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Visualizing quaternions: An explorable video series
Explaining how quaternions, a four-dimensional number system, describe 3d rotation.
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 494 posts.
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Intuitive Understanding of Euler’s Formula
Euler's identity seems baffling: It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow…
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The StarCraft path-finding hack
StarCraft pathfinding looked impossible to fix until one ugly hack changed everything. Here's the trick that shipped.
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Recursive Functions of Symbolic Expressions and Their Computation by Machine
This paper appeared in Communications of the ACM in April 1960. It is the original paper on Lisp.
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Operational Transformation (algorithm behind Wave, Google Docs)
Almost exactly a year ago, Google made one of the most remarkable press releases in the Web 2.0 era. Of course, by "press release", I actually mean keynote at their own conference, and by "remarkable" I mean potentially-transformative and groundbreaking. I am referring of course to the announcement of Google Wave , a…
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Evolution Strategies as a Scalable Alternative to Reinforcement Learning
Stay up to speed on the rapid advancement of AI technology and the benefits it offers to humanity.
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Quines: self-replicating programs (2005)
A “ quine ” (or “ selfrep ”) is a computer program which prints its own listing. This may sound either impossible, or trivial, or completely uninteresting, depending on your temper and your knowledge of computer science. Actually, it is possible, and there are some interesting ideas involved (in particular, writing a…
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Discrete Analysis – A new type of math journal
This post is to announce the start of a new mathematics journal, to be called Discrete Analysis. While in most respects it will be just like any other journal, it will be unusual in one important w…
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An Intuitive (and Short) Explanation of Bayes’ Theorem
Bayes’ theorem was the subject of a detailed article. The essay is good, but over 15,000 words long — here’s the condensed version for Bayesian newcomers like…
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Surprisingly Turing-Complete
A catalogue of software constructs, languages, or APIs which are unexpectedly Turing-complete; implications for security and reliability.
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Google, D-Wave, and the case of the factor-10^8 speedup
Update (Dec. 16): If you’re still following this, please check out an important comment by Alex Selby, the discoverer of Selby’s algorithm, which I discussed in the post. Selby querie…
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How Shor's Algorithm works (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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Learning more about Claude's mathematical capabilities
An unreleased Claude model improved the lower bound for the fraction of zeroes of the Riemann zeta function that satisfy the hypothesis, raising it from 41.6% to 67.2%.
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Donut math - How donut.c works
There has been a sudden resurgence of interest in my "donut" code from 2006 , and I've had a couple requests to explain this one. It's been five years now, so it's not exactly fresh in my memory, so I will reconstruct it from scratch, in great detail, and hopefully get approximately the same result.
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Seemingly impossible functional programs (2007)
Andrej has invited me to write about certain surprising functional programs. The first program, due to Ulrich Berger (1990), performs exhaustive search over the “ Cantor space ” of infinite sequences of binary digits. I have included references at the end. A weak form of exhaustive search amounts to checking whether…
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Zero Knowledge Proofs: An illustrated primer (2014)
One of the best things about modern cryptography is the beautiful terminology. You could start any number of punk bands (or Tumblrs) named after cryptography terms like ‘hard-core predicate&#…
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Magic: The Gathering Is Turing Complete (v5)
I'm assuming you're familiar with the basic idea of what a Turing machine is . There are a few parts to building one in Magic: a tape extending in two directions storing cells of a few different "colours"; a processing head that can read the current cell's colour, write a new colour and change its state if necessary…
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Colorized Math Equations
Years ago, while working on an explanation of the Fourier Transform, I found this diagram: Most ideas aren't inherently confusing, but their technical…
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Do C compilers disprove Fermat's Last Theorem?
(You probably already have realized this.) The reason the compiler produced code that terminated was based upon the fact that the values being altered (i.e., a, b and c) within the “while” loop are never referenced again after the loop. But, once you added code which printed the resulting values it was forced to…
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Efficiently Generating a Number in a Range
The vast majority of my posts about random number generation have focused on looking at the properties of different generation schemes. But, perhaps surprisingly, the performance of your randomized a
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The Aggregate Magic Algorithms (2015)
There are lots of people and places that create and collect algorithms of all types ( here are a few WWW sites). Unfortunately, in building systems hardware and software, we in The Aggregate often have found it necessary to do relatively obscure low-level things very efficiently. Many of the tricks we've devised or…
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Why FTL implies time travel
In science fiction, it is pretty standard fare to introduce some form of faster-than-light communication or travel. After all, space is big, and you can't write your swashbuckling Hornblower-in-space novel if you have to wait for a generation ship to crawl painfully slowly between the nearest stars,
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P.S.: The Best of Intentions
Operations in Lisp, Scheme, and other dynamically-typed languages typically dispatch on representational type information rather than intentional type information. Several broad classes of bugs and confusions can be traced to improper attempts to recover intentional type information from representation types.
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The Relativistic Rocket
Thanks to Alan R. Wilson for correcting a two-word typo involving GPS satellites. Thanks to Aron Yoffe for suggesting the newtonian comparison. Thanks to Tom Fuchs for suggesting the analogy to the stone thrown upwards. Fuel numbers added by Don Koks, 2004. Updated by Phil Gibbs, 1998. Thanks to Bill Woods for…
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Suspicious Discontinuities
If you read any personal finance forums late last year, there's a decent chance you ran across a question from someone who was desperately trying to lose money before the end of the year. There are a number of ways someone could do this; one commonly suggested scheme was to buy put options that were expected to expire…
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