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Knuth changes his mind on Bernoulli number B_1
(I thank Yoshiyuki Kotani for this identity. We'll be able to use the same idea again in 2024 and 2027!)
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A high-dimensional sphere spilling out of a high-dimensional cube
Make a square, split each side into two halves, producing four cells. Put a circle into each cell such that it fills it completely. There is a small gap right in the middle of the square. Put a circle there again such that it touches the other four circles. The central circle is obviously inside the square, right?…
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Algorithms you should know before you take system design interviews
I put together a list and explained why they are important.
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Exponentially Better Rotations
If you’ve done any 3D programming, you’ve likely encountered the zoo of techniques and representations used when working with 3D rotations. Some of them are better than others, depending on the situation.
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Signed distance functions in 46 lines of Python
A walkthrough of 46 lines of code that render a 3D ASCII donut using signed distance functions.
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On AlphaTensor’s new matrix multiplication algorithms
Two acquaintances independently asked about this today, so it seems worth a write-up: recently (as of this writing), DeepMind published a new paper about a new practical fast matrix multiplication …
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Demystifying Fourier analysis
I've started learning about spectrum analysis a long time ago at university, but failed to gain a true insight in how it works under the hood. For me, it was just some magical math formulas I had to memorize and know about. I knew the theory but it didn't really click .
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How to get rid of gerrymandering: the math is surprising
Gerrymandering distorts the will of the people, but the basic math of geographic representation makes it difficult to get rid of, unless we get rid of representation for geographic districts.
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The claimed effect size is about a zillion times higher than is plausible
Some years ago, you blogged about a research article by Hasan and colleagues (2013) .
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Algorithms to Live By – The Computer Science of Human Decisions
What does the math tell us about how many job applicants we should look at before hiring one? While onboarding our new employees, how can ideas from the TCP networking protocol help us to identify the optimal workload for them? Why would giving employees unlimited vacation days most likely lead to less vacation being…
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The Kelly criterion: How to size bets (2019)
Let’s play a game. I have a coin that lands heads 60% of the time and tails 40% of the time. You have $25 and can bet on either side of the coin — every time you’re right you double your bet, and every time you are wrong you lose it.
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The big six matrix factorizations
Six matrix factorizations dominate in numerical linear algebra and matrix analysis: for most purposes one of them is sufficient for the task at hand. We summarize them here. For each factorization …
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Is the Kalman filter just a low-pass filter?
Exploring the relationship between Kalman filtering and optimal frequency domain filtering.
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Implementing Cosine in C from Scratch (2020)
How does the math library's trigonometric function work?
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What is an eigenvalue?
An eigenvalue of a square matrix $LATEX A$ is a scalar $latex \lambda$ such that $latex Ax = \lambda x$ for some nonzero vector $latex x$. The vector $latex x$ is an eigenvector of $LATEX A$ and it…
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Math on GitHub: The Good, the Bad and the Ugly
Finally! On May 19, 2022, GitHub released one of its most requested features: Math support on Markdown! Many had been anticipating this for years, including myself, so much in fact that I wrote a browser extension to get there early. (xhub, it can do a few other things so check it out.)
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A 9-line summary of textbook physics
All their wonders in one table
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Wordle-solving state of the art: all optimality results so far
Almost all mathematical questions one could have about Wordle are settled by now. I summarize here what is known so far.
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What do new Sudoku techniques teach us about real-world problem solving?
What do revolutionary new Sudoku techniques teach us about real-world problem solving?
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Finding Waldo in π
The SMBC comic Convert included as the “Votey button” a challenge to extract a Waldo visual from the digits of π.
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Markov chain Monte Carlo without the bullshit (2015)
I have a little secret: I don’t like the terminology, notation, and style of writing in statistics. I find it unnecessarily complicated. This shows up when trying to read about Markov Chain Monte Carlo methods. Take, for example, the abstract to the Markov Chain Monte Carlo article in the Encyclopedia of…
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Fast midpoint between two integers without overflow
Let us say that I ask you to find the number I am thinking about between -1000 and 1000, by repeatedly guessing a number. With each guess, I tell you whether your guess is correct, smaller or larger than my number. A binary search algorithm tries to find a value in an interval by repeating … Continue reading Fast…
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29-year-old Conway conjecture settled
Ilkka Törmä and Ville Salo, a pair of researchers at the University of Turku in Finland, have found a finite configuration in Conway’s Game of Life such that, if it occurs within a universe a…
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Parallel Curves of Cubic Béziers
Blog of Raph Levien.
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What is the inverse of a circle?
In this post we’ll try to make sense of what we’re seeing, and try to understand what it means to invert a circle!
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Markov Chains for programmers
Materials for book: “Markov Chains for programmers”