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

The space is the whole point.

[MjDascombe]

It's supposed to make individuals think. Without the space it's just an optical illusion. Whats next, threories explaining Mona Lisa using computers? Morphing?

What?! They've already done that. Well, fuck it, I'll go back to coding...

Re:The space is the whole point.

[MaxwellStreet]

I believe that just about everything creative is about thinking (often about established things), and broadcasting your ideas.

I can understand the desire to leave things just as the artist left them - but creating these derivative works doesn't diminish the value of the original. Quite the opposite ... people are looking and considering today what they might not have before.

And he's only rendered and published some solutions that appeal to him ... not -the- solution, and not even just -a- solution.

I contend that by publishing several, it challenges the viewer even more to think about why these are good, and what changes we (the viewer) might make. The fact that they are based on mathematical principles - extrapolating the center and all - only serve to make his 'solutions' more compelling viewing.

absurd

[Anonymous Coward]

This is perhaps one of my favorite drawings by esher and has been so for many years. Oddly enough, when I first saw the picture I was sorely pissed off because the picture didn't seem complete. What the hell was in that spot? I wanted to know badly and I couldn't possibly like the drawing until I did.

It was only when I came back to the picture years later. I tried to figure out what I would put in the spot that I realized how excellent the drawing is. It is a stunning metamorphosis between images and I believe the spot only serves to compound that perfectly. If the spot was there you would spend more time staring at the spot them following the transforming images around the outside. The subtly of the picture would be lost on people who were fascinated by the damn spot in the middle (as it was with me).

I'm not denouncing their work. It is very impressive and interesting to read. However I have no intentions of ever hanging a print up without that damn spot. (insert appropriate Shakespeare joke here)

Re:Re:Hmm

[Anonymous Coward]

This makes me wonder/question/revisit the old debate of nature/nurture? Is it possible that Escher's brain was wired in a particular way that allowed him to create works of art based on mathetic principles without knowing the underlying structure? I.e. was escher following principles of cellular automata or something like it?

This brings up another debate which is more interesting than why did escher leave a hole in the picture. What constitutes genius or brilliance? Is the artist who draws instinctively a genuis? Or is the mathematician who applies complex theories to pictures and natural patterns a genuis? Are both the artist and scientist manifestations of two sides of a coin? Or are we just playing into stupid labels? In the end, does it really matter that escher left a hole in the picture, or that people wonder why the hole is there?

Focal point

[reelbk]

I think Escher meant to leave that area blank since it seems like the rest of the drawing is being drawn towards it. It's the focal point of the picture since that's where the picture in the gallery actually connects to the gallery. If you look closely, you'll notice that the frame of the picture is on it's way to meet with the picture itself, which is infact the gallery (woah, I've fallen into the loop). I think this dot was left up to the imagination. There is no correct solution, but this method is a terrific idea.

Re:Wish I could do that...

[squaretorus]

Degrees are handy if you want to work for others, as it makes it easier for them to believe you when you say "Im worth hiring". But makes not one ounce of difference when you want to do things for yourself.

Just get out there and do what you want, measure your own success by your own values - not by the size of your car - and you'll be happy.

Forget all societal measures of your worth - they mean nothing. Except karma of course - anything less than excellent and your a twat!

Re:Re:Hmm

[Pig Hogger]

This makes me wonder/question/revisit the old debate of nature/nurture? Is it possible that Escher's brain was wired in a particular way that allowed him to create works of art based on mathetic principles without knowing the underlying structure? I.e. was escher following principles of cellular automata or something like it?

Perhaps. Some people's brains are better than other at extrapolating phenomenon after a cursory glance; mathematics simply attempts to formally describe the extrapolation, so people who are unable to extrapolate by themselves can do so by applying the formal principles. Maurits Cornellis Escher was amongst the former people, and university gratuates are amongst the latter.

Art or math

[gilroy]

I've read a bunch of comments along the lines of "Oh, that's interesting. But those mathematicians, with their formality, are killing Escher's art". Bullwash. There is beauty in the math, too, and grace, and yes, even art. Sure, these researchers are using a different brush and a different canvas. But in number theory there are intricacies and elegances to break your heart. It's no less "art" because it's done through math.

I don't think they've improved on Escher, any more than I think they've "ruined" him. They've just used his artwork as a springboard for their own. For a community that likes to rhapsodize about the value of the public domain and the intellectual commons, an awful lot of slashdotters seem to object to this.