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.

Re:Hmm

[ThogScully]

No one reallly knows except Escher, who's unlikely this late in the game the disclose it (ie. he's passed on).

Some theories are that he wanted people to look at it and wonder. Some say he just didn't quite know how to proceed. This guy seems to think he's done what Escher meant to do, but perhaps didn't quite have the mathematical understanding to complete. Escher was always known for not being very book-smart and sort of amazed at what mathemeticians found in his works. He knew he was making them with some structural intent, but never really knew the theories behind what made them seem to click.

OK, I'm impressed.

[6Yankee]

Wow. I've loved Escher's work for as long as I can remember. But I never knew it was this complex. Guess I always imagined he was smoking something interesting.

My favourite Escher work is his "Three Spheres". I dabble with raytracing and regularly give my P4 a headache. Anyone who can do this sort of thing without a computer is a genius in my book.

Going to go back and look at the diagrams properly now, see if I can learn something.

Wow.

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...

Re:The space is the whole point.

[Anonymous Coward]

But it did make them think! Just like most people looking at the picture they probably wondered what would go in the white space if one would fill it appropriately. And in stead of photocopying the picture and start drawing by hand, they used more contemporary means and a bunch of maths.

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.

Just because you can...

[Anonymous Coward]

...doesn't mean you should.

a few things

[kimota]

1. Okay, haven't read the article yet because the site is effectively Slashdotted, but... During the Renaissance, I think it was (or maybe the 1700s), there was a genre of painting/drawing that involved using a highly reflective cylinder in the center of the work while creating it. The result was that the picture was distorted until you placed the cylinder in the center--then you could see in the cylinder itself the undistorted image. I've seen photos, but can't recall the name or the source. Anyhoo, that's what I thought of as soon as I saw this work. I'd be curious to know what shape cylinder you'd need, though, to make these images look normal!

2. My favorite work of art is Durer's (I don't know how to get an umlaut on that u) Melancholia I. It's got an angel of indiscriminate gender (when did we shift from thinking of angels as exclusively male to using the name "Angel" for women?), platonic solids, a magic square, a comet, a rainbow, drawing/navigational tools, and carpentry tools. But what is it about? I find it endlessly fascinating. Check it out at
http://www-groups.dcs.st-and.ac.uk/~history/Misc el laneous/Durer/Melancholia.html

3. Another example of mathematical interesting... um, -ness, in art is Celtic design. You can check out a discussion of it in "Turbulent Mirror" (http://www.amazon.com/exec/obidos/ASIN/0060916966 /qid=1028039334/sr=1-1/ref=sr_1_1/103-5270790-4159 806)

Escher put himself in the center

[nucal]

Isn't the artist where art and reality meet? Maybe that's what Escher was getting at ... after all - it's not just a blank spot, he put his signature there. If so, then filling in the "spot" may actually change the point of the drawing.

Re:The space is the whole point.

[Steve Franklin]

You don't suppose Escher just didn't KNOW what went in the middle? The guy wasn't a mathematician, after all. He was an artist. From the grid he used it's apparent that what he came up with doesn't agree absolutely with the extracted mathematical representation, so it's pretty clear he was just doing art and not making a mathematical statement. Martin Gardner and others make this mistake about Escher. His art may represent certain mathematical principles, but it doesn't necessarily derive therefrom. The center also would have been very difficult to paint, since it gets progressively more detailed, almost fractalized, at the center: Take a close look at the very center of this image. It keeps going!

the spot

[John Harrison]

I agree that the spot adds to the intrigue of the work. I doubt that Escher "couldn't figure out what to put there" as others have suggested. He left it there on purpose, since he had already implied what the contents of it are.

Looking at the print you are drawn to the spot, just as the person depicted on the right side of the work seems to be. What does he see? If you think about it you realize that he sees the same thing you do, the back of his head. He is observing the same work you are. By including the spot Escher makes you part of the picture. If the person on the right is "Observer #1" and the person he is looking at is "Observer #2" you are "Observer #0". If the spot were filled in it wouldn't have the same effect. Go to the site and spend some time looking at the original work and the filled in version. I find that the original give a different sense of wonder and point of view than the new one.