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Demos, equations, and facts from math, physics, and computer science. One per screen.
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notesoncomputing.com
Demos, equations, and facts from math, physics, and computer science. One per screen.
Boolean algebra
| A | B | AND |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
10120
The Shannon number — a lower bound on the count of distinct chess games. Roughly 1040 times the number of atoms in the observable universe.
Combinatorics →One pendulum hanging off another. Same starting angles within a hair, totally different paths a few seconds later.
Three equations modeling convection. The path never repeats, never settles, and switches between two looping wings forever.
0.1 + 0.2
= 0.300…04
In nearly every programming language. Floating-point can't store 0.1 exactly — it picks the nearest binary fraction, and the rounding error survives the add.
Floating-point →Solving an ODE
A stochastic differential equation: tiny step in the drift direction, then a random kick. Sixty paths from the same start, and the spread has shape.
For each point: iterate z = z² + c. Does it stay bounded? The boundary is the Mandelbrot set, and it repeats at every zoom level forever.
Run Newton's method on z³ = 1 from every starting point. Three cube roots. The boundary between basins is fractal all the way down.
N² → N log N
The fast Fourier transform (Cooley & Tukey, 1965). Without that speedup, no MRI, no JPEG, no MP3, no Wi-Fi, no 5G. Same math — better bookkeeping.
Fourier series →Fourier series
Live cell with 2 or 3 neighbors lives. Dead cell with exactly 3 comes alive. That's it. From those rules: gliders, oscillators, structures that look like organisms.
Two chemicals, same rules everywhere. Tiny noise breaks the symmetry and the mix self-organizes into spots, stripes, coral. Turing 1952.
Every agent: match neighbors' direction, drift toward the local center, keep your distance. No leader. Coherent flocking emerges.
P ≟ NP
If a solution is easy to verify, is it easy to find? Open since 1971. The Clay Institute will pay one million dollars for the answer.
Algorithms →Monte Carlo
Pick three corners. Start anywhere. Repeatedly pick a random corner and step halfway toward it. You'd expect a blob. You get a Sierpinski triangle every time.
Dots in a spiral with 137.5° between each. The pattern sunflowers, pinecones, and pineapples settle on. The tightest packing of points on a disc.
Fifteen pendulums, each slightly longer than the last. They drift out of phase, into a snake, a wave, a braid — then snap back to a line.
The 2D wave equation on a drumhead. Pulse spreads out, bounces off the walls, passes through other pulses. Tap anywhere on the canvas.
Every periodic signal is a sum of rotating circles. Stack enough and you can trace any curve. Complicated things, secretly simple.
Excitatory cells recruit inhibitory cells. Inhibition suppresses excitation. The feedback loop produces the brain's gamma rhythm.
Stronger E↔I coupling and τ_I > τ_E gives a Hopf bifurcation. E ramps up, recruits I, I suppresses E, both fall, E recovers, cycle repeats. This is the canonical mechanism for cortical gamma rhythms (~40 Hz).
The electron in hydrogen isn't a point — it's a cloud. The density peaks at a distance set by one parameter ζ. Drag it.
Exact 1s ground state of hydrogen — ζ = Z = 1. The peak of the radial probability sits at r* = 1/ζ = 1 Bohr radius, the textbook Bohr radius result.
2136,279,841 − 1
The largest known prime (Mersenne, 2024). Around 41 million digits long — printed in 9-pt type it would span two football fields end to end.
Number theory →end of feed
Each demo on this feed is a single page. There are around 420 more — on math, physics, neuroscience, CS, and circuits.
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