### Doodling in Math Class

#### Posted by John Baez

Vi Hart has a great video called Doodling in Math Class: Snakes + Graphs. She has others, too. Some people are watching her stuff and enjoying math for the first time. Something about the hyperactive, slightly cynical delivery combined with pictures drawn in real-time reaches certain folks who are completely immune to the charms of Martin Gardner.

But some people here will notice that she’s gently teaching us about alternating knots and Seifert surfaces. Alternating knots (and links) have long been used in art:

and Christian Mercat has a nice tutorial on them, in English, French and German. Bathsheeba Grossman has made sculptures that are Seifert surfaces whose boundaries are Borromean rings:

and to create it, she used Jack van Wijk’s free program that draws Seifert surfaces in a variety of styles:

Van Wijk says someone asked him to draw some pictures of Seifert surfaces, and “things got totally out of hand”. If you let the math of alternating knots and Seifert surfaces get out of hand, before you know it you may find yourself studying the Tait conjectures or instanton Floer homology. The subject is so addictive that even Seifert’s pet border colllie got involved: when shown a knot it would either bark or not bark, following some mysterious rule — and thus the Arf invariant was born.

I also enjoyed Vi Hart’s video Doodling in Math Class: Infinity Elephants. It actually manages to explain an infinite amount of material in a finite time! It’s about Apollonian gaskets. Here’s one by Todd Stedl, with the circles labelled by their curvatures:

## Re: Doodling in Math Class

That… is… insane.