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Main Area and Open Discussion => Living Room => Topic started by: mouser on July 25, 2006, 01:04 PM

Title: Mathematical Imagery Gallery
Post by: mouser on July 25, 2006, 01:04 PM


The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.
Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.


http://www.ams.org/mathimagery/

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Title: Re: Mathematical Imagery Gallery
Post by: f0dder on July 25, 2006, 01:20 PM
Nifty - I've always been fascinated of "applied mathematics" especially when used for graphics/animation/etc. That's why I like www.pouet.net I guess ^_^

Oh, and I absolutely love Escher's works.
Title: Re: Mathematical Imagery Gallery
Post by: housetier on July 25, 2006, 02:00 PM
I was in a mathematical research group at one of the Fraunhofer Institutes. Some fellow members worked on a project to improve ray tracing techniques; my "project" dealt with the Lyapunov exponent, basically a number that says how robust a certain iterative function is towards perturbation.

I'll skip the nasty details (http://en.wikipedia.org/wiki/Lyapunov_exponent) to say that one could normalize this exponent and make a color value of it. Chosing a nice color space, one could get amazing results. Some images where incredibly dynamic, others looked like they had a third dimension (which was impossible from the way the data was calculated and projected). Results also depend on the specific formular and chose perturbation pattern: slight variations of either could lead to completely different results.

Results (http://plus.maths.org/issue9/features/lyapunov/ba256.jpg) are often (http://www.cis.syr.edu/~blair/Lyap.gif) spectacular (http://plus.maths.org/issue9/features/lyapunov/lya3.jpg)!

Title: Re: Mathematical Imagery Gallery
Post by: f0dder on July 26, 2006, 02:40 AM
Very nice, housetier - fractal'ish. (I seem to recall the name Lyapunov from various fractal programs... coincidence? :))
Title: Re: Mathematical Imagery Gallery
Post by: housetier on July 26, 2006, 03:37 AM
Fractals and Lyapunov are often seen together, both are used to create pretty images. I find the usual Mandelbrot and Julia sets somewhat boring. The graphic representation of an array of Lyapunov exponents is not a fractal though.

There used to be a program for x windows by name "xlyap", I can still find its man-pages (http://www.linuxcommand.org/man_pages/xlyap1.html) on the net; it seems to (have) be(en) a part of a collection of xscreensaver programs. For windows I found this (http://www.efg2.com/Lab/FractalsAndChaos/Lyapunov.htm), and this seems to be a matlab program (http://www.math.tamu.edu/~mpilant/math614/Matlab/lyapunov.m).

I was hoping Gnofract4D (http://gnofract4d.sourceforge.net/) would have a Lyapunov function, but I couldn't find the answer on the site of the projet; I am guessing it is primarily for fractals. Another program I have to write myself, it seems...
Title: Re: Mathematical Imagery Gallery
Post by: JavaJones on July 26, 2006, 10:31 PM
Very nice. My dad was really into fractals so I've also been into them for years. I used to play for hours with Fractint, an early DOS-based fractal app that was extremely comprehensive in terms of the number of algorithms implemented (several hundred). Modern programs like UltraFractal make much prettier stuff, but it was pretty awesome at the time.

- Oshyan