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Notes on the complex roots of unity
Eli Bendersky works through the complex roots of unity from first principles: deriving them with polar coordinates, rotations on the unit circle and regular polygons, conjugates, the cyclic group and primitive roots, their sums, and how their powers form an orthogonal basis, the Fourier matrix behind the DFT.
11 min · 2,423 words
Demystifying Spherical Harmonics: An Optimization Approach
Andrew Helmer teaches spherical harmonics for computer graphics by starting from a familiar engineering problem, approximating a spherical lighting function, and solving it with plain least-squares regression in Python. The result turns out equivalent to L1 and L2 SH, and it exposes the bias and ringing pitfalls of naive SH use.
28 min · 6,522 wordsagent-assisted
You could have invented transformers from memory
Alex Wang builds intuition for linear maps, MLPs and self-attention as different kinds of memory stores, using the interference-to-signal ratio to estimate each one's capacity. With 64-dimensional keys, capacity grows from about 64 memories for a linear layer to about 2,700 for an MLP and over a million for self-attention.
21 min · 4,738 words
Eli Bendersky introduces Monte Carlo simulations, which apply randomness and the law of large numbers to scientific and engineering problems, through worked examples: estimating pi by throwing darts, a combinatorial simulation of the card game SET, and a retirement portfolio projection, with code.
8 min · 1,757 words
A Visual Intro to Complex Numbers
A visual, intuition-first tutorial on complex numbers: starting from an unsolvable system of equations, it builds up the imaginary unit as a 90-degree rotation, the complex plane, multiplication as rotation and scaling, Euler’s formula, and applications in AC circuits, signals, control and MRI.
16 min · 3,760 words
Danielle Navarro warns against dplyr::ntile() in regulatory analyses: the SQL-inspired ranking helper often fails to produce equal-sized groups or the quantile splits analysts think they are getting.
29 min · 6,556 words
In Search of PCA's Lost Variance
A geometric derivation of why PCA reconstruction mean-squared error equals the sum of discarded eigenvalues, using orthogonal projections and a clear 2D example to make the lost-variance identity intuitive.
6 min · 1,308 words
Thomas Ahle presents a technique for evaluating polynomials in roughly half the multiplications that Horner's method requires, by preprocessing the coefficients offline. An interactive tool lets users input a polynomial and field to see the optimised evaluation chain.
1 min · 234 wordsagent-written
Luke Haas derives music theory from first principles — starting from the physics of sound and working up through frequencies, harmonics, the twelve-tone scale, intervals, chords, and progressions — entirely through code examples. The tutorial requires no instrument and takes nothing on faith, making it accessible to programmers who found conventional music education unsatisfying.
1 min · 282 wordsagent-written
Simon Sapin models Factorio Space Age quality upcycling with transition matrices and equilibrium equations, then builds an interactive TypeScript calculator for gambling, washing, upcycling, and asteroid reprocessing loops.
15 min · 3,372 words
Gregory Gundersen reconstructs backpropagation from first principles, explaining why the algorithm needs a backward pass to compute neural network gradients efficiently.
6 min · 1,490 words
Jeremy Kun explains hybrid images, pictures that read differently up close and from afar, using Fourier analysis: an intuitive tour of the discrete Fourier transform, low- and high-pass Gaussian filtering, and Python code that blends Marilyn Monroe and Albert Einstein into one image, with ideas for further experiments.
10 min · 2,292 words