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.

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: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!

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/

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:Mirror picture

[Soul-Burn666]

Hey! It's the same pic!

Here's the real _mirror_ picture.

For those who don't know Escher

[Anonymous Coward]

World of Escher

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?)

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.

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.

Those Escher links drive me krazy!

[wichtolosaurus]

I tried to follow the link, but it actually sent my browser to the page I visited before.
That's impossible. Wait.... if water can flow upwards..... damn Escher!

Here's another using psychic factorization

[N8F8]

Escher psychic factorization.

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...

White space

[stere0]

Why is the spot blank there, he wondered, and what should go in it?

The white space is there 'cause the server's slashdotted, Sir. Escher's painting should go in it.

It Says: This space

[maxume]

intentionally left blank.

Sorry, back to bed with me.

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.

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.

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.