Dan Bach
Hi, Dan Bach here (@dansmath on twitter), former college mathematics teacher from California. I used Mathematica for over 25 years in the classroom, for teacher workshops, and at conference talks. My students were known to create mathematical graphics, movies, and sounds; either as class assignments or purely for pleasure. I'm now a 3D math artist and interactive book author. The present work involves simple trigonometry, xyz-coordinates, linear interpolation, geometry, and visual composition. Now go out and create some of your own math art!
Starting with a basic red saddle, curves are drawn in space that lie at different weighted averages of pairs of points on the saddle. The sampled pairs get farther apart until they are diametrically opposite. They chase each other around the curve and create trace curves. And after all that chasing, the three shapes hang out together at the beach! Drawn in Mathematica, using trigonometry and simple linear functions. (This scene is also viewable in 3D/VR/AR; follow the "More Info" link.)
The underlying polyhedral shapes are an octahedron, a cuboctahedron, and one made from 18 squares and eight triangles. Each polyhedron has a path that visits all the edges once and only once, returning to its original vertex. A rainbow of line segments then tracks its progress along this Eulerian circuit! Follow the journey with your eyes. Can you see the squares that lie on a cube, or the triangles sitting at the corners of a cube? (This scene is viewable in 3D or VR by going to the "More Info" link.)