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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."
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've nothing to say here...
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)
Hey! It's the same pic!
Here's the real _mirror_ picture.
^_^
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.
In Capitalist America, bank robs you!
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!
Escher psychic factorization.
"God fights on the side with the best artillery." - Napoleon, Marshal of France - speaking truth to power
The white space is there 'cause the server's slashdotted, Sir. Escher's painting should go in it.
Trollem mirabilem hanc subnotationis exigiutas non caperet
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?)
"Pinky, you've left the lens cap of your mind on again." - P&TB
"I can see my house from here!" - ST:
intentionally left blank.
Sorry, back to bed with me.
Nerd rage is the funniest rage.
...and I found this: /.ed. Just my $0.02.
"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
We're Doomed
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.