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The number line freaks me out (2016)
“Welcome to math class, boys and girls! Today we’ll be drawing a picture that gestures beyond the limits of mortal comprehension.”
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Lace: A New Kind of Cellular Automata Where Links Matter
This article is about a new kind of simple computational rule ("LACE rules" running on LACE, the Link Automata Computing Engine platform) which, when applied locally on a grid of cells, demonstrates fascinating emergent "artificial life" behavior.
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Random Attractors – Found using Lyapunov Exponents (2001)
Contribution by Philip Ham: attractor.basic and Python implementation by Johan Bichel Lindegaard.
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An illustrated guide to automatic sparse differentiation
In numerous applications of machine learning, Hessians and Jacobians exhibit sparsity, a property that can be leveraged to vastly accelerate their computation. While the usage of automatic differentiation in machine learning is ubiquitous, automatic sparse differentiation (ASD) remains largely unknown. This post…
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Sharing everything I could understand about gradient noise
You've most likely heard about gradient noise through the name Perlin noise , which refers to one particular implementation with various CPU optimizations. Because it's an incredible tool for creative work, it's used virtually everywhere: visual effects, video games, procedural mathematical art, etc. While getting it…
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God created the real numbers
W. Jherek Swanger writes in his introduction to Camillo 7 Agrippa's "A Treatise on the Science of Arms, with a philosophical Dialogue (1553)" that:
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A triangle whose interior angles sum to zero
In hyperbolic geometry, the sum of a the interior angles of a triangle can be any value less than π. If the sum is zero the triangle has area π.
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The Block Stacking Problem
John D. Norton Department of History and Philosophy of Science University of Pittsburgh https://sites.pitt.edu/~jdnorton/jdnorton.html
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Solving LinkedIn Queens with SMT
For sure easier than solving it in SAT!
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The Flummoxagon
We’re excited to announce Flummoxagon, a new geometric puzzle game that might just leave you flummoxed. Imagine a mash-up of Pentominoes and Sudoku. Your goal is to fit all the pieces into the hexa…
162
Five Kinds of Nondeterminism
Or four kinds, or six kinds, I'm not picky about how you count them
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An overview of gradient descent optimization algorithms (2016)
Gradient descent is the preferred way to optimize neural networks and many other machine learning algorithms but is often used as a black box. This post explores how many of the most popular gradient-based optimization algorithms such as Momentum, Adagrad, and Adam actually work.
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The Ruliology of Lambdas
Stephen Wolfram explains the rich ruliology of lambdas, made particularly significant by their connection to practical computing. Covers basic computations to undecidability to multiway graphs and evaluation strategies.
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Lemma for the Fundamental Theorem of Galois Theory
This post illustrates a key lemma that is used in proving the fundamental theorem of Galois theory (FTGT). Note that FTGT is not covered in this post. The focus of this post is on understanding and proving this lemma only. Here is the lemma from the book Galois Theory , 5th ed. by Stewart (2023):
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A shower thought turned into a Collatz visualization
I recently went on a nice long SCUBA diving trip with my wife and daughters. Lots of diving implies lots of showers, and lots of showers means lots of shower-thoughts! [1] An especially interesting one I had turned into a nice way to visualize some aspects of the Collatz Conjecture.
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A Random Walk in 10 Dimensions (2021)
The geometry of random walks in high dimensions provides the power behind deep learning and may be the secret to intelligence.
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Using secondary school maths to demystify AI
Educators can show in secondary school maths that AI systems don’t think, making maths more interesting while teaching core concepts of AI.
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There Is No Diffie-Hellman but Elliptic Curve Diffie-Hellman
When I first learned about Diffie-Hellman and especially elliptic curve Diffie-Hellman, I had one rather obvious question: Why elliptic curves? Why use this strange group that seems rather arbitrar…
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Quanta to publish popular math and physics books by Terence Tao and David Tong
Quanta Books is delighted to announce two new upcoming books by mathematician Terence Tao and theoretical physicist David Tong. Six Math Essentials will be Tao’s first math book written for a popular…
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Tversky Neural Networks
Psychologically Plausible Deep Learning with Differentiable Tversky Similarity
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Scarcity, Inventory, and Inequity: A Deep Dive into Airline Fare Buckets
Airline pricing may seem mystifying, but behind every airfare is a complex system of fare buckets and inventory controls. Airlines don't just sell seats - they manage a dynamic inventory of fares, divided into booking classes (fare buckets) with different prices and rules. For the technically curious, understanding…
174
The inspection paradox is everywhere (2015)
I have updated this article with new data, better code, and friendlier data visualization. You can read the new version here . Th...
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Revisiting Moneyball
Data, sports, payrolls, and memes
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Basketball has evolved into a game of calculated decision-making
Basketball has evolved into a game of calculated decision-making. From a game of points, assists, and rebounds, it has progressed into using thousands of data points to optimize every element of the game.
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You need much less memory than time
Just as I was complaining that we haven't seen many surprising breakthroughs in complexity recently, we get an earthquake of a result to st...
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Geometrically understanding calculus of inverse functions (2023)
The Legendre transformation tells you how the inverse map acts on integrals
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Rediscovering Quaternions
How do we represent a rotation in 3 dimensions? This is a surprisingly deep question, and one we can learn a lot from. Our goal will be to “rediscover” quaternions from first principles, building them up by exploring different representations and fixing the failures along the way.
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Solving First Order Differential Equations with Julia
In this tutorial, we will learn about solving Differential Equations with Julia. Differential Equations are everywhere in Science and Engineering problems. And, being able to solve Differential Equations using computers is very convenient. We will use the DifferentialEquations.jl package for solving Differential…