Algorithms & math trivia
A common theme on this blog are write-ups about various mathematical curiosities. Although these are covered in countless other placed on the internet, I try to do go above and beyond: instead of regurgitating formulas, I research the “why” and explain it in a way that should click even if you skipped the calculus class.
Infinity and computability
I’m especially fond of a four-piece anthology of articles about infinity. It kicks off with the well-known 0.999… = 1 equality as an excuse to explore the meaning of the symbol “∞” and some alternative arithmetic systems we could use:
This is followed by a more general article that looks at the construction of natural numbers from the principles of formal logic. The article shows that this approach naturally lends itself to the development of infinite quantities with some unexpected properties:
The third article zeroes in on the intersection of real numbers and the theory of computability. It explains how real numbers can be constructed and why they’re more unusual than it may seem:
The final piece explores the limits of computational knowledge, including an outline of Gödel’s incompleteness theorems and their relationship to busy beavers:
I think these articles are exceptionally accessible, but they get relatively few views. If you end up reading any of them, I’d love to hear your thoughts.
Math for engineers
If you’re working with software or electronic circuits, you’ll sooner or later need to make sense of an algorithm known as the discrete Fourier transform. I think it’s one of the worst-explained concepts in computer science, so here’s my attempt to do it right:
Relatedly — does the Fourier transform reveal some deeper truth about reality, or are there completely different frequency domains that play by their own rules? I explore this in the following article:
Another piece of math many engineers can’t escape are complex numbers. They’re not hard to grasp, but most introductory texts don’t really provide a good motivation for why they exist. I approach this question from geometry:
If you liked the introduction to complex numbers, you might also get a kick out of deriving their less useful, higher-dimensional equivalent:
Last but not least, in the context of complex numbers, many textbooks wheel out a somewhat inexplicable equation known as Euler’s identity. The following article sheds some light on what the equation means:
Mathematical trivia
I also have a number of more whimsical articles that deal with recreational mathematics. For example, have you ever wondered what does it mean that a shape has n dimensions, and how to actually measure it?
If you liked that, you might also enjoy a more lighthearted article about the surprising finding that our 𝜋 is better than their 𝜋:
Are rational numbers difficult to approximate? Find out here:
Next, if you enjoy geometry or computer graphics, you might have a blast trying to visualize the hypercube:
If you’re a nerd, you might have seen an old troll proof that 𝜋 = 4. Most YouTube explanations of why the proof is wrong are… not great. Here’s my take:
Almost every geek knows about fractals, but it’s sometimes hard to understand how they come to be. The following article looks into why a familiar fractal pops up as a consequence of a simple bit-twiddling hack:
It’s a lot harder to explain the Mandelbrot fractal, but there’s a number of interesting observations we can make about its structure:
Computer algorithm fun
Have you wondered if blurring can be reversed without serious math? The answer is kinda, sorta, maybe yes:
Oh — and have you ever wondered if it’s possible to smuggle goods in the frequency domain?
Last but not least — what happens if you pitch-shift an image or blur an audio track?
This is just a sampling; for more, visit this page. And if you have suggestions or other feedback, you can reach me at lcamtuf@coredump.cx.