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Math Graphics Software

Art with a Mathematical Twist 69

Euler points out a story about art created through mathematics. The Science News article covers selections from a recent exhibit, where over 40 artists gathered to show their work and the math behind it. The rest of the pieces are also viewable at the exhibit's website. "Michael Field, a mathematics professor at the University of Houston, finds artistic inspiration in his work on dynamical systems. A mathematical dynamical system is just any rule that determines how a point moves around a plane. Field uses an equation that takes any point on a piece of paper and moves it to a different spot. Field repeats this process over and over again--around 5 billion times--and keeps track of how often each pixel-sized spot in the plane gets landed on. The more often a pixel gets hit, the deeper the shade Field colors it."
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Art with a Mathematical Twist

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  • When it comes to the relationship between mathematics and the arts, my favourite example is the music of Per Norgard. In 1959 Norgard discovered a way of serializing melody that resulted in endless self-similarity, a type of fractal. He termed it the infinity series [], and though the two-tone infinity series had already been discovered by mathematicians, the application of the principle to chromatic and diatonic scales resulted in a series no mathematician had discovered before. The infinity series is a fascinating concept, and in Norgard's works like the Symphony No. 3 [] it proves immensely beautiful.

    Other composers have, of course, made use of mathematical processes. The golden section is often heard in Bartók, for example, though who knows if it was done consciously.

    • Re: (Score:2, Interesting)

      by opec ( 755488 ) *
      I'm a musician and nerd, so I had to look up Norgard. Those crazy Danish, I found out that his name is fully Nørgard. Lucky me, I'm sitting working at the library and my search tells me we have recordings of his in the collection. Sweet. Fractal art is good stuff.
      • by jhol13 ( 1087781 )
        I remember one of the first graphics gurus to say "I wish I'd never see another fractal". That was about 20 years ago. Oh boy do (and did) I agree with him.

        Unfortunately I do not remember his name.
    • Re: (Score:3, Interesting)

      by Coryoth ( 254751 )
      For an interesting take on mathematical analysis of music, you could try The Topos of Music []. It sets out to apply deep modern mathematics to issues of musical composition. Starting with a base in category theory and topos theory (hence the title), it can then spiral down to using differential geometry and algebraic geometry. Personally I don't know enough music theory to know if it really stacks up, but it is certainly mathematically very interesting (and goes well beyond the basic mathematical dabbling of
      • Re: (Score:2, Interesting)

        by popmaker ( 570147 )
        But what's the point? Is it achivement in itself to make use of mathematics in music? I would think that the real justification for the whole thing was musical value, not mathematical. The whole idea should be that by bringing mathematics to music, you would be able to create music that sounds truly fascinating, but it sound from you that being able to use the mathematics at all is enough.

        I am a little skeptic about bringing mathematics to music - sometimes it seems to be the end in itself, which it shoul
        • by Threni ( 635302 )
          > But on the other hand, if the results are MUSICALLY interesting, that's another story.

          Now all you have to do is define "musically interesting". Shouldn't be too hard. After that you can help the AI guys out with a workable definition of either "conciousness" or "intelligence".

          • Think about: "Does it still sound good if you don't know the mathematics"? The piano (and most modern instruments) are tuned using an exponential function with base 2. Most people like music regardless of weather they know this or not. So, in this way the exponential is "musically interesting"... take it as "definition through examples" - I'll provide more of them if you want.

            This is a very stretchable defintion and I am very fond of any kind of a mathematical experiment which might provide musical ideas.
            • by mpiktas ( 740253 )
              Aah, you've hit the head of the nail. Math and music do have something in common, but there is nothing mistical going on. But that does not stop people from creating various comparisons and drawing far reaching conclusions. The sad part that these people usually are neither mathematicians nor musicians.
            • Re: (Score:2, Insightful)

              by Threni ( 635302 )
              > The piano (and most modern instruments) are tuned using an exponential function with base 2.

              It's not quite that simple.

              > So, in this way the exponential is "musically interesting"... take it as "definition through examples" -
              > I'll provide more of them if you want.

              The point is, it's all subjective. Some people make music using this or that system (improvisation, strict counterpoint, using elements of chance, partly composing but leaving decisions to the performer, algorithmically defined music(w
        • by Coryoth ( 254751 )
          Oh I didn't say it wasn't musically interesting, just that I (being a mathematician and not a musician) am probably not the best person to judge the musical quality. From the mathematial point of view, however, it provides a very elegant and deep mathematical framework from which to build a theory of composition and a theory of performance. DO these theories, put into practice produce great music? Well it sounds good to me, but then I'm hardly a good judge... and that was all I was saying.
          • Well, all criticism aside... I am actually deeply intrigued. I've always liked the mathematics of music.
    • What a waste of time. I just studied the entire site you linked to and there was nothing to see. It's to early to read about math. I'm looking for my Monday AM inspiration.
    • I'm surprised that no-one has mentioned that Salvador Dali's works were heavily influences by his fascination with science and mathematics. Several examples exist but one that sticks out immediately is Crucifixion (Corpus Hypercubus) [] which points to the paradox his own mind must of dwelled in supported by what I heard him say in an interview (sorry this is from memory so it may not be exact) "The mathematical and scientific evidence I've observed tells me that God exists, but I don't believe it". This from
  • If you have the photorealism of the Rennaisance, you get all of the math involved in regular life (e.g. the golden ratio). With various less realistic artists (e.g. Pollock, Van Gogh), haven't mathematicians found various deep mathematical patterns in their work? This is what you get when you start out with pure math, and turn it into art, whereas most of art is what you get when you have an intuitive understanding of math (i.e. what looks good) and go with that. All art has math in it.
    • That's all very well if you define math to be "what looks good". But most mathematicians consider it to be something a little bit more rigorous. Incidentally, if you think all is beautiful, try solving systems of linear equations or ODE's. Really freaking ugly maths.
      • Beauty is when all the pieces fall together to create something new. ODEs and Linear Algebra are both beautiful in that respect.
  • by paroneayea ( 642895 ) on Sunday February 17, 2008 @05:45PM (#22456238) Homepage
    If you're interested in pretty, shiny, mathematical things that you can run on Linux, check out:
      - electricsheep: animated fractal flames: [] (I highly recommend running this as your screensaver, though it takes a bit for the first sheep to download)
      - Jenn: pretty, shiny, blue(?) polytopes, rendered on your computer: []

    Anyone have any others?
  • ....oh well, so much for mathematical art on the web...
  • IFS, fractal flames (Score:4, Informative)

    by Anonymous Coward on Sunday February 17, 2008 @05:52PM (#22456280)
    The images described in the summary (which are not really representative of most of the stuff in the gallery, just Fields's stuff) are generally known as iterated function systems [], and perhaps belong to the subset known as fractal flames. The description is fairly accurate, but the images he has made are rather unimpressive compared to ones I've seen (and made myself). Probably the best known example of a fractal flame program is Electric Sheep []. However, another good program for making fractal flames is called Apophysis [] (regretfully, it's Windows only, but does work fairly well under Wine). I've been working with Apophysis for about 3 years now, and trust me, there's a lot of more artistic stuff out there that uses fractal flames. Even some of the stuff on Wikimedia Commons [] is better than his stuff.

    Coincidentally, my captcha was "artful".
    • But, Electric Sheep was released in 1999.

      Field's book, 'Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature' was released in 1992. He is one of the early mathematicians doing work in iterated symmetric systems.

      His work might be unimpressive to you, and Mandelbrot's set might seem old hat, but they were the guys who did the math you borrow..
    • Incidentally, Field's image on the first link is a strange attractor rather than an IFS. It's an Icon, which is one of the few equation to produce symmetrical sets. The degree of symmetry is one of the equation constants, so I'm sure he won't have any trouble creating a new image for each of his wife's birthday.
  • Constructivism, Suprematism, Cubism, Bauhaus, Serial compositions, twelve-tone theory. Math in Art is so not new or news. kthxbye
  • New and yet not new (Score:4, Informative)

    by fractalus ( 322043 ) on Sunday February 17, 2008 @06:10PM (#22456404) Homepage
    It's true that mathematical proportions and structures have been found in artwork for centuries, but what's different about these things is the role of the algorithm and raw computational power in producing this artwork. These are works that could not have been done before the availability of computers. The artist directs and controls the mathematics, using them like other artists use different kinds of paints, brushes, and canvas. But the computer does the mind-numbingly tedious work of billions of computations to render it on-screen. This is not all that different from artists using 3D sculpting and rendering tools; it's just a different set of algorithms.

    Others have pointed out Electric Sheep and Apophysis; these focus on one particular type of non-linear iterated function system, the "fractal flame". There are many other fractal rendering tools out there, some free, some not. Wikipedia has a list if you're interested. This is a medium that has been in constant change for twenty years and doesn't look like it's ready to settle down any time soon.
  • Mathematical Music (Score:3, Interesting)

    by ilikepi314 ( 1217898 ) on Sunday February 17, 2008 @06:24PM (#22456488)
    What I found more interesting than mathematical art was the music produced from differential equations and such.

    I really wish I remember more details but a few years ago I saw a presentation by a mathematician in which he had a little program that solved some sort of equations. Grr, I'm going to hate myself now for not remembering. Well, regardless the details, it solved something and assigned the solution values specific notes/chords from a piano, so that whenever a value was obtained, the computer played that note. Thus, the time evolution gave a sequence of notes, and so he recorded this sequence.

    He played a few excerpts, I tell you what, it sounded like Mozart or Beethoven. Well, certain parts you could pick up a very forced/electronic feel to it, but other parts glided so beautifully that it sounded like a master pianist was playing.

    That was an incredible lecture. Perhaps anyone else knows what I speak of? I'd like to find out what program and equations were used, it was fascinating.
  • by MillionthMonkey ( 240664 ) on Sunday February 17, 2008 @06:44PM (#22456640)
    A few years ago I got the idea to write code that fed massive scene files into POV-Ray. There are probably better tools nowadays but POV-Ray had the virtue of a simple scene description language that I was already familiar with. It's easy to create code to generate it.

    I made a heart out of the sextic (huhhuhhuhhuh) polynomial

    (2xx+2yy+zz-1)^3 - xxzzz/10 - yyzzz = 0

    and had POV-Ray create a bunch of scene files by rotating this thing through 180 degrees to create an animated heart GIF. [] (This was back in the Dark Ages when the web was full of animated GIFs.) There were probably a thousand other animated hearts out there but this one was mine.

    I got the idea to do space filling of the unit sphere with thousands and thousands of small boxes [] or smaller spheres, [] playing around with the lighting to see if I could create something vaguely moonlike [] with inside-out craters. I tried doing this with thousands of hearts [] but got bitten in the ass by a bug in POV-Ray's polynomial rendering code where it trips over a planar singularity in the heart equation, so every little heart ends up with an unromantic slit running across its equator. There were just too many to fix by hand.

    The most interesting image from this technique came from a routine that recursively generated spheres, invoking itself six times per sphere to create smaller spheres on the top, bottom, left, right, front, and back, each of which then does the same thing, to a depth of 5 or 6. You end up with a Sierpinski octahedron. []

    All this stuff has been done to death by others. I wish I were good at drawing comics.
  • For an interesting and entertaining experience with fractal art, see also [] Bahman Kalantari, the creator of the site, has been exploring the artistic side of math for some time now. Have fun!
  • Roman Verostko (Score:3, Interesting)

    by raddan ( 519638 ) on Sunday February 17, 2008 @07:25PM (#22456964)
    Roman Verostko and others have been doing something he calls algorithmic art [] for awhile. E.g., put a paintbrush in a pen plotter and then write an algorithm to paint on canvas. Although sometimes I feel like artists like Verostko (who call themselves algorists) are tremendously arrogant sometimes (which I suppose makes them like many other artists), a lot of their stuff seems really beautiful to me. In particular, Verostko's pseudo-calligraphy is just mesmerizing to me-- it looks sort of like a written language, but it's not.

    And of course, you can't forget the grandmaster of algorithmic art: Bach. Bach was a master of counterpoint, and the mathematical beauty of some of his works (e.g., The Art of Fugue) is readily apparent. If he indeed did not generate his works in an algorithmic way, well, that's surprising to me. Listen to Glenn Could play Bach, Partitas 1,2, and 3 [] being my favorite...
    • Re: (Score:2, Informative)

      by theazreal ( 1015759 )
      Fugues are inherently algorithmic. You take a theme, invert, reserve, invert-reverse, modulate... Bach just did this in a particularly beautiful and inventive way. You'll find his counterpoint and stretto are also somewhat regular. []
  • procedural art (Score:2, Interesting)

    by vesabios ( 1149567 )
    I did some mart art work awhile ago, based on Daubechies' scaling functions. Check it out: The Strangers Series [].
  • The site's running very slow, guess it fell victim to the infamous Slashdot Effect :)
  • by iminplaya ( 723125 ) on Sunday February 17, 2008 @07:29PM (#22456992) Journal
    Don't say that! next thing you know somebody is going to sue Pirate Bay for linking to pi. If that was to happen maybe we can determine how many digits are within "fair use". As far as I know, nobody has uploaded the whole thing yet.
    • That was exactly my thought. There is still no conclusive proof that uploading the whole Pi to The Pirate Bay would not constitute a copyright violation of all known works of art []. And a few unknown ones. Imagine that - Prince and Village People suing [] you for infringement of the songs they might will have recorded(*) with their gerontal voices some time in the future! Wait...? Damned, let's pray that Prostetnic Vogon Jeltz doesn't know how to count to five. Hmm, damned again, Pi is less then four, right? Duc
  • Personally I rather some of the work by artist Benar Venet [], in which equations and commutative diagrams are rendered as wall sized paintings. They can be surprisingly striking and beautiful. Unfortunately the website is flash, so I can't link directly to examples; you can find them under "Painting->Wall paintings".
  • is filled with all sorts of fractal and computer generated art.

    My own page,

    Includes many pieces created with the help of home-grown Java filters and tools.

  • Escher (Score:1, Informative)

    by Anonymous Coward
    How can this thread go on this long without the obvious comment about MC Escher's hyperbolic tilings (such as the Circle Limit images)? It's not like there wasn't a world renowned mathematician (H. S. M. Coxeter) helping him work out the details... (and let's not forgot some of the inspiration he also got from Roger Penrose and his father L. S. Penrose)
  • Ric Werme wrote a program to draw roses on the Graphic Wonder that was entered in the Three Rivers Arts Festival back in the 70s. There is python code [] and some history for it here.
  • Wow!

    We had shadebobs in the Amiga demoscene back in the 90s.

    Good to see oldskool effects making it into the mainstream.
  • It seems nobody has yet mentioned the work of Paul Bourke [] (if that name seems familiar, he hosted the POV-Ray short code competition recently featured on Slashdot []). I'm a fan of his work on fractals [] (scroll down, there's a *lot* of stuff on that page), especially slices of four-dimensional Julia sets []. Definitely mathematical art of the highest order.

    ... well, that is, unless you're a fan of Ken Perlin [] instead ;)

  • I'm surprised there hasn't been mention of Bathsheba's work [], "exploring how math, science and sculpture meet".

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