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What I've learned about flow fields so far
Noise functions and the tiniest bit of trigonometry can take you a long way towards making art.
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Tog's Paradox
Tog’s Paradox (also known as The Complexity Paradox or Tog’s ComplexityParadox) is an observation that products aiming to simplify a task for users tend toinspire new, more complex tasks. It’s one ...
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An RNG that runs in your brain
Humans are notoriously bad at coming up with random numbers. I wanted to be able to quickly generate “random enough” numbers. I’m not looking for anything that great, I just want to be able to come up with the random digits in half a minute. Some looking around brought me to an old usenet post by George Marsaglia…
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Why Aren't We Sieve-Ing?
Long-time readers of this blog will know that I have mixed feelings about caches. One on hand, caching is critical to the performance of systems at every layer, from CPUs to storage to whole distributed architectures. On the other hand, caching being this critical means that designers need to carefully consider what…
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Is the frequency domain a real place?
Dissing Fourier, exploring the Hadamard-Walch transform, and making square-wave spectrograms.
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A practical introduction to constraint programming using CP-SAT and Python
We discuss constraint programming, an approach to solve discrete optimization problems, and illustrate how it works with a scheduling code example implemented in Python with the CP-SAT solver.
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Category Theory Illustrated: Logic (2021)
Now let’s talk about one more seemingly unrelated topic just so we can “surprise” ourselves when we realize it’s category theory. By the way, in this chapter there will be another surprise in addition to that, so don’t fall asleep.
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The zombie misconception of theoretical computer science
In Michael Sipser’s Introduction to the Theory of Computation textbook, he has one Platonically perfect homework exercise, so perfect that I can reconstruct it from memory despite not having …
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AlphaProof's Greatest Hits
Here I’ll try to explain the coolest ideas in each of AlphaProof’s IMO 2024 solutions. AlphaProof produces proofs in Lean, and each Lean proof is composed of a series of tactics. So I’ll pick out the tactics that correspond to these ideas in the proofs for problems 1, 2 and 6 (the three problems that AlphaProof…
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Counterintuitive Properties of High Dimensional Space (2018)
Our geometric intuition developed in our three-dimensional world often fails us in higher dimensions. Many properties of even simple objects, such as higher dimensional analogs of cubes and spheres, are very counterintuitive. Below we discuss just a few of these properties in an attempt to convey some of the weirdness…
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Finding near-duplicates with Jaccard similarity and MinHash
How do you find near-duplicates in a massive collection of documents? An exploration of the Jaccard similarity metric, and the MinHash hashing trick used to efficiently approximate it at web scale.
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Clean your codebase with basic information theory
Cut out everything that's not surprising.
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Doing is normally distributed, learning is log-normal
There are few things I think about more than the essays on gwern.net, and there are few with as satisfying a theoretical payout to contemplate in my orb as his essay on “leaky pipelines”, aka log-normal distributions. The skulk: Say you’re working on a Laravel web app. You’re about 90% sure you know how to start the…
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Hurried Thoughts: You're Wrong About Tidal-Locking
Space Engine ...
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What's Going on in Machine Learning? Some Minimal Models
Stephen Wolfram explores minimal models and their visualizations, aiming to explain the underneath functionality of neural nets and ultimately machine learning.
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Alice's adventures in a differentiable wonderland
A free, self-contained introduction to neural networks and differentiable programming, from automatic differentiation to transformers, with additional chapters on advanced topics.
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Collision Detection (2015)
An online book about collision detection using Processing.
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Physics for Mathematicians – Introduction
A blog about math by Nic Ford
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Building a data compression utility in Haskell using Huffman codes
In this post we will implement a data compression program in about 150 lines of Haskell. It will use Huffman coding and handle arbitrary binary files using constant memory for encoding and decoding.
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The Curious Case of MD5
Recently I came across a puzzling fact: the International Criminal Court hashes electronic evidence with MD5, even though MD5 is badly broken. So, why are lawyers using broken, outdated technology? The answer involves the common law system, cultural isolation, and a single man named Don L. Lewis.
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Using a Markov chain to generate readable nonsense with 20 lines of Python
Describes a simple Markov chain algorithm to generate reasonable-sounding but utterly nonsensical text, and presents some example outputs as well as a Python implementation.
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Convolutions, Fast Fourier Transform and polynomials (2022)
You may remember from high school what a polynomial is. If so, you may also remember how to multiply two of them. But what if I told you that the method you were taught is slow as F? In this post we will connect polynomials with the Fourier Transform and convolutions, and show you how to multiply polynomials with…
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Possibly all the ways to get loop-finding in graphs wrong
In my puzzle collection , there are many games in which you have to connect points together by edges, making a graph, and the puzzle rules say you must avoid making any loop in the graph. Examples are Net , Slant , and some configurations of Bridges (although not the default one). Loopy and Pearl also care about…
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Generating SVG for the prime knots
How I created a wall poster of knot diagrams.
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Poor Foundations in Geometric Algebra
EDIT: Be warned that this post is angry. If you’re easily offended when someone goes on a rant, then leave now because I’m very unforgiving in several places below. The tone of my writing is unusually harsh, but the people whose work I’m talking about deserve it, and here’s why. Even though I had been working with…
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Perceptually lossless (talking head) video compression at 22kbit/s
Achieving extreme video compression rates using the LivePortrait model for talking head videos.
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Math writing is dull when it neglects the human dimension
Note: These pages make extensive use of the latest XHTML and CSS Standards . They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in…
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Diffusion Is Spectral Autoregression
A deep dive into spectral analysis of diffusion models of images, revealing how they implicitly perform a form of autoregression in the frequency domain.