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Knuth's Art of Computer Programming, V 4B, has gone into print
(I'm still looking for the catchiest way to appreciate the number MMXXVI. Exercise 7.2.1.6–122 gives 1234+.5*6+789 and -1+2345*6/7+8+9 and 1+(2/3)*45*67.8-9 ; other suggestions are welcome. The most elegant that I've seen so far is a beauty that was suggested independently by Arnaud Lefebvre and by Evangelos…
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Turns are better than radians
Switching away from radians makes code simpler, faster, and more precise.
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The fastest GIF does not exist
Some games on a website
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I can't let go of “The Dunning-Kruger Effect is Autocorrelation”
Organizing my thoughts regarding an argument about the Dunning-Kruger study and statistics in general
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An update on the campaign to defend serious math education in California
Update (April 27): Boaz Barak—Harvard CS professor, longtime friend-of-the-blog, and coauthor of my previous guest post on this topic—has just written an awesome FAQ, providing his pers…
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Someone’s Been Messing with My Subnormals
TL;DR: After noticing an annoying warning, I went on an absurd yak shave, and discovered that because of a tiny handful of Python packages b...
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You need to know what right-half-plane zeros are
Why some systems -- as a consequence of their intrinsic dynamics -- require a control action that first pushes them the wrong way and makes things worse before it gets better.
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Challenging algorithms and data structures every programmer should try
Bored of algorithms? Here are my recommendations that are exciting, understandable, and sometimes even useful!
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SHA-3 Buffer Overflow
Update: This result appeared in the proceedings of CT-RSA 2023. A preprint is available.
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Why do arrays start at 0?
It's not the reason you think. No, it's not that reason either.
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How does perspective work in pictures?
Why don’t photographs make distant objects big enough?
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“Burning Ship” fractal
The images described here are generated by analysing the following series for each point (c x ,c y ) on the plane.
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A neat XOR trick
Looking over the solutions mega-thread , it seems like not many people discovered this trick; when I posted about it , people found it to be noteworthy.
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Visual Sum of Cubes
This article discusses a ‘visual’ derivation of the formula for 1³+2³+…+n³.
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CRDT: Fractional Indexing
Collaborative peer-to-peer applications sometimes need to operate on sequences of objects with a consistent order across all peers. For example, a peer-to-peer photo album application might need to sync the order in which photos appear in an album.
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Seven sins of numerical linear algebra
In numerical linear algebra we are concerned with solving linear algebra problems accurately and efficiently and understanding the sensitivity of the problems to perturbations. We describe seven si…
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The Color of Infinite Temperature
This is the color of something infinitely hot. Of course you’d instantly be fried by gamma rays of arbitrarily high frequency, but this would be its spectrum in the visible range. This…
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Demystifying financial leverage
Leverage is actually reasonably easy to understand, both in the math and in the implications for financial firms.
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Shor, I’ll do it (2007)
I’ve been talking a lot recently about how quantum algorithms don’t work. But last week JR Minkel, an editor at Scientific American, asked me to write a brief essay about how quantum al…
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Masterclass on mathematical thinking
About a year ago, I was contacted by Masterclass (a subscription-based online education company) on the possibility of producing a series of classes with the premise of explaining mathematical ways…
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Why isn't there a replication crisis in math?
The replication crisis is a major problem in medicine and social science; we know that a huge fraction of the published literature is outright wrong. But in math we don't seem to have a similar crisis, despite reasonably frequent minor errors in published papers. Why not, and what can this tell us about the fields…
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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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