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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."
Mirror picture 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/
World of Escher
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!
A little history on MC Escher here.
HTH HAND
I got a 1600 on the SATs.
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...
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 those days, High School generally would include three languages, loads of classic literature, and math up to integral calculus.
This whole thing has been discussed with a lot of insightful details in6 567/ ref%3Dnosim/robotbooks-20/104-8470745-1045520
http://www.amazon.com/exec/obidos/ASIN/046502
Born June 1898, Died March 1972. The work in question was created in 1956.
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."
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.
These are my friends, See how they glisten. See this one shine, how he smiles in the light.
I have a complete-works-of-Escher book, and another on centuries-old Celtic and Arabic pattern art. Quite interesting to compare, as they all use many similar techniques.
~REZ~ #43301. Who'd fake being me anyway?