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

Wish I could do that...

[ThogScully]

...with only a high school education. I've already been brainwashed into thinking a degree is necessary to get anywhere though.

I have trouble believing anyone will take tech people seriously these days without a degree, but I think it's great to see that there's still an opportunity for a true genius to break that belief.

I can't believe it!

[Spackler]

This is one of the few articles where the troll responses made more sense than the real ones.
1. It's art. Just enjoy it.
2. Not everything needs a higher meaning

My opinion is that it is the drain that the world is circling around, but that is just MY opinion.

Re:I can't believe it!

[Masem]

I would argue that the researcher that undertook this work was not trying to depreciate the value of the art at all by doing this analysis: he was simply interested in seeing if he could 'finish' the work by using elliptical curves and image manipulation.

First, I do think that Escher left that space blank intentionally partially to help the eye follow the 'progression' of the illusion, but also, it would be impossible to draw out the center with 'dull' tools like pencils and pens. On this latter point, the researcher's site points out that the image would be infinitely recursive into the center; to draw it out completely would be neigh impossible. Escher probably realized this when drawing it (and without knowing exactly what elliptical curves were), and concidering the overall positive effect of the white space, left that area blank when he couldn't effectively draw any finer detail than his usual style.

So what is of interest of this research is more of what we can do with image manipulation and mathematics to 'extrapolate' art, rather than to say that Escher was lazy and could have finished that work. There was an article almost a year ago here on a program that 'analyzed' the style of one image and applied that to a second image, one example being of Monet's dot style applied to photos and other classic artwork. This falls in the same line; the group had to extrapolate a few parts of the picture that fell outside Escher's original, then used complex math to rebuild it in a number of ways. The results are certainly not 'new' artwork in anyway, but they do show what we can do in "Computational Art".

(Hmm, I wonder, before it was /.ed, did they try to take this procedure in reverse; that is, take a photo that has sufficiently similar properties like the print itself, after it was deconvoluted into the simple image, and reapply the elliptical curve as to generate the same optical illusion as the original had?)

Do you need Mathematics ....

[os2fan]

Seriously.

The point is, that you can perfectly see the sort of space that Escher draws, or that I dabble in, without too much mathematics.

I quite often see the curves that Escher drew in his pictures.

Also, one can even understand hyperbolic geometry without any great understanding of the mathematics. I have even made new discoveries out there.

The thing is, that the relations that describe these things can be found quite intuitively. In this light, one does not need a "formal education" to see them.

His circle-limits, for example, were gleaned from a drawing in H.S.M. Coxeters' book, of the symmetry group of a {6,4}. My understanding comes from a similar drawing of a {7,3}.

Also, there are some of Escher's drawings where he assembled ideas into distinctly non-mathematical drawings, such as his final lithograph, Snakes [which is a poincine projection, coupled with one that bends inwards as well].

The fact is, that Escher understood certian constructs of absolute geometry, and was also an artist. Having read a number of his notes, I can understand how he came to devise his drawings.

I can draw reasonably accurate projections in hyperbolic geometry even without any understanding of hyperbolic trig, etc...

I did a little research...

[Quantum Singularity]

...and I found this:
"The secret of its making can be rendered somewhat less obscure by examining the grid-paper sketch the artist made in preparation for this lithograph. (picture here)Note how the scale of the grid grows continuously in a clockwise direction. And note especially what this trick entails: A hole in the middle. A mathematician would call this a singularity, a place where the fabric of the space no longer holds together. There is just no way to knit this bizarre space into a seamless whole, and Escher, rather than try to obscure it in some way, has put his trademark initials smack in the center of it."
The whole article can be found here. I didn't see the site, apparently /.ed. Just my $0.02.

Re:Re:Hmm

[pmz]

Are both the artist and scientist manifestations of two sides of a coin?

Of the people I've known, a brilliant scientist and a brilliant artist are most frequently found in the same person. It really isn't two sides of something but two different words for the same thing.

It is unfortunate that our culture has separated art and science, because both are manifestations of knowledge, critical thinking, and ingenuity. For example, Ludwig van Beethoven and Sigmund Freud each had profound insight into human psychology, but they employed different vocabularies and reached different audiences.