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Airfoil
Interactive article explaining the physics of an airfoil and what makes airplanes fly
Top posts
Every blog post that took off on Hacker News, ranked by the votes of its best submission. 237 posts.
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Visualizing Attention, a Transformer's Heart [video]
Demystifying attention, the key mechanism inside transformers and LLMs.
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Too much efficiency makes everything worse (2022)
This blog is intended to be a place to share ideas and results that are too weird, incomplete, or off-topic to turn into an academic paper, but that I think may be important. Let me know what you think! Contact links to the left.
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A visual proof that a^2 – b^2 = (a + b)(a – b)
A visual proof that a2 – b2 = (a + b)(a – b). Sophie Germain wrote, “It has been said that algebra is but written geometry and geometry is but diagrammatic algebra.”
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The hunt for the missing data type
A (directed) graph is a set of nodes, connected by arrows (edges). The nodes and edges may contain data. Here are some graphs: All graphs made with graphviz (source) Graphs are ubiquitous in software engineering: Package dependencies form directed graphs, as do module imports. The internet is a graph of links between…
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An animated introduction to Fourier series
This article will be part of an extensive series in which I plan to explore various aspects of Fourier Mathematics. I will take notes, create some visuals (a good pretext to learn more about graphics), and hope that it will be helpful to anyone looking for a clear, visual introduction to these concepts. The article…
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An intuitive guide to Maxwell's equations (2020)
An Intuitive and Visual Guide to Maxwell's Equations
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Double-entry bookkeeping as a directed graph
In the past couple of years, I’ve been working in billing and payments at Justworks . This experience introduced me to the world of bookkeeping and accounting in ways I didn’t expect: I took a course and read a textbook in accounting, adopted plain text accounting for my personal finances, and worked in a double-entry…
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Look, ma, no matrices
Since the 2019 SIGGRAPH course [1], Geometric Algebra, and Euclidean PGA (plane-based or projective geometric algebra) in particular, has been gaining traction within the computer graphics and machine learning communities [2, 3, 4]. Despite its broad applicability, including for higher dimensional geometry and…
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A* tricks for videogame path finding
A* Tricks for Videogame Path Finding, A* (A-star) Search Algorithm
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Understand how transformers work by demystifying the math behind them
Understand how transformers work by demystifying all the math behind them
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Dynamic programming is not black magic
This year’s Advent of Code has been brutal (compare the stats of 2023 with that of 2022, especially day 1 part 1 vs. day 1 part 2). It included a problem to solve with dynamic programming as soon as day 12, which discouraged some people I know. This specific problem was particularly gnarly for Advent […]
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Sir, there's a cat in your mirror dimension
Pets do the darndest things, especially if you teach them a bit of math.
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Relearning math as an adult
I recently passed 100 days of practicing math every single day 💯 I’ve wanted to beef up my math chops for a while, but I needed a good reason that would justify the time investment. Plus…
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Fermat's Last Theorem – how it’s going
So I’m two months into trying to teach a proof of Fermat’s Last Theorem to a computer. We already have one interesting story, which I felt was worth sharing.
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When do we stop finding new music?
When does our taste in music stagnate?
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Conway's Gradient of Life
What happens if we start from this configuration and take a single step of the game? Here is a snapshot of the next generation:
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Can AI do maths yet? Thoughts from a mathematician
So the big news this week is that o3, OpenAI’s new language model, got 25% on FrontierMath. Let’s start by explaining what this means.
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Gödel, Escher, Bach is the most influential book in my life (2022)
Musings on Hofstadter’s phenomenal book about computation, consciousness, and epistemology.
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Calculus with Julia
This is a set of notes for learning calculus using the Julia language. Julia is an open-source programming language with an easy to learn syntax that is well suited for this task.
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Diffusion models from scratch, from a new theoretical perspective
This tutorial aims to give a gentle introduction to diffusion models, with a running example to illustrate how to build, train and sample from a simple diffusion model from scratch.
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When Random Isn't
This post is an anecdote from over a decade ago, of which I lost the actual code. So please forgive me if I do not accurately remember all the details. Some details are also simplified so that anyone that likes computer security can enjoy this article, not just those who have played World of Warcraft (although the…
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My Favorite Algorithm: Linear Time Median Finding (2018)
Finding the median in a list seems like a trivial problem, but doing so in linear time turns out to be tricky. In this post I’m going to walk through one of my favorite algorithms, the median-of-medians approach to find the median of a list in deterministic linear time. Although proving that this algorithm runs in…
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Kalman Filter Explained Simply
Tired of equations and matrices? Ready to learn the easy way? This post explains the Kalman Filter simply with pictures and examples!
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Using Fibonacci numbers to convert from miles to kilometers and vice versa
I recently learned an interesting fact about Fibonacci numbers while watching a random number theory video on YouTube. Fibonacci numbers can be used to approximately convert from miles to kilometers and back. Here is how. Take two consecutive Fibonacci numbers, for example 5 and 8. And you're done converting. No...
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Generating sudokus for fun and no profit
A guide on how to generate sudokus of any difficulty.
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Feynman’s Razor
If an expert can't understand your explanation, you've dumbed it down too much!
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Math from Three to Seven
Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers, Alexander Zvonkin (Moscow Center for Continuing Mathematical Education, 2007).