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How Many People? Ancient Demography
One of the first questions anyone asks about past societies is, ‘how many people were there?’ It is one of those basic bits of information and generally speaking those of us who teach t…
Topics /
Math & Algorithms
The most popular blog posts about math & algorithms on Hacker News. 189 posts.
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Four Kinds of Optimisation
Premature optimisation might be the root of all evil, but overdue optimisation is the root of all frustration. No matter how fast hardware becomes, we find it easy to write programs which run too slow. Often this is not immediately apparent. Users can go for years without considering a program’s performance to be an…
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Algebraic graph calculus (2021)
We describe a graph-theoretic analogue of vector calculus. The linear operators of vector calculus (gradient, divergence, laplacian) correspond to the matrices naturally associated to graphs (incidence matrix, adjacency matrix). This analogy is useful for formalizing the discretization of some problems in image and…
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Painting with math: A gentle study of raymarching
A compilation of my Raymarching and Signed Distance Field learnings and work. In it, I teach you how to leverage those techniques along with lighting models, smoothmin, soft shadows, and some math to create beautiful 3D raymarched scenes ranging from abstract shapes to infinite realistic-looking landscapes.
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A Lisp interpreter implemented in Conway's Game of Life (2021)
Lisp in Life is a Lisp interpreter implemented in Conway’s Game of Life.
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The Dome: A simple violation of determinism in Newtonian mechanics (2005)
John D. Norton Department of History and Philosophy of Science, University of Pittsburgh Pittsburgh PA 15260. Homepage: www.pitt.edu/~jdnorton This page is available at www.pitt.edu/~jdnorton/goodies
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Embeddings, vectors, and arithmetic
An embedding is the representation of a piece of text like a word, sentence, or paragraph. Traditionally this value is in the form of a mathematical vector - a point in space. You can think of it as analogous to coordinates on a map that just happens to have many, many dimensions.
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Higher quality random floats
Much has been written about how to generate random 64-bit integers; people might argue over which particular (CS)PRNG is best suited for a given task, but there is agreement on the general shape of the solution: keep some state around, and then extract 64 bits of randomness at a time, where each bit is equally likely…
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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…
161
Solving “Two Sum” in C with a tiny hash table
I came across a question: How does one efficiently solve Two Sum in C? There’s a naive quadratic time solution, but also an amortized linear time solution using a hash table. Without a built-in or standard library hash table, the latter sounds onerous. However, a mask-step-index table , a hash table construction…
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Open Questions
Some anomalies/questions which are not necessarily important, but do puzzle me or where I find existing explanations to be unsatisfying.
163
A Bayesian view of Amazon resellers (2011)
I was buying a used book through Amazon this evening. Three resellers offered the book at essentially the same price. Here were their ratings: 94% positive out of 85,193 reviews 98% positive out of 20,785 reviews 99% positive out of 840 reviews Which reseller is likely to give the best service? Before you assume it's
164
Bitwise Binary Search: Elegant and Fast
I recently read the article Beautiful Branchless Binary Search by Malte Skarupke. In it they discuss the merits of the following snippet of C++ code implementing a binary search :
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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 largest number representable in 64 bits
A rewrite of this blog post with many new insights and updates appeared in Jan 2026
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Math for kids outside of the Calculus Sequence
The Enrichment side of the "Accelerate vs Enrich" dichotomy in math education for kids who love math.
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Formalising modern research mathematics in real time
(This is a guest post by Bhavik Mehta) On March 16, 2023, a paper by Campos, Griffiths, Morris, and Sahasrabudhe appeared on the arXiv, announcing an exponential improvement to the upper bound on R…
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Occluding Contour Breakthroughs, Part 1: A Surprisingly Hard Problem
Exact line rendering of smooth surfaces is harder than anyone would think
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Tag Systems
I've tried to write a blog post on tag systems for years now. Literally years, I think I first started drafting it out in 2018 or so? The problem is that...
171
How Correlated Are You?
This post is an offshoot from Chapter 1 of Probably Overthinking It, which is available for pre-order now! Suppose you measure the arm and leg lengths of 4082 people. You would expect those measurements to be correlated, and you would be right. In the ANSUR-II dataset, among male members of the armed forces, this…
172
Q* Hypothesis: Enhancing Reasoning, Rewards, and Synthetic Data
Emergency special: The information we need to understand what Q* is was right in front of us, but the memes are more fun than reality.
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How to keep lambda calculus simple
Once upon a time, my curiosity of dependent types transitioned into an endeavour to learn how they work under the bonnet. Then I found the paper called “A Tutorial Implementation of a Dependently Typed Lambda Calculus” or just “Simply Easy” – and began reading it. However, the affairs happening there were so complex…
174
Stable Fiddusion: Frequency-domain blue noise generator
Frequency-domain blue noise generator
175
Marvelous Arithmetics of Distance
No reckoning allowed save the marvelous arithmetics of distance (from Smelling the Wind by Audre Lorde) Suppose a child comes up to you and says “I know 1 is odd and 2 is even, but I think 4 is mor…
176
Bayesian statistics and machine learning: How do they differ?
My colleagues and I are disagreeing on the differentiation between machine learning and Bayesian statistical approaches. I find them philosophically distinct, but there are some in our group who would like to lump them together as both examples of machine learning. I have been favoring a definition for Bayesian…
177
What Is Dimensional Analysis?
Mathematics is famously concerned with numbers . However, most of the physical quantities that we encounter in everyday experience cannot be expressed as numbers alone.
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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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