We’ve posted about mathematical images a few times on this blog, but recently I’ve been impressed with how many great math animations I’ve been seeing! So much of mathematics is about motion, and it’s nice to see visualizations that include that motion.

Last week, John Baez and Greg Egan blew my mind with a beautifully illustrated blog post about hypocycloids rolling around in other hypocycloids. The animations are gorgeous in and of themselves, but the explanations are also deeper than just the definition of the shape. There are connections between these hypocycloids and symmetry groups that are important in particle physics, and Baez, Egan, and some other commenters explore those connections in the post and the comments. (I wrote about these hypocycloids on my other blog, Roots of Unity, as well, but I loved the post so much that I want to share it everywhere I can!)

MathGifs is a fairly new blog by Virginia Military Institute faculty members Troy Siemers and Greg Hartman. The blog is about a lot more than just pretty animations. The exposition is excellent, and the animated GIFs help make the ideas clear. My two favorite posts have been Rotations Through Translations and Translations Through Rotations. In the former, collections of points seem to rotate, but each individual point just travels along a straight line. In the latter, collections of points seem to travel along horizontal paths, but each individual point moves along a circle or another closed path. You can follow mathgifs on Twitter to keep up with their latest posts.

Matthew Henderson’s blog matthen has quite a few cool GIFs as well, and he usually includes Mathematica code so you can play along at home if you’re so inclined. His posts are short but usually include links to further information on the problem or object discussed. I really enjoyed the post about the Haberdasher’s problem: can you cut an equilateral triangle into four pieces that can be rearranged to make a square? He includes a link to Mircea Pitici’s page about hinged dissections, which is quite fun! The Euler spiral post is another favorite of mine.

Finally, last week a friend shared David Madore’s hyperbolic maze with me. You use your keyboard to rotate and travel along a fundamental domain of a Riemann surface of genus 8812 with a few walls to make it interesting. The goal is to get from the starting point to a target point and back using a path that is not homotopic to the identity. You can choose your favorite model of the hyperbolic plane: Poincaré disc or Beltrami-Klein. (If you prefer the upper half-plane or Lorentz model, you’ll have to make your own maze!) I must admit that as someone who usually uses the Poincare disc model, I thought the Beltrami-Klein model felt pretty strange.

Have you seen any cool or enlightening math animations recently?

Some interactive animations of mine are at Euclids Muse. For example this one:

http://euclidsmuse.com/app?id=427

relates Pascal’s Limacon seen as a conchoid, as an isotomic and as an epotrochoid.

This one:

http://euclidsmuse.com/app?id=667

gives a mechanical simulation (using springs) of linear least squares regression

Since you mention my hyperbolic maze game, and since it’s also in the theme of “math that moves”, I feel that should mention this video I made illustrating various projections of the sphere and of the hyperbolic plane (and the analogies between them), and the effect of moving a regular tiling under these projections: http://www.youtube.com/watch?v=xHvAqDuWG2M