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Science

All Shapes in One Equation? 74

Posted by chrisd from the coolness dept.
asadodetira writes ""One simple equation can generate a vast diversity of natural shapes, a Belgian biologist has discovered. Nature has the story. "The Superformula" sounds impressive, apparently its only for shapes, i thought you could solve lots of PDE's or tensor integrals or something with this, but not, it's only for shapes."
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All Shapes in One Equation? More

All Shapes in One Equation?

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  • "The Superformula" sounds impressive, apparently its only for shapes"

    Yeah, what useless crap. It only applies directly to everything we know in the universe. Such crap! why doesn't it cook me breakfeast?

    • Yeah, what useless crap. It only applies directly to everything we know in the universe.

      No really, it isn't all that useful. If it were somehow applicalbe to partial differential equations, for example, it might offer a simple way of handling many engineering problems.

      Granted, being able to describe shapes with a function can be terribly useful, as it might then become possible to use linear algebra or calculus to directly solve various problems relating to that shape's geometry rather than approximatin

  • sounds familiar (Score:4, Interesting)

    by ddd2k ( 585046 ) on Monday April 07, 2003 @11:18PM (#5683860) Homepage
    Changing one term in the formula varies the proportions of the shape - moving from a round circle to a long and skinny ellipse.
    This reminds me of the eccentricity ratio, C, of a conic function. It relates the parabola, hyperbola, and elipse. (eg, the parabola is the perfect shape as it has a eccentricity of 1 and the hyperbola >1 while the elipse is 1) However, im curious to what he did to transform a circle into various other shapes, which he did not mention in the article. big secret? ;-)
    • Shit, it reminds me of Spirograph.

    • Re:sounds familiar (Score:3, Insightful)

      by eggstasy ( 458692 )
      He added a varying radius instead of a constant one and felt very manly about it. Woop-dee-doo.
      • Fascism:Extreme right-wing dictatorial government,belligerently nationalist,that merges state and business leadership

        1 - You know this is an exciting article when people are replying to sigs

        2 - Socialism: Moderate left-wing totalitarian government, usually post-nationalist, where people smarter than you spend your money to save you from yourself. [Usually has much better slogans than the above, although the phenomenon has not been studied]
        • 1 - LOL
          2 - Thanks. You know you finally found a decent sig when someone replies to it :)
          • Thanks for reminding me to update my meta-sig. Sigs about moderation are soooo passe, now. Bitching about or in favor of the war is the "in" thing, now.

        • Socialism: Moderate left-wing totalitarian government, usually post-nationalist, where people smarter than you spend your money to save you from yourself.

          No.

          Socialism: an economic system in which capital is controlled by workers (the people who use the capital to get stuff done). Comes in free-market and command-economy flavors. Contrast to capitalism, where capital is controlled by state-designated owners.

          It's sad that the legacy of Palmer, McCarthy, and Hoover thrives to the extent that most Ameri

          • most Americans think socialist -> totalitarian.

            In fact most Americans seem to think that socialism is the same as communism and that they're both Stalinism. The AC that replied to you earlier is a good example of this sort of retarded world-view.

            TWW

  • Shape eh... (Score:3, Funny)

    by Anonymous Coward on Monday April 07, 2003 @11:18PM (#5683863)

    As an average /.'r I'm definitely out of shape.

    Hopefully this will allow me to program a new shape.

  • One equation... (Score:4, Funny)

    by icemax ( 565022 ) <<moc.liamtoh> <ta> <enots_d_wehttam>> on Monday April 07, 2003 @11:20PM (#5683873) Homepage
    One equation to rule them all, one equation to find them one equation to bring them all and in the darkness bind them...

    now THAT's a nice ring-shape
    • One equation to rule them all, one equation to find them one equation to bring them all and in the darkness bind them...

      You laugh, but (from the article):

      Gielis has patented his discovery, and is developing computer software based on it. Using one formula to produce shapes will make graphics programs much more efficient, he says. It might also be useful in pattern recognition.

      On another "note":

      References

      1. Gielis, J. A generic geometric transformation that unifies a wide range of natural and abst

  • Shapes are cool (Score:2, Interesting)

    by Sevn ( 12012 )

    So like if it's only for shapes then I'm cool with
    it too cause yanno like shapes are cool and stuff.

    But seriously,
    Bummer. Graphics realism and speed could probably be
    greatly enhanced with a technology burned into the
    firmware that can make any shape with one equation.
    That could be a neat way to do a lot of things. In
    the very least it could be a new way to precache
    memory if you think about it. Or something.

    • Re:Shapes are cool (Score:3, Insightful)

      by eggstasy ( 458692 )
      Ever saw the "flower" visualization thingy for winamp?
      Its the same thing!
      Look, you can plot the graph of a function on a typical 2D cartesian X,Y reference frame or you can plot it in a circular reference frame where Y is the distance from the center at which you plot and X is the "degree" where you plot it.
      So if you plot a constant C in the 1st ref. frame you get a straight horizontal line at Y=C and on the 2nd one you get a straight circle where the radius = C.
      If you plot a function, like a sine, it will
      • Does anyone know where I can read the stupid paper without subscribing to that site?

        Atleaast I can view the paper [amjbot.org] without any subscription.

        Also the guy is a botanist, not a mathematician. And the journal he published in is a botanical paper. So I don't think it's surprising that it passed the checks.
    • Graphics realism and speed could probably be greatly enhanced with a technology burned into the firmware that can make any shape with one equation.

      It's called a "Pentium". It's a highly sophisticated, evolved equivalent of a recursive function. There is rumored to be a number of utilities for it that make it easier to input, output, and compose those functions.

  • For gods sake, a stupid little Qbasic program i wrote years ago could ALSO generate a lot of different shapes like those using modified circle equations. I called it a "2d renderer" and didnt think it was anything significant. I still dont think it is of any significance and wonder why the hell there are so many crappy trivialities being passed off as important research breakthroughs.
    • For gods sake, a stupid little Qbasic program i wrote years ago could ALSO generate a lot of different shapes like those using modified circle equations. I called it a "2d renderer" and didnt think it was anything significant

      It wasn't.

      I [snip] wonder why the hell there are so many crappy trivialities being passed off as important research breakthroughs

      From the article: "It's a new way of describing nature," and The Superformula might provide a single, simple framework for analysing and comparing the shap

      • Instead of fighting them, let's join them. I'll patent x^2 and you patent x^3. Just think about how rich we'll be.
      • Wow, this is one of the most negative threads I've ever read on SlashDot, and that is saying a lot. Yes, it does seem trivial. And Yes, he shouldn't have patented it. But let let me put my analysis into terms the the typical SlashDot reader should be able to appreciate:

        Using one formula to produce shapes will make graphics programs much more efficient,

        That means faster and more realistic video games and Sci-Fi/Fantasy movies folks, what could be bad about that?

        • Using one formula to produce shapes will make graphics programs much more efficient,

          Ah, but is this necessarily true? Using a single representation may be more elegant or more convenient, in the sense that it allows the creation of programs that allow you to manipulate shapes in "natural" or "intuitive" ways. For the highest performance, it may however be preferable to use an optimized equation for the specific shape that you are drawing.
        • That means faster and more realistic video games and Sci-Fi/Fantasy movies folks, what could be bad about that?
          Yeah but we don't actually know anything about the size or complexity of the formula. That's why most graphics use simplexes to model shapes (ie simplest shape in n -dimensions) like a triangle in 2-D or tetrahedron in 3-D depending on what you need. I can currently model a cube with 12 triangles or 5 tetrahedrons and only 8 vertices, how many would I need with this formula? I'm sure this formu
        • That means faster and more realistic video games and Sci-Fi/Fantasy movies folks, what could be bad about that?

          As someone who as done 3D graphics work, I found the article rather depressing. Getting different shapes based on variations of the circle equation is very old work, and very commonly known (as a child I first read about the idea in a Piers Anthony novel).

          And now this guy is patenting whatever he imagines he's discovered. Graphics Gems has some much more clever ideas in every book, and fort

    • I was doing stuff like this on my TI-85 in 9th grade (5+ years ago). How the heck does this guy get a patent on something so obvious that an only moderately advanced 9th grader could figure it out? I'm sure lots of young students have tried varing the terms of various equations to see what happens. What's the patent number, and what exactly does he claim?

  • Beautiful Shapes (Score:3, Funny)

    by mc_barron ( 546164 ) on Monday April 07, 2003 @11:54PM (#5684003) Homepage
    "When I found the formula, all these beautiful shapes came rolling out of my computer,"

    Ahh...reminds me of when I first got past the Trivia Quiz Age Check questions in Leisure Suit Larry.

    What's amazing is that this question was in the 1987 release (no joke):

    O.J. Simpson is
    a. an R & B singer.
    b. under indictment.
    c. embarrassed by his first name (Olivia).
    d. no one to fool with.
    Correct answer: d

    Duh.

  • What's New (Score:4, Interesting)

    by frantzdb ( 22281 ) on Monday April 07, 2003 @11:57PM (#5684019) Homepage
    The full text appears not to be available online. All of the examples look like simple polar functions. I find it hard to believe that someone discovered a fundamentally new equation for r(\theta).

    --Ben

  • Patented it? WTF? (Score:4, Interesting)

    by MacJedi ( 173 ) on Tuesday April 08, 2003 @12:02AM (#5684039) Homepage
    WTF? He found an equation that can describle all kinds of fundamental shapes and he PATENTED IT?!

    Call me old fashioned, but I don't think you should have the right to patent maths!

    /joeyo

  • Comment removed based on user account deletion
    • Roger Penrose patented his tesselations a few years ago. Yes, it's not ethical. Mathematics should not be patentable, otherwise we'll decend into a new Dark Age.
      • He invented those shapes, nerds were tiling their bathrooms with them. I don't see why he shouldn't be able to gain royalties from commercial exploitation of his idea. Patenting vs copyrighting is immaterial--isn't he simply protecting intellectual property?

  • Java applet visualisation (Score:5, Informative)

    by FrenZon ( 65408 ) on Tuesday April 08, 2003 @12:09AM (#5684076) Homepage
    I have a java applet online that allows you to fiddle with the values in the equation and generate the 'super'shapes in realtime:

    bodytag.org/supershapes1/ [bodytag.org]

  • Real world? fractals (Score:3, Informative)

    by ptaff ( 165113 ) on Tuesday April 08, 2003 @12:10AM (#5684080) Homepage
    I doubt we can do anything with that formula to express Nature's art. Sure, we can build simpler graphical engines, but that's it.

    I don't know how far this "transformed circle formula" is from a circle formula, but as long as it's an integer-dimension thing, we get nothing from it. It doesn't scale.

    The concept of locality is too important - the behaviour of a cell is really parametered by its neighbors; the same ADN is in your brain, your liver and your nails.

    Fractals are still a relatively simple method of describing 3D structures - sure it's really hard to start with a real object and map it to a fractal (but Nature works the other way around!). A very small disturbance can create as many shapes as you want; the number of different vegetal organisms showing very similar DNAs seem to support this.

    • The first thing I thought when I saw this article was, "sounds like this guy has discovered fractals". What he's describing would appear (from the limited information provided) to be a fractal equation. It will be interesting to see how easily it is incorporated into Fractint [triumf.ca]. Fractint currently has about 70 or so different types of fractals that you can tweak, play with, and zoom into to your heart's content.

      There is a lot of cool art on the fractint homepage as well as come descriptive information abou

      • My thoughts exactly...

        Hmm, he just now discovered fractals???

        I can still remember back in the days when I would spend countless hours watching a 386 render a fractal in Fractint.

        I would always show off all the cool (prerendered) images to my friends. The first thing that most people seemed to say is "kinda looks like a plant growing".....

        SuperGlueBooger

  • stop the presses (Score:5, Funny)

    by g4dget ( 579145 ) on Tuesday April 08, 2003 @02:02AM (#5684490)
    Belgian biologist discoveres mathematics

    After avoiding the subject for decades, a Belgian biologist discovers mathematics. One of the first areas he plays around with is "polar coordinates". "I never knew math could be this much fun", the biologist is quoted as saying. In his enthusiasm, the befuddled biologist decided to patent several formulas, following a recently fashionable trend of patenting the obvious.

  • Could this possibly be applied to that story from the other day about the more something compresses the better the odds it was produced in nature? And would it still be the image of the object, or could the formula expressing it somehow be compressed? I would assume it would already be expressed in it's simplest form. Hell, what do I know? I count boxes.
  • Homepage describing the discovery (which I am still a little skeptical about): Geniaal.be (translated: "brilliant.be") [geniaal.be]

    P.S.: I find it somewhat amusing that the fortune quote at the bottom of the SlashDot page I'm using to engter this happens to be

    "Confound these ancestors.... They've stolen our best ideas!" - Ben Jonson

    :-)

  • More technical information (Score:3, Informative)

    by Elbelow ( 176227 ) on Tuesday April 08, 2003 @05:40AM (#5685023) Homepage
    Besides the biologist's own homepage [geniaal.be], there is also the Genicap homepage [point1design.com], featuring a link to a PDF whitepaper [point1design.com].
    It describes the superformula as a "generalized superellipse equation". The 3D version is based on superquadrics.

  • My God! (Score:1, Redundant)

    by superdoo ( 13097 )
    He's discovered the Spirograph [spirograph.com]!

  • My favourite quote (Score:4, Interesting)

    by SolemnDragon ( 593956 ) <solemndragon@gm[ ].com ['ail' in gap]> on Tuesday April 08, 2003 @07:49AM (#5685383) Homepage Journal
    He specialises in Bamboo BioTechnical Rearch?

    But my favourite quote, from his homepage, is:

    "Moreover, well known equations from mathematics like the Theorem of Pythagoras, the equations for conics and conics sections and the equation from Fermat's last theorem, are all special cases of this formula."

    So... a guy who specialises in finding new ways to help bamboo propagate- and mind you, bamboo is pretty prolific on its own, don't let that 'lucky bamboo' (which is not actually bamboo, but a plant of another type entirely) fool you- has now found a new way to describe shapes. Yes, this is important, but it's not the next big thing. Folks have been trying to find ways to describe shapes by equations in images long before this, and while his rush to patent may cause some interesting snarls up ahead, i find it unlikely that he even understands Fermat's last theorem,

    Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.
    Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caparet.
    let alone knows the solution and has described it in shape-description formula format.

    But if he does, he'd better post something more mathematical on his website, because he's just landed himself into mathematician waters- and it's sink or swim there, buddy. You don't get to try it again next growing season (Andrew Wiles' revisions notwithstanding), and contrary to what laypeople tend to believe, they still require proof when you walk in and say something crazy like 'Pi is 3.' [smc.edu] Even mathemeticians are still arguing over the proofs available. And it's pretty cutthroat, with ten-day conferences, [ams.org] so i bet he's in for some entertaining phone calls.

    • Re:My favourite quote (Score:1, Funny)

      by Anonymous Coward
      Yeah, you're right. Because he raises bamboo, he's an idiot with no insight who understands nothing. And you're so clever for discovering him! Imagine, actual mathematicians and science publishers fooled by the chicanery of this foul grass gradener, but you, lowly Slashdot karma whore, have discovered him for who he truly is.

      Well done!
      • Obviously the parent poster was a mathematician.

        Here's how the saying goes:
        Biologists think they're Chemists.

        Chemists think they're Physicists.
        Physicists think they're Mathematicians.
        Mathematicians think they're God.

  • Don't be so condescending... (Score:4, Insightful)

    by robbo ( 4388 ) <slashdotNO@SPAMsimra.net> on Tuesday April 08, 2003 @10:13AM (#5686125)
    I can decide which comment to reply to so I'll just top-level post. Yes, polar coordinate equations are quite simple, and yes, there is prior art for a variety of shape generating equations (for example, superquadrics)... That being said, the fact that this research has been published in Nature is indicative that his work has generated at least *some* excitement among mathematicians. Sometimes the most compelling mathematical constructs are also the simplest. e=mc^2 anyone?
  • I've just invented a new universal theorem. It goes likes this :

    The theorem takes a natural value as parameter n, and n more parameters between 1 and 27. Then, you read out the theorem by replacing each parameter by a letter between 'a' and 'z', and space being 27.

    The really interesting point is that every known theorem, including Godel incompleteness theorem and general relativity are special cases of this theorem.

    Ok, I'll agree that writing out mathematical symbol is a bitch with it though. ;-)
  • I'm almost certain (this was like, over 15 years ago) that I discovered this same exact thing while experimenting with graphics on an apple II. Using sin and cos left something to be desired speed wise, so I used a completely integer algorithm I found in an issue of sync (the Sinclair magazine!). Then I tweeked it. I also was trying to do 3D, so I tried to make elipses and then elipses with perspective. Not knowing a thing about how 3D graphics were done, matrix equations, etc, I just messed around.

    I r
  • I have always wanted to know the formula for an egg shell...

  • This is going to ruin all those OOP textbook examples that show polymorphism on multiple shape sub-types. There is no longer multiple "types" of shapes, just parameters to a single shape equation :-P
  • Here is an old "hamburger drawer" G-BASIC program
    that uses a parametric equation to draw a
    hamburgers. The equation is basically:

    X = sin(tan(t))
    Y = cos(t)

    10 REM HAMBERGER DRAWER - VERSION 1
    PRINT " 1 - Hamberger"
    PRINT " 2 - Hogey "
    PRINT " 0 - quit"
    INPUT " Enter Choice:"; chc
    IF chc = 1 THEN fat = 170
    IF chc = 2 THEN fat = 80
    IF chc = 0 THEN GOTO 9999
    SCREEN 12
    CLS
    FOR T = 0 TO 6.32 STEP .0006 'define loop
    x = 300 + 230 * SIN(TAN(T)) 'calc x X = SIN(TAN(T))
    y = 225 + fat * COS(T) 'calc y Y = COS(T)
  • I seem to remember the circle command would erroniously take an extra parameter which would make these sort of shapes. Sort of like a Spirograph.

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