What a week.
George Floyd Protests
Shit’s going down. Last Monday, May 25 2020, a black man named George Floyd gets murdered by a cop in Minneapolis (after already restraining Floyd, the cop knealt on his neck for 9 minutes).
Since then, there’s been protests in 75 cities. Some people are protesting, some are looting. There’s cops getting violent, there’s civilians getting violent. Some people are showing up because they want to push things over into civil war against the police. All while we’re still in the middle of a pandemic. It’s crazy.
I’d recommend twitter for keeping tabs on what’s happening on the ground in your city.
Throughout all this, a blog post has been echoing in my mind. It’s part summary of Days of Rage (a book about the crazy political violence and domestic terrorism of the 60’s and 70’s), and part analysis of how different sides of the culture war use political violence differently.
Updates on Technical Projects!
When can you use a Linear replacement?
Differential Calculus is all about taking non-linear functions and creating linear approximations for them. The general definition of a higher dimensional derivative boils down to “the derivative of a f at a is the unique linear function that becomes an arbitrarily good approximation of your original function as you zoom in at point a”. On the Cartesian plane, linear functions are straight lines. In 3D Euclidean space, they’re planes. In higher dimensions, they’re hyper-planes. What’s great about this definition is it lets you do calculus on functions that aren’t from Euclidean Space to another. The function which squares a matrix has a derivative!
This definition also helps highlight why some functions aren’t differentiable. Check out the gif below. We’ve got three functions, X^2, |X|, and a piece-wise function.
As we literally zoom in, they all start of behave differently. X^2 flattens out and looks like a line. |X| doesn’t change at all, and no matter how much we zoom, it will always have that sharp bend. And our piece-wise function, which looked like a flat line at the beginning, turns out to be two different lines, with a discontinuous jump between them.
Having a jump or having holes (aka not being continuous) is the most egregious way a function can fail to be differentiable. Having “sharp” bends, like in |X| is a more common way. But the most interesting failure to be differentiable comes with fractal functions! The gif below is what it looks like to zoom in on the Weirestrass function (aka. The Monster).
Fractals are self-similar, which means that no matter how far you zoom in, you’ll just keep seeing the same structure, and the function can’t be approximated arbitrarily well with a linear version of itself. Pretty neat!
Visualizing Gauss-Jordan Elimination
A mantra of my current math studying is “find setting that allows for the best visualization, milk it for intuition, and project those learnings to the more abstract cases”. This video does an awesome job of showing what different algebraic manipulations of matrices are doing on a visual/geometric level.
Mechanical Computers
I found this amazing old navy video about how various computations needed for accurately firing anti-aircraft weapons are made my these mechanical apparati that computer different math functions. It’s a really interesting meditation on what math and computation really is.
(I found the video on the BetterExplained blog, a great source for all sorts of mathematical intuitions)
Gaining Clojure
I’m learning a new programming language! Clojure is my target, picked because I’ve got a friend that’s always lobbying for it, Paul Graham (look time Lisp proponent) recommends it, and it’s being used to build Roam. Clojure for the brave and true has been my 1st point of entry, accompanied by aggressive experimentation in the repl.
I’ve never done a Lisp before and am excited to see what magic comes from it. Dipped my toes into functional programming with SML in college. Now I’m hyped to try and learn enough to make something cool. I’ve already got schemes for a Chrome plugin I want to make, but details of that will be left for later.
Stay Lucky,
Hazard