# Art with a Mathematical Twist

#### Soulskill posted more than 6 years ago | from the just-like-on-your-graphing-calculator dept.

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."*

## Sometimes math is created through the arts (5, Interesting)

## CRCulver (715279) | more than 6 years ago | (#22456188)

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 [pernoergaard.dk] , 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 [amazon.com] 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:Sometimes math is created through the arts (2, Interesting)

## opec (755488) | more than 6 years ago | (#22456868)

## Re:Sometimes math is created through the arts (1)

## jhol13 (1087781) | more than 6 years ago | (#22467144)

Unfortunately I do not remember his name.

## Re:Sometimes math is created through the arts (0)

## Anonymous Coward | more than 6 years ago | (#22473056)

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.

Benoit Mandelbrot?

## Re:Sometimes math is created through the arts (2, Interesting)

## Coryoth (254751) | more than 6 years ago | (#22457082)

## Re:Sometimes math is created through the arts (2, Interesting)

## popmaker (570147) | more than 6 years ago | (#22457544)

I am a little skeptic about bringing mathematics to music - sometimes it seems to be the end in itself, which it shouldn't be. But on the other hand, if the results are MUSICALLY interesting, that's another story. Like the mathematical construction of a truly bizarre polyrhythm. But that still doesn't go beyond simple modular arithmetic.

Some mathematical stuff in music just sounds superficial... like (actual) the idea of writing a piece which shifts the tempo with a ratio of pi : e. You might think it's cool, I don't know, but no one really cares if the ratio is pi : e or 1.2 or "just slightly faster". There is no intrinsic musical value in the idea. So... is it really worth it?

## Re:Sometimes math is created through the arts (1)

## Threni (635302) | more than 6 years ago | (#22457760)

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".

## Re:Sometimes math is created through the arts (1)

## popmaker (570147) | more than 6 years ago | (#22457866)

This is a very stretchable defintion and I am very fond of any kind of a mathematical experiment which might provide musical ideas. But at some point it becomes just math geeks jerking off. If that isn't clear, I can define "jerking off" for you if you want.

The AI guys are on their own.

## Re:Sometimes math is created through the arts (1)

## mpiktas (740253) | more than 6 years ago | (#22460354)

## Re:Sometimes math is created through the arts (2, Insightful)

## Threni (635302) | more than 6 years ago | (#22460708)

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(wholly or partly)), and some people like it, and some don't. At least one person finds all sorts of music interesting, so it's not a very fruitful definition.

## Re:Sometimes math is created through the arts (1)

## Coryoth (254751) | more than 6 years ago | (#22462560)

## Re:Sometimes math is created through the arts (1)

## popmaker (570147) | more than 6 years ago | (#22469234)

## Where are the fractals? (1)

## BrunoUsesBBEdit (636379) | more than 6 years ago | (#22462886)

## Salvador Dali (1)

## MrKaos (858439) | more than 6 years ago | (#22469536)

## Doesn't most art have a mathematical twist? (2, Interesting)

## pipoca (1142297) | more than 6 years ago | (#22456208)

## Re:Doesn't most art have a mathematical twist? (1)

## Fallen Seraph4 (1186821) | more than 6 years ago | (#22456338)

## Re:Doesn't most art have a mathematical twist? (1)

## DeadChobi (740395) | more than 6 years ago | (#22458054)

## Some great examples of mathematical art (5, Informative)

## paroneayea (642895) | more than 6 years ago | (#22456238)

- electricsheep: animated fractal flames: http://www.electricsheep.org/ [electricsheep.org] (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: http://www.math.cmu.edu/~fho/jenn/ [cmu.edu]

Anyone have any others?

## Re:Some great examples of mathematical art (1)

## CRCulver (715279) | more than 6 years ago | (#22456252)

## Re:Some great examples of mathematical art (1)

## Janek Kozicki (722688) | more than 6 years ago | (#22456856)

## Re:Some great examples of mathematical art (1)

## CRCulver (715279) | more than 6 years ago | (#22456878)

## Re:Some great examples of mathematical art (2, Informative)

## baud123 (977365) | more than 6 years ago | (#22456388)

## Context Free (5, Informative)

## replicant108 (690832) | more than 6 years ago | (#22456406)

As you can see from the link below, some of the results from this project are stunning.

Context Free Art gallery [contextfreeart.org] .

## Re:Some great examples of mathematical art (2, Informative)

## maxume (22995) | more than 6 years ago | (#22456474)

via:

http://nedbatchelder.com/blog/200705/chaoscope.html [nedbatchelder.com]

he also mentioned Jenn:

http://nedbatchelder.com/blog/200802/jenn_visualizing_polytopes.html [nedbatchelder.com]

## Re:Some great examples of mathematical art (1)

## paroneayea (642895) | more than 6 years ago | (#22456498)

## Re:Some great examples of mathematical art (1)

## maxume (22995) | more than 6 years ago | (#22456542)

http://www.btinternet.com/~ndesprez/faq.htm [btinternet.com]

## Re:Some great examples of mathematical art (0)

## Anonymous Coward | more than 6 years ago | (#22456896)

## Processing (4, Informative)

## mingrassia (49175) | more than 6 years ago | (#22456948)

Perhaps the king of all environments (at least in my mind) is Processing [processing.org] . It is a Java based environment created by Ben Fry [benfry.com] and Casey Reas [reas.com] . It's open source, has a huge active community [processing.org] , and plenty of 3rd party libraries [processing.org] for exploring things like computer vision, audio, physics, ray tracing, AI, etc.

There are a ton of really talented people doing cool things in Processing. Too many to list here, check out the Exhibition [processing.org] page for things to play around with.

## Re:Some great examples of mathematical art (1)

## cycoj (1010923) | more than 6 years ago | (#22457006)

## Re:Some great examples of mathematical art (1)

## platyk (696356) | more than 6 years ago | (#22457502)

I started a web site along these lines--only had time to implement one idea so far. It's an applet that looks at the Mandelbrot set in a different way from the usual approach:

http://platy.org/ [platy.org]

## already slashdoted second site (1)

## 3seas (184403) | more than 6 years ago | (#22456242)

## IFS, fractal flames (4, Informative)

## Anonymous Coward | more than 6 years ago | (#22456280)

Coincidentally, my captcha was "artful".

## Re:IFS, fractal flames (1)

## bgspence (155914) | more than 6 years ago | (#22461038)

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..

## Re:IFS, fractal flames (0)

## Anonymous Coward | more than 6 years ago | (#22463220)

Benoit Mandelbrot was exploring generating julia set images using IFSs in 1980 (see his paper "Fractal Aspects of Complex Iteration"), based on the theories of Gaston Julia from 1918.

## Re:IFS, fractal flames (1)

## Nicolas Desprez (712482) | more than 6 years ago | (#22461332)

## Been done (1)

## Arnlod (1103739) | more than 6 years ago | (#22456308)

## Re:Been done (0)

## Anonymous Coward | more than 6 years ago | (#22457588)

## I have this thought... (1)

## coolhaus (186994) | more than 6 years ago | (#22456392)

## New and yet not new (3, Informative)

## fractalus (322043) | more than 6 years ago | (#22456404)

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.

## Other resources (0)

## Anonymous Coward | more than 6 years ago | (#22456428)

## Generative art. (0)

## Anonymous Coward | more than 6 years ago | (#22456440)

My favourite: http://www.complexification.net/ [complexification.net]

## Fractint? (0)

## Anonymous Coward | more than 6 years ago | (#22456442)

## Lag (0, Redundant)

## junner518 (1235322) | more than 6 years ago | (#22456466)

## Re:Lag (1)

## gardyloo (512791) | more than 6 years ago | (#22456750)

## Mathematical Music (3, Interesting)

## ilikepi314 (1217898) | more than 6 years ago | (#22456488)

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.

## Re:Mathematical Music (2, Informative)

## Ambitwistor (1041236) | more than 6 years ago | (#22456664)

## This is the only kind of art I can do (5, Interesting)

## MillionthMonkey (240664) | more than 6 years ago | (#22456640)

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. [photobucket.com] (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 [photobucket.com] or smaller spheres, [photobucket.com] playing around with the lighting to see if I could create something vaguely moonlike [photobucket.com] with inside-out craters. I tried doing this with thousands of hearts [photobucket.com] 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. [photobucket.com]

All this stuff has been done to death by others. I wish I were good at drawing comics.

## Re:This is the only kind of art I can do (-1, Offtopic)

## Sanat (702) | more than 6 years ago | (#22456760)

## Brian Eno, Laurie Anderson, et al (0)

## Anonymous Coward | more than 6 years ago | (#22456752)

http://www.inmotionmagazine.com/eno1.html [inmotionmagazine.com]

Eno has done quite a lot of work using various fractal composition programs.

## Re:Brian Eno, Laurie Anderson, et al (1)

## popmaker (570147) | more than 6 years ago | (#22457776)

One notable difference is of course, that you can also have this happening in your living room with a computer system. It just surprises me that he didn't talk about jazz AT ALL. Isn't he supposed to know stuff about music or something?

## Nothing new... (0)

## Anonymous Coward | more than 6 years ago | (#22456862)

Check out:

http://apophysis.deviantart.com/ [deviantart.com]

## polynomiography (1)

## 1 a bee (817783) | more than 6 years ago | (#22456902)

## Oblig xkcd (1)

## Bluewraith (1226564) | more than 6 years ago | (#22456906)

## Roman Verostko (2, Interesting)

## raddan (519638) | more than 6 years ago | (#22456964)

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 [amazon.com] being my favorite...## Re:Roman Verostko (2, Informative)

## theazreal (1015759) | more than 6 years ago | (#22462310)

## procedural art (2, Interesting)

## vesabios (1149567) | more than 6 years ago | (#22456974)

## Slashdot effect (1)

## Doug52392 (1094585) | more than 6 years ago | (#22456988)

## Math and Art? (4, Funny)

## iminplaya (723125) | more than 6 years ago | (#22456992)

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.## Re:Math and Art? (1)

## K. S. Kyosuke (729550) | more than 6 years ago | (#22457670)

notconstitute a copyright violation of all known works of art [wikipedia.org] . And a few unknown ones. Imagine that - Prince and Village People suing [slashdot.org] 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? Duck and cover! ~_~(* Huh? As there is no such thing as 'perfect future', I'm not sure if this makes sense! See Dr. Dan Streetmentioner's work for more information.)

## Mathematics with an artistic twist (1)

## Coryoth (254751) | more than 6 years ago | (#22457026)

## www.deviantart.com (1)

## BlueBoxSW.com (745855) | more than 6 years ago | (#22457074)

My own page,

thefusa.deviantart.com

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

## This is really Michael Henon's work (0)

## Anonymous Coward | more than 6 years ago | (#22457182)

## Escher (1, Informative)

## Anonymous Coward | more than 6 years ago | (#22457642)

## roses (1)

## Smallpond (221300) | more than 6 years ago | (#22458846)

## Shadebobs.. (1)

## vjouppi (621333) | more than 6 years ago | (#22460030)

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

Good to see oldskool effects making it into the mainstream.

## Paul Bourke (1)

## Ari Rahikkala (608969) | more than 6 years ago | (#22461974)

It seems nobody has yet mentioned the work of Paul Bourke [uwa.edu.au] (if that name seems familiar, he hosted the POV-Ray short code competition recently featured on Slashdot [slashdot.org] ). I'm a fan of his work on fractals [uwa.edu.au] (scroll down, there's a *lot* of stuff on that page), especially slices of four-dimensional Julia sets [uwa.edu.au] . Definitely mathematical art of the highest order.

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

## Smoking clover (0)

## Anonymous Coward | more than 6 years ago | (#22462602)

## Dynamical Systems?? (0, Troll)

## AP31R0N (723649) | more than 6 years ago | (#22463540)

What the fuck is a dynamicAL systems? When did people go so stupid with making up words? "Hey Jeff, how do I make this boring thing sound more interesting?" "Oh, um, make up some words, use a noun as a verb, or add inappropriate suffixes to a perfectly good word!"

## Bathsheba Grossman (1)

## Onymous Coward (97719) | more than 6 years ago | (#22467496)