This is what about ten minutes every other day or so for a year gets you.
Apollo. adj 1. brave, without fear (from L. ab- not + pollo chicken).
Say you’re me and you just watched Vi Hart’s video on infinity elephants and you totally missed the joke about Mr. Tusks even though you read Dinosaur Comics all the time but you liked the bit about Apollonian gaskets, which don’t blow out in Battle Mountain like the one in your car did but rather on the way to the L1 point and then you need Richard Feynman to tell you why. You thought she was going to draw the tiniest camels going through the eye of a needle, but you suppose that would ruin the hyperbole in the parable, so the ellipsis was justified. Anyway, you decide to avoid circular reasoning and doodle rectangles instead, filling them up with squares.
Eventually you wonder which ones you can fill up with finitely many squares and which ones you need infinitely many for, and so you start with squares and build up some rectangles. One square can only make one rectangle, itself. Two squares can only make one rectangle, but it can be lying down or standing up so you decide to say they’re different. Three squares make four rectangles and four squares make eight rectangles, and then you start thinking about Vi Hart’s video on binary trees. So you put the numbers into a tree, but it looks kind of stern, so you add some fareys to cheer it up. Then you see that the height of a rectangle is just the sum of its neighbors’ heights, and similarly for the width. You see lots of nice patterns in the dimensions involving flipping things over or running them backwards, kind of like the Blues Brothers’ police car when it was being chased by the Neo Nazis or that V6 racecar that Johann sent to Frederick.
Now instead of breadth, you decide to go for depth. Making the rectangles very long or very tall is too boring, so you add one square each time, alternately making it longer and taller. 1 1 2 3 5 8 13 21… You get the Fibonacci numbers; the limiting ratio is the golden ratio, [1+sqrt(5)]/2 to 1. This rectangle is the worst at being approximated by repeated squares so it shows up in systems where repetition is bad, like the angle at which plant leaves grow so they overlap the least and gather the most sunlight or how sunflowers pack the most seeds into a flowerhead and Roger Penrose thinks the Fibonacci spirals in the microtubules in the neurons in your brain are doing quantum error correction.
You decide to look at other irrational numbers to see if they have any nice patterns. e does. pi doesn’t. square roots of positive integers give repeating palindromes! You wonder whether all palindromes occur and if not which of the lyrics to Weird Al’s song Bob are special that way. And maybe then you make up a palindrome with vi hart’s name in it and turn it into a square root.
Maybe you decide that you want to doodle some circles after all, so you start with this gasket and figure out where the circles touch the line. The numbers look very familiar. You wonder what the areas of the circles are and how the gasket relates to the modular group and Poincare’s half-plane model of the hyperbolic plane and wish you had time to just sit in math class and doodle…