, and how he drafted out and worked on these designs, they were incredibly labor intensive. He could have chosen to draw in any style he wanted, and he chose this mathematically precise style because it was fun to him. Anyone who suggests that Escher was in any way lazy or allowed a work to be "uncompleted" simply does not know the first damn thing about the man.