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Category Theory for Programmers (2014)
Table of Contents Part One Category: The Essence of Composition Types and Functions Categories Great and Small Kleisli Categories Products and Coproducts Simple Algebraic Data Types Functors Functo…
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 38 posts.
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A Primer on Bézier Curves
A detailed explanation of Bézier curves, and how to do the many things that we commonly want to do with them.
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Understanding LSTM Networks (2015)
Humans don’t start their thinking from scratch every second. As you read this essay, you understand each word based on your understanding of previous words. You don’t throw everything away and start thinking from scratch again. Your thoughts have persistence.
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The Patriot Missile Failure
On February 25, 1991, during the Gulf War, an American Patriot Missile battery in Dharan, Saudi Arabia, failed to track and intercept an incoming Iraqi Scud missile. The Scud struck an American Army barracks, killing 28 soldiers and injuring around 100 other people. A report of the General Accounting office…
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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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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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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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Down with Determinants (1995)
This paper was published in the American Mathematical Monthly 102 (1995), 139-154.
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I Wrote My Fastest Hashtable
I had to get there eventually. I had a blog post called “I Wrote a Fast Hashtable” and another blog post called “I Wrote a Faster Hashtable.” Now I finally wrote the fastest…
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Algorithmic Solution to My Insomnia
I’ve struggled with insomnia for all of my adult life. It began in college and has waxed and waned in severity ever since, correlating with stress levels…
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The ABC conjecture has still not been proved
Five years ago, Cathy O’Neil laid out a perfectly cogent case for why the (at that point recent) claims by Shinichi Mochizuki should not (yet) be regarded as constituting a proof of the ABC conjecture. I have nothing further to add on the sociological aspects of mathematics discussed in that post, but I just wanted to…
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Floating Point Visually Explained
While I was writing the Wolfenstein 3D book [1] , I wanted to demonstrate how much of a handicap it was to work without floating points. My attempts at understanding floating points using canonical [2] articles [3] were met with resistance from my brain. I tried to find a different way. Something far from $(-1)^S *…
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Principal Component Analysis Explained Visually (2015)
Principal component analysis (PCA) is a technique used to emphasize variation and bring out strong patterns in a dataset. It's often used to make data easy to explore and visualize.
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TreeMaker: Origami Design Software
In 1989, I wrote an article for the magazine Engineering & Science about the state of technical folding, which, even then, seemed to be progressing by leaps and bounds due to an infusion of scientific and mathematical principles. In recounting some of the connections between origami, math, and technology, I wrote:
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Optimizing things in the USSR (2016)
A review of Red Plenty, by Francis Spufford
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Three-Valued Logic
Making decision despite missing SQL data (NULL): How to avoid OR…IS NULL in the WHERE clause.
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Insert D-Wave Post Here
In the two months since I last blogged, the US has continued its descent into madness. Yet even while so many certainties have proven ephemeral as the morning dew—the US’…
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Deep image prior 'learns' on just one image
Deep convolutional networks have become a popular tool for image generation and restoration. Generally, their excellent performance is imputed to their ability to learn realistic image priors from a large number of example images. In this paper, we show that, on the contrary, the structure of a generator network is…
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Reverse Engineering the Hacker News Ranking Algorithm
A data-driven exploration of how the Hacker News ranking algorithm works.
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Understanding the math behind 0x5f3759df and the fast inverse square root (2012)
This post is about the magic constant 0x5f3759df and an extremely neat hack, fast inverse square root, which is where the constant comes from.
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How many floating-point numbers are in the interval [0,1]?
Most commodity processors support single-precision IEEE 754 floating-point numbers. Though they are ubiquitous, they are often misunderstood. One of my readers left a comment suggesting that picking an integer in [0,232) at random and dividing it by 232, was equivalent to picking a number at random in [0,1). I am not…
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The Amazing Power of Word Vectors (2016)
For today’s post, I’ve drawn material not just from one paper, but from five! The subject matter is ‘word2vec’ – the work of Mikolov et al. at Google on efficient vect…
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The Scientific Case for P≠NP
Out there in the wider world—OK, OK, among Luboš Motl, and a few others who comment on this blog—there appears to be a widespread opinion that P≠NP is just “a fashionable dogma of…
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Oh My Gosh, It’s Covered in Rule 30s
New British train station design identified as cellular automata rule 30. Generating rule-based patterns. And a website that lets you try it yourself.
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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 …