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MathCode, Mathematical Coding Agent
A terminal AI coding agent that formalizes math problems into Lean 4 theorems and proves them.
182
One night in Uzbekistan: Why was this one data point so influential?
We have a comment coming out in Nature next week that is going to cause the retraction of a high-profile paper by Kotz et al. from last year (the second most cited climate paper in the news in 2024).
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Category Theory Illustrated – Types
In this chapter we will talk about types. This might be disappointing for you, if you expected to learn about as many new categories as possible (which you don’t even suspect are really categories till the unexpected reveal)—we’ve been talking about the category of types in a given programming language ever since the…
184
Cosmology with Geometry Nodes
Learn about the real-life scientific applications of GeoNodes in Cosmology research.
185
Six (and a half) intuitions for KL divergence
KL-divergence is a topic which crops up in a ton of different places in information theory and machine learning, so it's important to understand well. Unfortunately, it has some properties which seem confusing at a first pass (e.g. it isn't symmetric like we would expect from most distance measures, and it can be…
186
Egyptian Fractions (2006)
From the highly eclectic blog of Mark Dominus
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(Post-Quantum) Cryptographic Right Answers
In the tradition started by Colin Percival and Thomas Ptacek, a set of “Cryptographic Right Answers” updated for the accelerated post-quantum transition timeline.
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The lattice of sets of natural numbers is rich (2021)
$\newcommand\of{\subseteq}\newcommand\N{\mathbb{N}}\newcommand\R{\mathbb{R}}\def\<#1>{\left\langle#1\right\rangle}\newcommand\Z{\mathbb{Z}}\newcommand\Q{\mathbb{Q}}\newcommand\ltomega{{{<}…
189
Squares in Squares
The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). If a pictured packing has multiple numbers in its label above, the picture represents the largest; each smaller is represented by removing any square. For the $n ≤ 324$ not pictured, the trivial packing (with no…
190
Truth is not a direction: a Tarski attack on LLM probes
Abel Jansma is a scientist studying emergence, higher-order interactions, information theory, and complex systems across biology and artificial intelligence.
191
Mathematics Distillation Challenge – Equational Theories
Mathematical research traditionally involves a small number of professional mathematicians working closely on difficult problems. However, I have long believed that there is a complementary way to …
192
The missing digit of Stela C
One bad thing about archeologists is that some of the successful ones get a big head. People used to think the Olmecs, who made these colossal stone heads, were contemporary with the Mayans. But in…
193
What it takes to transpose a matrix
Classical CPU architecture is a poor choice for performing matrix-oriented computations. Developers have to spend much effort to create efficient algorithms even for the problems appearing trivial on the surface. Matrix transpose is probably the most striking representative of such problems. It exposes numerous…
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Illustration Beyond Proof
[This is a guest post by Kate Stange and Edmund Harriss. This blog post was initially written in a different file format and converted using AI. — T.] In January a group of mathematicians, hi…
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A Mathematical Framework for Transformer Circuits (2021)
Transformer language models are an emerging technology that is gaining increasingly broad real-world use, for example in systems like GPT-3 , LaMDA , Codex , Meena , Gopher , and similar models. However, as these models scale, their open-endedness and high capacity creates an increasing scope for unexpected and…
196
There Is(Ǝ) – Such That (∋)
Let me tell you about a language. But if you wish to go play instead go here. It's going to get a little abstract below. There is a Canvas to Compose Upon The canvas is a square with the largest circle that has a radius of ρ. There are
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Efficient algorithms, the Exponential Time Hypothesis, and losing the plot
On efficient algorithms, the exponential time hypothesis, and losing the plot.
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Fast and Easy Levenshtein distance using a Trie (2011)
<img src="http://stevehanov.ca/blog/images/acer.jpg"/> If you have a web site with a search function, you will rapidly realize that most mortals are terrible typists. Many searches contain mispelled words, and users will expect these searches to magically work. This magic is often done using levenshtein distance. In…
199
Chess invariants
Chess is a lot trickier than it looks. It has so many rules: castling, en passant, pawn promotion, pinning, the discovered check, and the de...
200
You gotta think outside the hypercube
A closer look at the tesseract and the ways we can render it on the screen.
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Floating point from scratch: Hard Mode
Actually building floating point from scratch!
202
Jacobian Conjecture for Baby
An illustrated guide that starts from the basics, explains the Jacobian Conjecture, and follows what happened to it in July 2026.
203
Side-stepping the Secretary Problem, unwittingly
Having played both parts in the kabuki play that is employee-employer matchmaking, I feel the way we play it is a zero-sum game. I wish it were not so. When this post started life in 2024, as a wall of text chat message, it was brutal out there, on both sides of the software industry interview table. The ZIRP had…
204
Analytic Fog Rendering with Volumetric Primitives (2025)
In this post I’m going to cover a neat technique for rendering regions of fog with varying density. I’ll start by covering some of the basic principles behind fog rendering and a few co…