There is a popular misconception that math and art use separate halves of the brain, but that might be a feature of math art, not a bug. Many math concepts come to life when expressed visually, especially in realistic 3D. My work is done in Mathematica and Pixelmator, with some touchup in Cheetah3D.
I was a community college math teacher for decades, and my students kept telling me my graphs and pictures looked good, and not just good for a blackboard, but perhaps for an art gallery. After retiring as a teacher, I have pursued my goal of delivering math content to an unsuspecting audience. I want viewers to say, "That's cool! That's math?"
In a 1983 paper, Andrzej Szulkin showed how to fill R^3 with a union of disjoint circles. In this artwork I have organized the circles into spherical shells and slope, and a series of vertical rings. This gives every (x, y, z) point in Euclidean 3-space a new triple of numbers: (r, m, t) where r = distance from the origin, m = slope of plane containing a circle, t = angle along the individual circle. The spherical shells are shown, with each nuclear family of circles avoiding the two intersection points with the vertical red ring. (Source: Amer Math Monthly Nov 1983 pp 640-41.) Follow the link for a 3D interactive view of this piece!
The basic saddle curve (x, y, z) = (cos t, sin t, 0.5 cos 2t) can be adorned in many ways. I have often used the {tangent, normal, binormal} frame and curvature formulas to demonstrate properties of such a loop. Here there are seven styles of embellishment, with Osculating Circles in the center, surrounded by (clockwise, from top): Saddle Firebird, Oscu-Bubble, Color Brush, Normal Curves, Normal Bubble, and Wire Bundle. Do you know how these were made? With Mathematica and a lot of trigonometry.
