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A Fast 64-Bit Date Algorithm (30–40% faster by counting dates backwards)
Counting years backwards unlocks faster speed. The first date algorithm to use only 4 multiplications, instead of 7+
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The latest blog posts to take off on Hacker News. 2,391 posts.
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Set theory with types
It is known that mathematics is heavily reliant on set theory, but no one can agree on what set theory is. Many people today understand that we have a choice between set theory and type theory , but they don’t know what type theory is either. Many think that type theory refers to some sort of dependent type theory, as…
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Is Matrix Multiplication Ugly?
A few weeks ago I was minding my own business, peacefully reading a well-written and informative article about artificial intelligence, when I was ambushed by a passage in the article that aroused …
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Solving Fizz Buzz with Cosines
Fizz Buzz is a counting game that has become oddly popular in the world of computer programming as a simple test of basic programming skills. The rules of the game are straightforward. Players say the numbers aloud in order beginning with one. Whenever a number is divisible by 3, they say 'Fizz' instead. If it is…
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210 IQ Is Not Enough
You will never have enough IQ, nor money, nor time, nor energy.
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How Slide Rules Work
The survival of our species owes much to our brain, specifically, its ability to observe, analyse, and plan. Planting crops and storing grains for the winter were some of the earliest uses of these abilities. Measuring and calculating are foundational elements of observation, analysis, and planning. Computation, upon…
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Weighting an average to minimize variance
How would you allocate money to a set of independent assets with to minimize volatility if some assets are more volatile than others?
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Linear algebra explains why some words are effectively untranslatable
A modest mathematical framing of language
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IBM Patented Euler's 200 Year Old Math Technique for 'AI Interpretability'
IBM Slapped the Buzzwords 'AI Interpretability' on Generalized Continued Fractions and their Series Transformations and was awarded a Patent
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Hilbert space: Treating functions as vectors
The tools of linear algebra are extremely useful when working in Euclidean space (e.g. \mathbb{R}^3 ). Wouldn’t it be great if we could apply these tools to additional mathematical constructs, such as functions and sequences? Hilbert space allows us to do exactly this - apply linear algebra to functions.
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Cache-friendly, low-memory Lanczos algorithm in Rust
How a bit of algorithm engineering and low-level details can alter what seems like a straightforward trade-off on a blackboard.
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Think in math, write in code (2019)
Programmers love to discuss programming languages. We not only debate their technical merits and aesthetic qualities, but they become integrated into our personal identities, along with the values and traits that we associate with them. Some even defend a form of Linguistic Determinism that thinking is confined to…
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Mathematical exploration and discovery at scale
Bogdan Georgiev, Javier Gómez-Serrano, Adam Zsolt Wagner, and I have uploaded to the arXiv our paper “Mathematical exploration and discovery at scale”. This is a longer report on…
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Fourier Transforms
Fourier Transform
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The shadows lurking in the equations
If you've ever graphed equations, chances are you have only ever graphed equations in what I refer to as "Binary mode" - which draws a line where the equation is EXACTLY equal, and leaves white everywhere else.
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Why don't you use dependent types?
To be fair, nobody asks me this exact question. But people have regularly asked why Isabelle dispenses with proof objects. The two questions are essentially the same, because proof objects are intrinsic to all the usual type theories. They are also completely unnecessary and a huge waste of space. As described in an…
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Why Busy Beaver hunters fear the Antihydra
In which I explore the biggest barrier in the busy beaver game. What is Antihydra, what is the Collatz conjecture, how are they connected, and what makes them so daunting?
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987654321 / 123456789
The ratio 987654321 / 123456789 is very nearly 8. What about the analogous ratio in other bases?
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How to make a Smith chart
Deriving the basic mathematical properties of the Smith chart from electrical engineering
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Programming with Less Than Nothing
Lambda calculus is for wimps. Real programmers use combinatory logic.
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SATisfying Solutions to Difficult Problems
This post covers the same material as my !!Con 2024 talk , for which the slides are here .
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Why formalize mathematics – more than catching errors
I read a good post by one of the authors of the Isabelle theorem prover, that got me thinking. The author, Lawrence Paulson, observed that most math proofs are trivial, but writing them (preferably with a proof assistant) is a worthwhile activity, for reasons similar to safety checklists - “Not every obvious statement…
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Picturing Mathematics
I’m a great believer in low-tech math. I don’t like to rely on things a computer tells me; what if there’s a bug in the code? I prefer trusting things that I can check for myself. At the same…
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Fast calculation of the distance to cubic Bezier curves on the GPU
Bézier curves are a core building block of text and 2D shapes rendering. There are several approaches to rendering them, but one especially challenging problem, both mathematically and technically, is computing the distance to a Bézier curve. For quadratic curves (one control point), this is fairly accessible, but for…
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Lace: A New Kind of Cellular Automata Where Links Matter
This article is about a new kind of simple computational rule ("LACE rules" running on LACE, the Link Automata Computing Engine platform) which, when applied locally on a grid of cells, demonstrates fascinating emergent "artificial life" behavior.
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The Unknotting Number Is Not Additive
On June 30, 2025, Mark Brittenham and Susan Hermiller uploaded a preprint to the arXiv called “Unknotting number is not additive under connected sum” (and an updated version on Septembe…
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Less is more: Recursive reasoning with tiny networks
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