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I notice confusion in myself over the swiftly emergent complexity of mathematics. How the heck does the concept of multiplication lead so quickly into the Ulam spiral? Knowing how to take the square root of a negative number (though you don't even need that—complex multiplication can be thought of completely geometrically) easily lets you construct the Mandelbrot set, etc. It feels impossible or magical that something so infinitely complex can just exist inherent in the basic rules of grade-school math, and so "close to the surface." I would be less surprised if something with Mandelbrot-level complexity only showed up when doing extremely complex calculations (or otherwise highly detailed "starting rules"), but something like the 3x+1 problem shows this sort of thing happening in the freaking number line!
I'm confused not only at how or why this happens, but also at why I find this so mysterious (or even disturbing).
I was listening to a podcast the other day Lex Friedman interviewing Michael Littman and Charles Isbell, and Charles told an interesting anecdote.
He was asked to teach an 'introduction to CS' class as a favor to someone, and he found himself thinking, "how am I going to fill an hour and a half of time going over just variables, or just 'for' loops?" and every time he would realize an hour and a half wasn't enough time to go over those 'basic' concepts in detail.
He goes on to say that programming is reading a variable, writing a variable, and conditional br... (read more)
