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Science

Escher and Elliptic Curves 198

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."
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Escher and Elliptic Curves

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  • by MjDascombe ( 549226 ) on Tuesday July 30, 2002 @08:21AM (#3977256) Journal
    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...
    • by 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.
      • 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 [leidenuniv.nl]. It keeps going!
    • So, you're saying that these people shouldn't have thought about what goes in the space?
      • He's saying to just think about the space, but don't come up with an answer for what goes in the space, because then you won't be able to think about what goes into the space. Thinking about the space is almost an answer to thinking about the space too.
  • Mirror picture (Score:3, Informative)

    by bodin ( 2097 ) on Tuesday July 30, 2002 @08:30AM (#3977282) Homepage
    Mirror picture here [x42.com]
  • by ThogScully ( 589935 ) <neilsd@neilschelly.com> on Tuesday July 30, 2002 @08:31AM (#3977289) Homepage
    ...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.

    • I have trouble believing anyone will take tech people seriously these days without a degree

      AFAIK M.C. Escher was not a tech person ...

      but I think it's great to see that there's still an opportunity for a true genius to break that belief.

      1) Escher lived ~100 years ago. Things (for scientists as well as for artists) are MUCH harder now.

      2) Perhaps you really are that true genius. But even if you are, you'll probably need to study a LOT before you'll be able to make a deep impression.
      Studying is much easier when making a degree in a serious institution, than when studying alone. A degree is not a must, but it makes you study (professional) things you don't know about, and sometimes don't like very much, but will be beneficial for your future career.

      in short: studying is a must. formal education is not the only education form in existance, but it has many advantages.

      Besides, if you're the Autodeductive type, you can try studying at the open university.

    • by squaretorus ( 459130 ) on Tuesday July 30, 2002 @09:16AM (#3977425) Homepage Journal
      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!
      • "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."

        That's funny, because as a draftsman, I am stuck working for architects. I am going for my architecture degree because I want to DO THINGS FOR MYSELF. I think there are so many exceptions to what you just said, as to invalidate it. Degrees empower people to do things for themselves, though not everyone uses a degree for that purpose. In a broad amount of fields, it's not important if you can do something, but rather, that you can show credentials to prove it to some regulatory board.

        "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."

        I wanna be a doctor. I want to help people by prescribing medicine. It doesn't matter how much I learn or know, if I follow your suggestion, I'll be happy for the short while before I'm dragged off to jail, even if I'm more competent than your average doctor. For a huge amount of the professional world, this is the same.

        Sorry bud, degrees are handy when you want to do big, meaningful things, for others or for yourself.
        • Okay - you want to be a doctor, you have to work for a health authority, you need a degree.

          You want to be an architect, wow - guess what - you need a degree. Lawyers too? No shit sherlock!

          All these are professions which are exclusive and involve working wither for a boss or for the Man. These do not constitute doing something for yourself, creating something new, bringing a fresh view to the world. These involve TAKING A JOB. I dont think the original poster was talking about simply getting a job - he was talking about doing something special.

          If you think being a architect or doctor is something special, I'd have to disagree - there are plenty more of THEM out there - who gives a fuck if you turn into another?

          • Let me guess... With that type of emotional reaction, either you got kicked out of university for low marks, or you were never accepted in the first place.

            Any how, you make it seem like the only respectable job is being self-employed... Doctors don't have patients, they work for the health authority. Architects don't have clients, they answer to the regulatory board. Lawyers don't have clients, they have the bar assotiation to work for. I could go on.

            You make it sound like the only thing worthwhile in life is homesteading and making everything for yourself, while coding open source software.

            Everyone works for an employer, whether it be one, or a crowd of clients/customers. Degrees help you get established. What do YOU do?
            • If you must know I have an Honours degree in Chemistry. I loved my time at Uni, it set me up for life - but the time spent drinking beer and playing cricket was as important as the time spent in the labs.

              I've had a few jobs, and currently work for myself producing niche software for a small number of clients - its nothing ground breaking but it pays the bills.

              I don't think I have the ideal life - but it suits me. Being a doctor or lawyer or whatever is too restrictive for me - there is a strict code of conduct, a stricter pecking order, and a defined 'career path' to be followed.

              I have never been asked if I had a degree, and I've never volunteered the information to any potential employer or client. I never said medicine wasn't a respectable profession - I simply said its nothing special in most cases.

              The original poster didn't say 'wish I had a respectable job without a degree' he said 'wish I could do something important with my life without a degree'.

              The coolest job I know of is a guy who builds hedges - he started from scratch without a qualification to his name - but he is changing the landscape around here! By making traditional hedges a possibility at a reasonable cost he is changing the way the place works. He's WAY more important, and more interesting than 99.9% of doctors.

              I see no contradiction!!
      • 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

        I disagree. Degrees are earned by taking curriculum. Curriculum forces you to take subject that you may not be interested in but have to anyways because they are part of the standard. Because of this a degreed technical person can have concepts that a nondegreed person would never think or want to know. But these concepts are helpful in programming, even business DB work.

        It's a rare nondegreed person I talk to who understands order notation. To have to teach them that inorder to begin to address algorithmic efficiency is a task that doesn't have to be done with a degreed person. Rarer than that is one who knows what a state machine is and how it applies to parts of their work. Rarer still is one who understands stochastic algorithms and what variance is.

        But even if you decide to bone up on these things there will still be things you don't learn because you haven't been forced to do so. Getting a degree from a good university really does make you a better person. When working on a tough problem at work and going through pages of results related to it in Google, the curriculum you know will help make the pages make more sence. I doubt that before my degree I could answer many of the questions on this page [berkeley.edu]. I answered all but 3, and the 3 I didn't answer were because they would take more programming time than I cared to put in and 2 were covered in courses I took (the other is the spiral one which is relatively trivial). I'm fairly sure if I didn't go to university, I'd have been lost at O(n).

        And no matter what you are "doing for yourself", you are working for someone else if it's income related. Be it the client as an independent contractor, the bank for a private company, or shareholders for a public company, everyone is accountable to someone. Having a CS degree will help you in programming, design, and architecture jobs. A CTO really should have an understanding of CS concepts in order to be an effective decision maker.

        And if what you're doing for yourself isn't income related, then an educated background gives you a richer understanding of the things you do.
        • Jesus! How far out of context can I be taken!!!

          All I said, was that a degree doesn't help you do special things. ALL your examples are about fulfilling existing requirements through acquiring existing knowledge in a structured manner.

          Now, I don't dispute the value of that but the question is WHATS SO SPECIAL ABOUT THAT???

          I have a degree. I did pretty well. I learned a shit-load of stuff. But it in no way makes me more likely to CREATE something NEW.

          Think outside the box. The whole point of life is not to create neat code, or to effectively manage coders, or do draw shapes. Sometimes people create something new, like Escher, and in doing so they dont just fall back on stuff learned at 'Creating 301'.

          I'm not putting down degrees - even tho they let any old chimp on a degree course nowadays!
    • I was hired back in the tech boom, back when people would hire someone who had the skill to turn a computer on, just because they were so desperate. I never did finish my degree.

      I can say that at my company, I'm highly valued for my 10 years of experience, and my skills. But when I've attempted to find jobs elsewhere, I have trouble getting people interested in paying me more than half of what I'm making now. It's just a huge credibility jump for them, despite the fact that I've been in the same position in the same company, obviously not sucking at my job, getting fairly well paid - but without that piece of paper, you're just a pariah.
  • by Pxtl ( 151020 )
    I couldn't find in the article - what is the actual story of the hole in the picture? Was it deliberate? Was it a puzzle for fellow artists (as approached here)? Was it an error? Was it just to make people think? Was it damaged after completion? What?

    Btw, my fave escher has always been the hand drawing a hand. Relativity would make a great Q3 level tho.
    • Re:Hmm (Score:3, Interesting)

      by ThogScully ( 589935 )
      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.

      • Re:Re:Hmm (Score:3, Insightful)

        by Anonymous Coward
        This makes me wonder/question/revisit the old debate of nature/nurture? Is it possible that Escher's brain was wired in a particular way that allowed him to create works of art based on mathetic principles without knowing the underlying structure? I.e. was escher following principles of cellular automata or something like it?

        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?

        • Genius is the ability to make the heretofore unknown and render it blindingly obvious. Zmai
          • errr....

            How about....

            Genius is the ability to TAKE the heretofore unknown and render it blindingly obvious.

            Darned preview button. Theres a joke in this comment somewhere, but I can't find it.

            Zmai
          • 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
        • Re:Re:Hmm (Score:3, Insightful)

          by Pig Hogger ( 10379 )


          This makes me wonder/question/revisit the old debate of nature/nurture? Is it possible that Escher's brain was wired in a particular way that allowed him to create works of art based on mathetic principles without knowing the underlying structure? I.e. was escher following principles of cellular automata or something like it?

          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.

        • Re:Re:Hmm (Score:4, Interesting)

          by pmz ( 462998 ) on Tuesday July 30, 2002 @10:07AM (#3977691) Homepage
          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.
          • Richard Feynman is another fabulous example of this. Here's a link to Amazon [amazon.com] for a book about him and his art. He became interested in drawing and got very good at it. If you can find this book, it's worth a look, if just for all the pictures and the entertaining introduction by Mr Feynman himself.
        • Einstein had nominal education and apparently flunked math his first time through. Then he went and worked in a patent office... or so I've heard. I'm no expert.

          M.C. Escher made money designing stamps and advertising pamphlets for a while. Seriously.

          So maybe Genius is described as "intelligence" that is taught to one self. A kind of unexpected smarts that surpasses the expected level of smarts? But there are many other people who are geniuses that have had formal learning.

          Maybe genius is defined as "intelligence at a level that is commonly accepted to be extraordinary". Lets see what the dictionary has for us: ::SNIP::
          Main Entry: genius
          Pronunciation: 'jEn-y&s, 'jE-nE-&s
          Function: noun
          Inflected Form(s): plural geniuses or genii /-nE-"I/
          Etymology: Latin, tutelary spirit, natural inclinations, from gignere to beget
          Date: 1513
          1 a plural genii : an attendant spirit of a person or place b plural usually genii : a person who influences another for good or bad
          2 : a strong leaning or inclination : PENCHANT
          3 a : a peculiar, distinctive, or identifying character or spirit b : the associations and traditions of a place c : a personification or embodiment especially of a quality or condition
          4 plural usually genii : SPIRIT, JINNI
          5 plural usually geniuses a : a single strongly marked capacity or aptitude b : extraordinary intellectual power especially as manifested in creative activity c : a person endowed with transcendent mental superiority; especially : a person with a very high intelligence quotient ::SNIP::
        • It's quite possible a percentage of the population can intuitivly apply some mathamtical concepts without indepth analysis.

          On the other hand, we try to beat that out of them as students in elementry school.
        • Is it possible that Escher's brain was wired in a particular way that allowed him to create works of art based on mathetic principles without knowing the underlying structure?

          Yeah, same as everyone else's. Ever draw a circle? Know all the properties of a circle? Ever draw a cloud? Know all the properties of the fractal structure of clouds?

          In the end, does it really matter that escher left a hole in the picture, or that people wonder why the hole is there?
          In the end is the heat death of the universe. Sooner than that, in the end is our own death. In the end, none of it matters. But it's kinda cool no matter what.
      • Re:Hmm (Score:2, Funny)

        by eam ( 192101 )
        > No one reallly knows except Escher, who's
        > unlikely this late in the game the disclose it
        > (ie. he's passed on).

        Hey, it never hurts to ask. I lit a candle. Now, everybody hold hands, close your eye's and concentrate.
      • I think that Escher knew what perfectly well what went in the center. But what goes in the middle is an infinite regress in which objects get indistinguishably small very fast. Ultimately, you have to give up and stop drawing at some point. Escher just decided to stop drawing before everything started repeating, leaving that mysterious blank spot--and a nice prominent place to put his signature.

        The only reason the images on the site work is that they blow up the center so that you can see what's really going on. Without the zoom, you can't really make much sense of the center of the picture, anyway.

    • 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.
  • by Anonymous Coward
    Easy. Because it was too frigging difficult to draw what would go inside it. Look at the computer generated blank spot replacements. Escher would have had a very hard time to calculate and draw that.
    • "because it was too frigging difficult" This was my first thought too. I suspect he tried a few times, decided it was to much, then thought about the advantages of leaving it out. In the end he probably chose to leave it out for reasons other than the difficulty of drawing it, although that would br high up on his list. BLM
  • by tolleyl ( 580010 ) on Tuesday July 30, 2002 @08:42AM (#3977323)
    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 (Score:5, Insightful)

    by Anonymous Coward on Tuesday July 30, 2002 @08:44AM (#3977330)
    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)
    • Agreed. The way the work is drawn, the view is subtly transformed from inside the hanging print to the outside world and back as you move your eyes around the spot. If the spot were not there, it would force inside/outside to meet at some specific point. As it is now, it's a smooth transition.
    • Not Shakespeare, but how about "Give me no spot, or give me death!"?
    • the spot (Score:3, Interesting)

      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.

    • Escher knew what he was doing -- and so did the people who did this research. I don't think they were trying to improve on the drawing; they just entered into the intellectually playful spirit of Escher's art, and started asking questions and making cool pictures.

      As the NYT article points out, Escher himself was always thrilled when people found his artwork to be a springboard into new research and new ideas. He was always very derisive of highbrow artistic purism, and I imagine he would have been delighted with these new images.

      The original print is great, and so is this research.
  • OK, I'm impressed. (Score:1, Interesting)

    by 6Yankee ( 597075 )

    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.
  • by Anonymous Coward on Tuesday July 30, 2002 @08:45AM (#3977335)
    World of Escher [worldofescher.com]
  • but it's art, not math... I think it's a good practice, but I wouldn't try to take anything away from Escher but publishing it.

  • I can't believe it! (Score:4, Interesting)

    by Spackler ( 223562 ) on Tuesday July 30, 2002 @08:52AM (#3977353) Journal
    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.
    • > 2. Not everything needs a higher meaning True, but simply glancing at other work of Escher's reveals at once the the artist's fascination for self-replicating patterns. This discovery only verifies that even this picture can be extended in Escher's spirit. That can hardly be mere coincidence. Now, by not completing the picture ad infinitum makes it of course into a marvellous piece of art.
    • For very interesting interpretations, read Godel Escher Bach [amazon.com] by Douglas R. Hofstadter.

      While I agree that sometimes a cake is just a cake, this book may change your mind on the subject of Escher.
    • by Masem ( 1171 ) on Tuesday July 30, 2002 @09:40AM (#3977545)
      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?)

      • by jafac ( 1449 ) on Tuesday July 30, 2002 @12:54PM (#3978981) Homepage
        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.
  • I have been watching the animation full screen on repeat for 15 minutes.

    I now have a splitting headache and am dizzy and nauseous.
  • For the curious: (Score:5, Informative)

    by colmore ( 56499 ) on Tuesday July 30, 2002 @08:56AM (#3977369) Journal
    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 [vwh.net] pages [niu.edu] out [std.com] there [geocities.com].

    • Oh, well. Analysis is not the most popular area for elliptic curves right now.
      Try algebraic geometry, algebraic topology or crypto.
      In cryptography, one usually work over finite fields so everything becomes descrete and easy to implement.
      Generally, you can make sense of elliptic curves over any field.


      • I know, I was just about to write up a whole thing about how they can work on any field, and how elliptic curves themselves make pretty interesting groups, how they are topologically equivalent to toruses, etc. but I stopped myself, the article only mentioned applications of - curves that exist in 2 real or 2 complex dimentions.
  • by wichtolosaurus ( 558778 ) on Tuesday July 30, 2002 @08:56AM (#3977373) Homepage
    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!
  • Background (Score:2, Informative)

    A little history on MC Escher here [st-and.ac.uk].
    HTH HAND
  • Focal point (Score:3, Insightful)

    by reelbk ( 213809 ) on Tuesday July 30, 2002 @08:58AM (#3977376)
    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.
  • by MuMart ( 537836 )
    Now where can I find a program to Escherize(tm) any Droste-effect type picture :)
  • by Lev13than ( 581686 ) on Tuesday July 30, 2002 @09:16AM (#3977428) Homepage
    Lenstra gave a talk on the subject at the HP Research Labs Colloquium last July:

    http://www.hpl.hp.com/infotheory/lenstra071101.htm [hp.com]

    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'.
  • by Pig Hogger ( 10379 ) <pig...hogger@@@gmail...com> on Tuesday July 30, 2002 @09:17AM (#3977432) Journal
    Just got the time to save everything and mirror it here [emdx.org] before the Slashdot effect doomed the whole thing...
  • check out the animations [leidenuniv.nl]

    having downloaded a few loops (they'll make great screensavers), i'm now torn between two equal desires: part of me wants to research the proof of fermat's last theoren, while another part of me wants to put on some hendrix, drop some acid and stare at an endless escher loop for a few days

  • by os2fan ( 254461 ) on Tuesday July 30, 2002 @09:21AM (#3977451) Homepage
    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...
    • 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.

  • White space (Score:5, Funny)

    by stere0 ( 526823 ) <`ul.oerets' `ta' `liamtodhsals'> on Tuesday July 30, 2002 @09:39AM (#3977539) Homepage
    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.

  • by maxume ( 22995 ) on Tuesday July 30, 2002 @09:46AM (#3977577)
    intentionally left blank.

    Sorry, back to bed with me.
  • ...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 [mathacademy.com])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. [mathacademy.com] I didn't see the site, apparently /.ed. Just my $0.02.
  • by Anonymous Coward
    ...doesn't mean you should.
  • From here [nga.gov] it reads:
    Born in Leeuwarden, Holland, the son of a civil engineer, Escher spent most of his childhood in Arnhem. Aspiring to be an architect, Escher enrolled in the School for Architecture and Decorative Arts in Haarlem.
    I know that today, architecture schools require calculus and several basic structural engineering courses. I wonder what was covered back then.

    Add M.C. Escher to the list of people who got out of architecture and into something better, like Rick White of Pink Floyd.

    • As a recent graduate from one of the most prominent architecture schools in the US, I can say that your assumptions about architecture programs are as flawed as mine were before entering the program. Calculus is not required; in fact all that is required in my program is the basic university math requirement (pre-calc) and thats for graduation, not admission. That and the structural engineer courses are extremely basic, everything simplified into basic vector problems that can be solved graphically.

      That may scare you into running away from any built structure, but, keep in mind typical wood frame houseing is a well-established industry and the "rules of thumb" eliminate most need for structural calculations. Anything more complex than that (concrete or steel frame buildings) and the architect normally hires a structural engineer (Civil Engineering grad) as a consultant.
      • The University of Maryland School of Architecture requires Calculus I and Physics I & II to graduate. But no CAD classes required...("it's just a tool").

        BTW, are you sure you really want to be an architect? Do you really want to earn less than a school teacher and spend 50-60 hours per week at your job without getting paid for overtime? (Or if you do get paid hourly, it is de-rated so you still make very little)

        Do you want to spend endless days of CAD doing drafting of the princicpal's "designs" and CD's (construction directives, i.e. changes)?

        Do you want to go back to grad school, get your Master's, and then not make any more when you go back to work?

        Or do you have the needed capital and sales skills to start your own practice? (Despite the fact that most people and companies don't give a dang about architecture, i.e. Ryland Homes and office parks).

        Maybe you REALLY love construction. That might be a saving grace.

        Don't expect architecture to be about creativity. That day is long gone. Even the "rock stars" like Gehry are basically reduced to doing near copies of their existing work.

        Just a warning!
  • Art or math (Score:4, Insightful)

    by gilroy ( 155262 ) on Tuesday July 30, 2002 @10:25AM (#3977792) Homepage Journal
    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.

    • "There is beauty in the math, too, and grace, and yes, even art."

      Valid point. To me, just the bare grids have a beauty all their own. The remind me of the structure of a glass roof. And to think they've been there underneath, all this time, and I've never known.

      Of course, now I'm going to go looking for them, and I will probably have some strange vision/psychiatric disorder by the end of the week...

    • To those who think there is no art in this mathematical analysis:

      "Euclid alone has looked on beauty bare."
  • a few things (Score:2, Interesting)

    by kimota ( 136493 )
    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)
  • by mwhansen ( 597114 ) <mwhansenAThmcDOTedu> on Tuesday July 30, 2002 @11:12AM (#3978099)

    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.

    • Perhaps the mathematical relationship between the pieces of the picture were left intact, but it destroys the self-contained nature of the piece. The idea is that the boy is looking at the same picture he's standing within. Lenstra has created nothing more than a "Droste" picture with elliptical distortions. (If you look at the zoomed versions of the filled-in drawing, there's another copy of the boy in it.) If that's all Escher had wanted to do, he would have selected a different grid as the foundation of his drawing; filled in like this the grid is nonsensical.
  • What great timing! The Bridges [sckans.edu] conference was just held this weekend. It's all about work like this--stuff that bridges between math, art, music, and science. Neat stuff!
  • by kirkb ( 158552 )
    Wow, I can't beleive that no slashdotter has mentioned Douglas Hofstadter's excellent book "Godel, Escher, Bach: An Eternal Golden Braid"

    The book is over 20 years old and still a must-read for anyone who is interested in CS and/or art and/or music. It's fun, funny, and will make your brain hurt.

    Grab it at your favorite book store|website.
  • 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."
  • These are lithographs, stone etchings that were
    used to print onto ink soluble media, right?

    I hope the masters are safe somewhere, and that the technique for printing with them isn't a lost art already.

    A gif of the image or an iron-on t-shirt simply doesn't have a certain quality that seeing an original print of this type of work does have.

    I remember two life-changing moments in the development of my art appreciation:

    1. Seeing a Van Gogh painting in a museuem gallery.

    I had seen this picture many, many times, in books, on posters, etc. It is a very simple painting of sunflowers. But when I saw the real thing, it absolutely took my breath away. There is a third dimension to the paint that is utterly lost in a photograph of it, and there are qualities to the color of the original that would be impossible to describe, and a photograph only comes so close to capturing this, and only for one angle, one set of light conditions, etc.

    2. Salvador Dali's Psychedelic Toreador.
    I have had a print of this painting for years, and see something different in the detail every time I look at it. However, I never realized how very large Dali's original paintings are! There is extraordinary detail that might be well-reproduced on a laser print, but even if the details are there, something is lost.

    I'm not an artist or an art student, so I surely lack the language to say what I'm trying to say here. I suppose it's that I would like to see this etching in person, and that I'm really interested in the process used to make this type of work. Oh, and that in 10,000 years, the alien archaeologists will find the stone originals intact.

  • That's what you get for assuming. It's not a Droste effect picture at all.

    It's the world projected flat as seen from the inside of a Klein bottle. The empty space in the middle is the mouth of the bottle.

    That's also why the signature, litho number, and other information is in the center: it's the only unused portion of the surface, which encompasses the entirety of the world.

    The clue is the embedding of the gallery in the outside world, but the outside world in the image in the gallery.

    Escher is well known for projections of images onto/into objects (e.g. his self portrait on the reflective sphere held at arm's length).

    It's amusing that some well-meaning person would come along and try to put a cap on Escher's Klein bottle.

    -- Terry

Programming is an unnatural act.

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