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Avoiding visible texture repetition (2015)
Tutorials and articles of Inigo Quilez on computer graphics, fractals, math, art, demoscene, shaders and more.
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The latest blog posts to take off on Hacker News. 221 posts.
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Dividing a Square into 7 Similar Rectangles
This is a continuation of my post Dividing a square into similar rectangles, in which I discussed this puzzle: if you partition a square into n similar rectangles, what proportions can these rectan…
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The Smallest Hash Table
This December I once again did the Advent of Code , in Rust. If you are interested, my solutions are on Github. I wanted to highlight one particular solution to the day 2 problem as it is both optimized completely beyond the point of reason yet contains a useful technique. For simplicity we’re only going to do part 1…
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Bitwise Division
Integer division is expensive. Much more expensive than multiplication. Much, much more expensive than addition. If you’re trying to make some integer operations really fast it’s bad news if you need to do division.
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SymPy makes math fun again
An introduction into symbolic computations in Python. Don't worry, it's much simpler than it sounds. It's about making Python do your math for you with very little investment in the technology.
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The unreasonable effectiveness of conditional probabilities
I have been steeping myself in probabilistic exercises recently, because it’s an area where I feel like I could benefit from building better intuition. I have found a recurring key to solving many of them: conditioning on an intermediary state, and then computing the value of that intermediary state separately. This…
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NP-complete isn't always hard
A common assumption I see on the ‘net is that NP-complete problems are impossible to solve. I recently read that dependency management in Python is hard because package resolution is NP-complete. This is true in principle, but the reality is more complicated. When we say “NP-complete is hard” we’re talking about…
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An interactive explanation of quadtrees (2014)
The space is divided into four rectangles . Each of those rectangles is divided such that it contains a maximum of four children . Each child is either a point or a smaller rectangle.
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Google’s fully homomorphic encryption compiler – a primer
Back in May of 2022 I transferred teams at Google to work on Fully Homomorphic Encryption (newsletter announcement). Since then I’ve been working on a variety of projects in the space, including being the primary maintainer on github.com/google/fully-homomorphic-encryption, which is an open source FHE compiler for…
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Problem solving with dimensional analysis
A dimension is a physical quantity that can be measured, while a unit is a specific amount of a physical quantity. For example, time is a dimension, which can be measured in temporal units such as hours or seconds.
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Creatively Misusing TLA+
Five not-quite-normal uses
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Why does 0.1 and 0.2 = 0.30000000000000004?
Why does 0.1 + 0.2 = 0.30000000000000004?
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Fundamental Math for Game Developers
Let's review the most important math topics used by game devs, reflect about math education, and discuss techniques that can be used to improve the way we learn new math topics.
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They say that stocks go down during the day and up at night
Prompted by your blog post this morning, I attach a plot from Figure 3 of They Still Haven’t Told You showing overnight and intraday returns to AIG (with logarithmic vertical scale, updated with data through the end of October).
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Math breakdown: Anime homing missiles
Tech Notes on Cubic Bezier Curves, Perlin Noise, and Minimal Rotation Frames
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Estimating square roots in your head
Imagine we want to compute the square root of a number n n n . The basic idea of Heron’s method , named after the mathematician and engineer, Heron of Alexandria, is to find a number g g g that is close to n \sqrt{n} n and to then average that with the number n / g n/g n / g , which corrects for the fact that g g g…
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Cistercian Numerals
Cistercian numerals were invented by the Cistercian order of monks in the 13th century. Giuseppe Frisella explains how the notation system works: A vertical str
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PageRank algorithm for graph databases
What is PageRank algorithm? How can it be used in various graph database use cases? How to use it in Memgraph? If these questions are keeping you up at night, here is a blog post that will finally put your mind at ease.
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A baffling scale transform on a chart of university course selection trends
Harvard Magazine printed the following chart that confirms a trend in undergraduate majors that we all know about: students are favoring STEM majors at the expense of humanities.
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An intutive counterexample to the axiom of choice
Time for some hard-core geeking-out. This comment on HN by /u/jiggawatts struck me as a brilliant idea: it's an intuitive counter-example t...
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The Point of the Banach-Tarski Theorem
One of my favourite Limited Audience Jokes (although technically it's actually a riddle) goes:
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Formalising a new proof that the square root of two is irrational
Recently, somebody shared on Twitter a new proof of the irrationality of √2. Its claimed advantage is that “it requires no knowledge of mathematics above the definition of what it means for a number to be irrational, and can be written almost in one line.” These claims are dubious: the usual proof, which I’ve…
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Reverse engineering a neural network's clever solution to binary addition
While training small neural networks to perform binary addition, a surprising solution emerged that allows the network to solve the problem very effectively. This post explores the mechanism behind that solution and how it relates to analog electronics.
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