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987654321 / 123456789
The ratio 987654321 / 123456789 is very nearly 8. What about the analogous ratio in other bases?
Top posts
Every blog post that took off on Hacker News, ranked by the votes of its best submission. 2,391 posts.
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AI isn’t outthinking mathematicians, it’s out-remembering them
The key advantage may not be superior reasoning, but a virtually unlimited symbolic working memory.
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Let's remove quaternions from every 3D engine (2018)
To represent 3D rotations graphics programmers use Quaternions . However, Quaternions are taught at face value . We just accept their odd multiplication tables and other arcane definitions and use them as black boxes that rotate vectors in the ways we want. Why does $\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=-1$ and…
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The Dunning-Kruger effect is autocorrelation
Do unskilled people actually underestimate their incompetence?
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An Interesting Pattern in the Prime Numbers: Parallax Compression
Early this year a software engineer, Shaun Gilchrist, reached out to me after reading a blog post of mine from many years ago, about my informal search for hidden patterns in the prime numbers. The Ulam Spiral revealed non-random patterns, but they didn’t quite match up. Both Shaun and I had long felt there was […]
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The maths you need to start understanding LLMs
A quick refresher on the maths behind LLMs: vectors, matrices, projections, embeddings, logits and softmax.
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The Matrix Calculus You Need for Deep Learning
Most of us last saw calculus in school, but derivatives are a critical part of machine learning, particularly deep neural networks, which are trained by optimizing a loss function. This article is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We…
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XOR
Recently I was called on to explain the ‘XOR’ operator to somebody who vaguely knew of its existence, but didn’t have a good intuition for what it was useful for and how it behaved.
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An animated introduction to Fourier series
This article will be part of an extensive series in which I plan to explore various aspects of Fourier Mathematics. I will take notes, create some visuals (a good pretext to learn more about graphics), and hope that it will be helpful to anyone looking for a clear, visual introduction to these concepts. The article…
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Mihai Pătraşcu 1982–2012
A pathbreaking researcher taken too soon from us (src) Mihai Pătraşcu passed away on Tuesday of brain cancer, having not yet reached age 30. We, Dick and Ken, did not know about his year-…
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Succinct data structures
An overview of a class of data structures I didn't know about!
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Why I Don't Love Gödel, Escher, Bach
Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid is a classic text that's had a strong influence on countle...
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Estimating the chances of something that hasn’t happened yet
A handy rule of thumb for creating a confidence interval for the probability of an event you haven't seen happen yet.
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4D Toys: a box of four-dimensional toys
So I have been working on Miegakure for a long time now, and I have created and accumulated many cool 4D things of all sorts. I think it’s about time that I share some of them, so…
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I don’t understand Graph Theory
Graph theory represents one of the most important and interesting areas in computer science. But at the same time it’s one of the most…
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Reverse engineering a neural network's clever solution to binary addition
While training small neural networks to perform binary addition, a surprising solution emerged that allows the network to solve the problem very effectively. This post explores the mechanism behind that solution and how it relates to analog electronics.
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Mitchell Feigenbaum (1944–2019), 4.66920160910299067185320382…
Stephen Wolfram shares his memories of mathematical physicist Mitchell Feigenbaum. Also a detailed discussion of his work and big discovery of a universal constant for functions approaching chaos via period doubling.
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An intuitive guide to Maxwell's equations (2020)
An Intuitive and Visual Guide to Maxwell's Equations
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Visual Information Theory (2015)
I love the feeling of having a new way to think about the world. I especially love when there’s some vague idea that gets formalized into a concrete concept. Information theory is a prime example of this.
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Double-entry bookkeeping as a directed graph
In the past couple of years, I’ve been working in billing and payments at Justworks . This experience introduced me to the world of bookkeeping and accounting in ways I didn’t expect: I took a course and read a textbook in accounting, adopted plain text accounting for my personal finances, and worked in a double-entry…
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The Kolmogorov option
Andrey Nikolaevich Kolmogorov was one of the giants of 20th-century mathematics. I’ve always found it amazing that the same man was responsible both for establishing the foundations of …
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Reservoir Sampling
Choosing fairly when you don't know how many you're choosing from.
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GPU Fluid
Benedikt Bitterli's Portfolio
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Eigenvectors from eigenvalues
Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues”. This note gives two proofs of a general eige…
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Look, ma, no matrices
Since the 2019 SIGGRAPH course [1], Geometric Algebra, and Euclidean PGA (plane-based or projective geometric algebra) in particular, has been gaining traction within the computer graphics and machine learning communities [2, 3, 4]. Despite its broad applicability, including for higher dimensional geometry and…