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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.”
New
The latest blog posts to take off on Hacker News. 2,851 posts.
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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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Why cryptography is not based on NP-complete problems
Cryptographic schemes are based on the computational difficulty of solving some ‘hard’ problem. For example, RSA (specifically, RSA-2048) is based on the difficulty of a problem like this:
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Why Does Integer Addition Approximate Float Multiplication?
Here is a rough approximation of float multiplication (source): float rough_float_multiply(float a, float b) { constexpr uint32_t bias = 0x3f76d000; return bit_cast<float>(bit_cast<uint32_…
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Generating Voronoi diagrams using Fortune's algorithm
This article is about using Fortune’s Algorithm to generate Voronoi Diagrams in O(nlogn) time. If I’d known how hard it would be I wouldn’t have started it. If you are intending to implement it with the intent to actually use it, rather than as an exercise, I would recommend not using Fortune’s. Unless you are doing a…
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Visual explanations of mathematics (2020)
It is thought that Euclid wrote Elements in about 300 BC, but Oliver Byrne turned it into one of the true gems of visualization — and made it about 100 times more readable. By seamlessly combining typeset text (Caslon, if you’re interested) with minimalist geometric drawings in primary col
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There's Math.random(), and then there's Math.random() (2015)
V8’s Math.random implementation now uses an algorithm called xorshift128+, improving the randomness compared to the old MWC1616 implementation.
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Quaternions and spherical trigonometry
Hamilton’s quaternion number system $latex {\mathbb{H}}&fg=000000$ is a non-commutative extension of the complex numbers, consisting of numbers of the form $latex {t + xi + yj + zk}&f…
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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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Where do those undergraduate divisibility problems come from?
Chris Grossack's math blog and professional website.
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Don't use cosine similarity carelessly
Cosine similarity - the duct tape of AI. Convenient but often misused. Let's find out how to use it better.
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Generating an infinite world with the Wave Function Collapse algorithm
Generating an infinite world with the Wave Function Collapse algorithm - Personal blog about game development, programming and making
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Nearly all binary searches and mergesorts are broken (2006)
Posted by Joshua Bloch, Software EngineerI remember vividly Jon Bentley's first Algorithms lecture at CMU, where he asked all of us incoming Ph.D. ...
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Very Wrong Math
The difference between misinformation and disinformation is the difference between ignorance and malice. Trolling is somewhat different, incorporating an element of provocation and narcissism. But what the hell is this?
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Who Can Understand the Proof? A Window on Formalized Mathematics
Stephen Wolfram discusses understanding proofs discovered using automated theorem proving. Based on Wolfram’s proof of the simplest axioms of logic of Boolean algebra.
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Entropy of a Large Language Model output
Large language models like ChatGPT & Claude have become ubiquitous in today’s world. I visited a friend for the New years eve and was impressed by how ChatGPT was being extensively used within the family. The friend’s little kid is going to grow up in a world, where turning to large language models (LLM) for quick…
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Spline Distance Fields
Earlier this year I decided to put my main project Tangerine on hold indefinitely and began prototyping a new renderer in an unfamiliar language using a totally new graphics API with the goal of overcoming all of Tangerine's technical shortcomings. The new renderer is an eccentric CPU ray tracer called Star Machine…
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Fold-... and Monoids
A blog about computers, functional languages, Lisp, and Scheme.
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Reverse Engineering the Constants in the Pentium FPU
Intel released the powerful Pentium processor in 1993, establishing a long-running brand of high-performance processors. 1 The Pentium incl...
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Mandelbrot deep zoom theory and practice (2021)
Perturbation and rescaling for efficient rendering of fractal images.
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Penrose Mazes (2020)
Mazes are typically constructed on regular grids: square, triangular, hexagonal, etc. But what if you generate a maze on top of a Penrose tiling, which is guaranteed to be irregular? (It is an aperiodic tiling .) The resulting mazes have a very characteristic appearance: they are filled with circles and stars, but the…
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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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Dividing unsigned 8-bit numbers
Division is quite an expensive operation. For instance, latency of the 32-bit division varies between 10 and 15 cycles on the Cannon Lake CPU, and for Zen4 this range is from 9 to 14 cycles. The latency of 32-bit multiplication is 3 or 4 cycles on both CPU models.
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Collatz’s Ant
if $n \equiv 0 \, (\text{mod}\, 2)$ the ant turns 90º clockwise, else the ant turns 90º counter-clockwise. On both accounts, the state of the cell is flipped and the ant moves forward one unit. This is repeated until $n = 1$.
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Turing Machines
An interactive introduction to Turing machines, the theoretical foundation of modern computing.
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Spherical Harmonics
3D visualization tool of spherical harmonics. Visualize and compare real, imaginary, and complex components by adjusting the degree l and order m parameters.
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Kelly Can't Fail
Introduction You may have heard of the Kelly bet allocation strategy. It is a system for correctly exploiting information or bias in a gambling situation. It is also known as a maximally aggressive…
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Music and Geometry: Intervals and Scales
An article about music and geometry, in particular about musical intervals and scales and their geometric equivalent ...