The latest blog posts to take off on Hacker News. 266 posts.
Mazes are typically constructed on regular grids: square, triangular, hexagonal, etc. But what if you generate a maze on top of a Penrose tiling, which is guaranteed to be irregular? (It is an aperiodic tiling .) The resulting mazes have a very characteristic appearance: they are filled with circles and stars, but the…
So the big news this week is that o3, OpenAI’s new language model, got 25% on FrontierMath. Let’s start by explaining what this means.
Division is quite an expensive operation. For instance, latency of the 32-bit division varies between 10 and 15 cycles on the Cannon Lake CPU, and for Zen4 this range is from 9 to 14 cycles. The latency of 32-bit multiplication is 3 or 4 cycles on both CPU models.
if $n \equiv 0 \, (\text{mod}\, 2)$ the ant turns 90º clockwise, else the ant turns 90º counter-clockwise. On both accounts, the state of the cell is flipped and the ant moves forward one unit. This is repeated until $n = 1$.
An interactive introduction to Turing machines, the theoretical foundation of modern computing.
3D visualization tool of spherical harmonics. Visualize and compare real, imaginary, and complex components by adjusting the degree l and order m parameters.
Introduction You may have heard of the Kelly bet allocation strategy. It is a system for correctly exploiting information or bias in a gambling situation. It is also known as a maximally aggressive…
An article about music and geometry, in particular about musical intervals and scales and their geometric equivalent ...
2 model 2 dreidel
When I started learning about loss functions, I could always understand the intuition behind them. For example, the mean squared error (MSE) for regression seemed logical—penalizing large deviations from the ground-truth makes sense. But one thing always bothered me: I could never come up with those loss functions on…
A visual proof that a2 – b2 = (a + b)(a – b). Sophie Germain wrote, “It has been said that algebra is but written geometry and geometry is but diagrammatic algebra.”
February 24, 2020. The computational power of a humble napkin is awesome. I discuss three napkin algorithms — dimensional analysis, Fermi estimates, and random walks — and use them to figure out why rain falls, the length of the E. coli genome, and the mass of a proton, among other things. These examples suggest a…
So I’m two months into trying to teach a proof of Fermat’s Last Theorem to a computer. We already have one interesting story, which I felt was worth sharing.
I’ve been on a weird kick lately making images using Excel. [Here’s one post. I hope to post more soon.] If you add a background color to each cell in a spreadsheet and resize the cell …
Alok Singh's Blog
A brief numerical and graphical check on a 3Blue1Brown video.
where my words occasionally escape /dev/null
Follow me on Bluesky! A Log-Structured Merge tree, or LSM, is a popular data structure for storage engines. It’s what is used by RocksDB, which is sort of...
You might think it’s unlikely for any interesting mathematics to arise from incense appreciation, but that’s only because you’re unfamiliar with the peculiar character of Muromachi (室町) era Japanese nobles. There has never been a group of people, in any time or place, who were so driven to display their sophistication…
The deficiencies of the Unit Sphere and an exploration of Lie groups.
Database queries are a pretty surprisingly powerful tool that can solve seemingly intractable problems.
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…
Yesterday John Graham-Cumming posted about “Steve Ballmer’s incorrect binary search interview question,” and in the Hacker News discussion, among the predictable “interview culture is flawed!!” complaints, there was some discussion of whether binary search was even the right way to solve Ballmer’s puzzle. Here’s a…
The underpinning of information theory is Shannon’s formula for the entropy H of a discrete probability distribution P. Derived in 1948 in the paper “A mathematical theory of communication”, this is given by \[ H[P] = -\sum_{i=1}^N p_i \log p_i \] where \( p_i = p(x_i) \) is the probability of sampling \( x_i \) from…
