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How to create a game using hyperbolic geometry? (2020)
puzzle roguelike on a hyperbolic plane
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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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Eytzinger Binary Search
This tutorial is loosely based on a 46-page paper by Paul-Virak Khuong and Pat Morin “Array layouts for comparison-based searching” and describes one particular way of performing efficient binary search by rearranging elements of a sorted array in a cache-friendly way.
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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…
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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.
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Chemists develop new way to split water
The news portal offers current information about studies, research and life at Münster University.
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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
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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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The Viterbi Algorithm at 50 (2017)
In a noisy world, it allows you to be heard.
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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...
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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…
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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…
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Stable Fiddusion: Frequency-domain blue noise generator
Frequency-domain blue noise generator
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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…
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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…
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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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Creatively Misusing TLA+
Five not-quite-normal uses