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I know that Math is a difficult subject, but there is an underlying beauty in it that I can’t really describe in words. So I’ll just point you to a site on Mathematical Art. The exhibit took place in San Diego, California last January 6-9, 2008 as part of the 2008 Joint Mathematical Meetings. In that site, you can choose to view all the artworks featured in the exhibit. I’ll just show you some of my favorites.

My favorite is Thomas Hull’s work entitled Spiraling 20-gon. And I quote:

An annular piece of paper in the shape of a 20-gon is pleated using concentric 20-gons. This was wet-folded into a double-spiral shape, creating an illusion as to what the original paper must have been.

I was totally impressed because it was made out of paper. Beat that.

I am currently working on differential equations, so this work by Mark Stock intrigued me. This is entitled Green Streamlines.

Green Streamlines depicts individual traces through a vector potential field. The field is populated with random thick-cored vortex particles, and each path is created by integrating the Biot-Savart equation in both directions from an initially-random point. Barnes and Hut’s multipole treecode algorithm is used to accelerate the calculations. My goal for this image was to allow appreciation of ubiquitous fluid forms by instantiating a small tangle of turbulence.

Who says random things have no pattern? Look at what happened to Flow 19 by Andy Lomas.

This image is composed of the layered trajectories followed by millions of particles. Each individual trajectory is essentially an independent random process, with the trail terminating when it reaches a deposition zone. Collectively the paths combine to form delicate complex shapes of filigree and shadow in the areas of negative space that the paths don’t reach.

You can even generate art using polynomials. This is Jeffrey S. Elly’s Newton Without Newton.

This 3-dimensional image depicts the modulus of the Newton process when applied four times to the 5th degree polynomial, $f(s) = (s-q)^5 - 1$, where $q = 0.7 + 0.3i$ and $s$ is the complex variable, $x + iy$.

More concretely, this 3-dimensional (x,y,z) surface is

$z = g(x,y) = \left| N(N(N(N(s)))) \right|$

where $N(s) = s - \frac{f(s)}{f'(s)}$ is the Newton operator and $\left| \cdot \right|$ denotes the absolute value (modulus) of a complex number.

The surface has been cropped by discarding any point whose z-value is greater than 2, allowing us to peer inside some of the poles of the surface, especially the large center one. The surface is grey except for those points whose z-values closely match the moduli of one of the five roots, hence the 5 different colors.

See? There’s art in math.