The Lab
ẋ = σ(y − x) ẏ = x(ρ − z) − y ż = xy − βz
Deterministic chaos - a system with no randomness that never repeats. Tiny changes at the start diverge into wildly different futures. It is why long-range weather is fundamentally hard.
x = A·sin(a·t + δ) y = B·sin(b·t)
Two perpendicular oscillations, one closed figure - exactly what an oscilloscope traces from two out-of-phase signals. Frequency ratios you can see.
x = (R+r)cos t − r·cos((R+r)/r · t) y = (R+r)sin t − r·sin((R+r)/r · t)
The path of a point on a circle rolling around another circle - the geometry behind gears, planetary orbits, and every Spirograph you ever drew.
r = e^{cos θ} − 2cos 4θ + sin⁵(θ/12)Six wings from a single transcendental equation, swept over twelve full turns. Proof that one line of math can hold a surprising amount of structure.
x = 16 sin³ t y = 13 cos t − 5 cos 2t − 2 cos 3t − cos 4t
Yes - affection has a closed form. A parametric curve that resolves, exactly, into a heart. My favorite reminder that math has a sense of humor.
r = a·cos(k·θ)
A single integer k sets the petal count - odd k gives k petals, even k gives 2k. Dial one number, redesign the whole flower. Symmetry as a parameter.
r = a + b·θ x = r·cos θ, y = r·sin θ
Radius grows in step with angle, so every turn is evenly spaced - the shape of record grooves, coiled rope, and the arms of a galaxy.
f(x) = Σ aₙ·sin(nx) + bₙ·cos(nx)
Any signal is a sum of sines. Add enough harmonics and you can build any waveform - the exact idea powering the audio-feature engine inside AutoMix.
The same math shows up everywhere I work: scoring audio features in AutoMix, driving the visualizers around this site, running the models in MoodScope. It’s the common thread.