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Why numbering should start at zero (1982)
To denote the subsequence of natural numbers 2, 3, ..., 12 without the pernicious three dots, four conventions are open to us
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 27 posts.
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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…
3
Chesterton’s Fence: A Lesson in Second Order Thinking
A core component of making great decisions is understanding previous decisions. If we don’t understand how we got “here,” we run the risk of making things much worse.
4
There are only four billion floats, so test them all (2014)
A few months ago I saw a blog post touting fancy new SSE3 functions for implementing vector floor, ceil, and round functions. There was the inevitable proud proclaiming of impressive performance an…
5
An Intuitive Guide to Linear Algebra (2012)
Despite two linear algebra classes, my knowledge consisted of "Matrices, determinants, eigen something something". Why? Well, let's try this course format:
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An Interactive Introduction to Fourier Transforms
Fourier transforms are a tool used in a whole bunch of different things. This is a explanation of what a Fourier transform does, and some different ways it can be useful.
7
Division by zero in type theory: a FAQ
How can a theorem prover use the convention that 1/0=0 and still be consistent?
8
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…
9
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…
10
Programming with Categories
In this course we explain how category theory—a branch of mathematics known for its ability to organize the key abstractions that structure much of the mathematical universe—has become useful for writing elegant and maintainable code. In particular, we'll use examples from the Haskell programming language to motivate…
11
Lambda Calculus Diagrams (2015)
The following table shows diagrams of identity, the booleans, some standard combinators, some Church numerals, the predecessor function on Church numerals, and Omega. term definition diagram alternative I/1 λx.x K/true λx.λy.x false/0 λx.λy.y S λx.λy.λz.(x z)(y z) Y λf.(λx.x x)(λx.f(x x)) 2 λf.λx.f(f x) 3 λf.λx.f(f(f…
12
Mathematics for the Adventurous Self-Learner
For over six years now, I've been studying mathematics on my own in my spare time - working my way through books, exercises, and online courses. In this post I'll share what books and resources I've worked through and recommend and also tips for anyone who wants to go on a similar adventure.
13
Introduction to Clifford Algebra (2006)
Clifford algebra gives a unified view of many things that otherwise would need to be understood separately.
14
Continuations by example: Exceptions, time-traveling search, threads, and more
Continuations are the least understood of all control-flow constructs. This lack of understanding (or awareness) is unfortunate, given that continuations permit the programmer to implement powerful language features and algorithms, including exceptions, backtracking search, threads, generators and coroutines.
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Using neural networks to solve advanced mathematics equations
Facebook AI has developed the first neural network that uses symbolic reasoning to solve advanced mathematics problems.
17
A Programmer’s Intuition for Matrix Multiplication
What does matrix multiplication mean? Here's a few common intuitions: This is useful when first learning about vectors: vectors go in, new ones come out.…
18
All Hail Geometric Algebra
Rotate a cube by 90 degrees around the \(x\)-axis, then rotate it by 90 degrees around the \(z\)-axis. This is equivalent to a single rotation. What is the axis and angle of this single rotation?
19
When Bloom filters don't bloom
Last month finally I had an opportunity to use Bloom filters. I became fascinated with the promise of this data structure, but I quickly realized it had some drawbacks. This blog post is the tale of my brief love affair with Bloom filters.
20
The Map Is Not the Territory (2015)
The map is not the territory reminds us that our mental models of the world are not the same as the world itself. It cautions against confusing our abstractions and representations with the complex, ever-shifting reality they aim to describe.
21
Calculate the day of the week for any date in your head
The Doomsday Algorithm gives the day of the week for any date (and you can do it in your head)
22
The Stack Monoid
(Updated 2020-09-06 with pointers to related work and a bit more explanation)
23
Projective Geometric Algebra Done Right
[This is part one of a two-part series. The second part can be found here: Symmetries in Projective Geometric Algebra .]
24
Convolution Is Fancy Multiplication
Like making engineering students squirm? Have them explain convolution and (if you're barbarous) the convolution theorem. They'll mutter something about…
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Transformers Are Graph Neural Networks
Engineer friends often ask me: Graph Deep Learning sounds great, but are there any big commercial success stories? Is it being deployed in practical applications? Besides the obvious ones–recommendation systems at Pinterest, Alibaba and Twitter–a slightly nuanced success story is the Transformer architecture, which…
27
Decision Table Patterns
Decision tables are easy, simple, and powerful. You can teach them in five minutes and write one in half that time. You can look at a table and understand what it’s saying, see if it matches your problem, and check for design flaws. So it’s kinda weird that there’s basically nothing about them online. I wrote an…