Slashdot Mirror
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."
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...
Mirror picture here
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...
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.
I've nothing to say here...
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/
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)
World of Escher
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?
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.
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!
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.
- A real programmer uses $ cat > a.out
Escher psychic factorization.
"God fights on the side with the best artillery." - Napoleon, Marshal of France - speaking truth to power
Lenstra gave a talk on the subject at the HP Research Labs Colloquium last July:
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http://www.hpl.hp.com/infotheory/lenstra071101.ht
Abstract:
Elliptic curves form one of the hottest topics in arithmetic algebraic geometry. Applications of elliptic curves range from a proof of Fermat's Last Theorem to the design of secure cryptosystems. In the lecture we present, as a novel application of elliptic curves, a mathematical analysis of Escher's lithograph `Print Gallery'.
When you have nothing left to burn you must set yourself on fire
Just got the time to save everything and mirror it here before the Slashdot effect doomed the whole thing...
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...
OS/2 - because choice is a terrible thing to waste.
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.
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
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
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.
Healthcare article at Kuro5hin
The difference between a madman and a genuius is that we force the madman to live in our world while the genius forces us to live in his.
--Karl Evander Kaufeld
The Mongrel Dogs Who Teach
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.
The Mongrel Dogs Who Teach
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.
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.
Lasers Controlled Games!
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.
In a flight to the Netherlands, Dr. Hendrik Lenstra, a mathematician, was leafing through an airline magazine when a picture of a lithograph by the Dutch artist M. C. Escher caught his eye. Titled "Print Gallery," it provides a glimpse through a row of arching windows into an art gallery, where a man is gazing at a picture on the wall. The picture depicts a row of Mediterranean-style buildings with turrets and balconies, fronting a quay on the island of Malta. As the viewer's eye follows the line of buildings to the right, it begins to bulge outward and twist downward, until it sweeps around to include the art gallery itself. In the center of the dizzying whorl of buildings, ships and sky, is a large, circular patch that Escher left blank. His signature is scrawled across it. As Dr. Lenstra studied the print he found his attention returning again and again to that central patch, puzzling over the reason Escher had not filled it in. "I wondered whether if you continue the lines inward, if there's a mathematical problem that cannot be solved," he said. "More generally, I also wondered what the structure is behind the picture: how would I, as a mathematician, make a picture like that?" Most people, having thought this far, might have turned the page, content to leave the puzzle unsolved. But to Dr. Lenstra, a professor at the University of California at Berkeley and the University of Leiden in the Netherlands, solving mathematical puzzles is as natural as breathing. He has been known, when walking to a friend's house, to factor the street address into prime numbers in order to better fix it in his mind. So Dr. Lenstra continued to mull over the mystery and, within a few days of his arrival, was able to answer the questions he had posed. Then, with students and colleagues in Leiden, he began a two-year side project, resulting in a precise mathematical version of the concept Escher seemed to be intuitively expressing in his picture. Maurits Cornelis Escher, who died in 1972, had only a high school education in mathematics and little interest in its formalities. Still, he was fascinated by visual mathematical concepts and often featured them in his art. One well-known print, for instance, shows a line of ants, crawling around a Moebius strip, a mathematical object with only one side. Another shows people marching around a circle of stairs that manage, through a trick of geometry, to always go up. The goal of his art, Escher once wrote in a letter, is not to create something beautiful, but to inspire wonder in his audience. Seeking insight into Escher's creative process, Dr. Lenstra turned to "The Magic Mirror of M. C. Escher," a book written (under the pen name of Bruno Ernst) by Hans de Rijk, a friend of Escher's, who visited the artist as he created "Print Gallery." Escher's goal, wrote Mr. de Rijk, was to create a cyclic bulge "having neither beginning nor end." To achieve this, Escher first created the desired distortion with a grid of crisscrossing lines, arranging them so that, moving clockwise around the center, they gradually spread farther apart. But the trick didn't quite work with straight lines, so he curved them. Then, starting with an undistorted rendition of the quayside scene, he used this curved grid to distort the scene one tiny square at a time. After examining the grid, Dr. Lenstra realized that carried to its logical extent, the process would have generated an image that continually repeats itself, a picture inside a picture and so on, like a set of nested Russian wooden dolls. Thus, the logical extension of the undistorted picture Escher started with would have shown a man in an art gallery looking at print on the wall of a quayside scene containing a smaller copy of the art gallery with the man looking at a print on the wall, and so on. The logical extension of "Print Gallery," too, would repeat itself, but in a more complicated way. As the viewer zooms in, the picture bulges outward and twists around onto itself before it repeats. Once Dr. Lenstra understood this basic structure, the task was clear: If he could find an exact mathematical formula for the repetitive pattern, he would have a recipe for making such a picture with the missing spot filled in. Measuring with a ruler and protractor, he was able to estimate the bulging and twisting. But to compute the distortion exactly, he resorted to elliptic curves, the hot topic of mathematical research that was behind the proof of Fermat's last theorem. Dr. Lenstra knew he could apply elliptic curve theory only after reading a crucial sentence in Mr. de Rijk's book. For esthetic reasons, Mr. de Rijk explains, Escher fashioned his grid in such a way that "the original small squares could better retain their square appearance." Otherwise, the distortion of the picture would become too extreme, smearing individual elements like windows and people to the point that they were no longer recognizable. "At first, I followed many false leads, but that sentence was the key," Dr. Lenstra said. "After I read that, I knew exactly what was happening." Escher was creating a distortion with a well-known mathematical property: if you look at small regions of the distorted picture, the angles between lines have been preserved. "Conformal maps," as such distortions are known, have been extensively studied by mathematicians. In practice, they are used in Mercator projection maps, which spread the rounded surface of the earth onto a piece of paper in such a way that although land masses are enlarged near the poles, compass directions are preserved. Conformal principles are also used to map the surface of the human brain with all the folds flattened out. Knowing that Escher's distortion followed this principle, Dr. Lenstra was able to use elliptic curves to convert his rough approximation of the distortion into an exact mathematical recipe. He then enlisted a Leiden colleague, Bart de Smit, to manage the project and several students to help him. First, the mathematicians had to unravel Escher's distortion to obtain the picture he started with. A student, Joost Batenburg, wrote a computer program that took Escher's picture and grid as input and reversed Escher's tedious procedure. Once the distortion was undone, the resulting picture was incomplete. Some of the blank patch in the center of "Print Gallery" translated into a blurred swath spiraling across the top of the picture. So, the researchers hired an artist to fill in the swath with buildings, pavement and water in the spirit of Escher. Starting with this completed picture, Dr. de Smit and Mr. Batenburg then used their computer program in a different way, to apply Dr. Lenstra's formula for generating the distortion. Finally, they achieved their goal: a completed, idealized version of Escher's "Print Gallery." In the center of the mathematician's version, the mysterious blank patch is filled with another, smaller copy of the distorted quayside scene, turned almost upside-down. Within that is a still smaller copy of the scene, and so on, with the remaining infinity of tiny copies disappearing into the center. Since Escher's distortion was not perfectly conformal, the mathematician's rendition differs slightly from his in other ways as well. Away from the center, for example, the lines of some of the buildings curve the opposite way. The researchers also used their program to create variations on Escher's idea: one in which the center bulges in the opposite direction, and even an animated version that corkscrews outward as the viewer seemingly falls into the center. After a recent talk Dr. Lenstra gave at Berkeley, the audience remained seated for several minutes, mesmerized by the spiraling scene. While Dr. Lenstra has solved the mystery of the blank patch and more, one question remains. Did Escher know what belonged in the center and choose not to represent it, or did he leave it blank because he didn't know what to put there? As a man of science, Dr. Lenstra said he found it impossible to put himself inside Escher's mind. "I find it most useful to identify Escher with nature," he said, "and myself with a physicist that tries to model nature." Mr. de Rijk, now in his 70's, said he believed Escher knew his picture could continue toward the center, but did not understand precisely what should go there. "He would be astonished to experience that his print was still much more interesting than was his intention," Mr. de Rijk said. He added that while he knew of another effort to fill in Escher's picture, it was not based on an understanding of the mathematics behind it. "He was always interested when somebody used his prints as a base for further study and applications," Mr. de Rijk said. "When they were too mathematical, he didn't understand them, but he was always proud when mathematicians did something with his work."