Marks without judgment

Most people stop drawing around age ten, at the exact moment drawing starts being evaluated. What gets lost isn't a hobby — it's a cheap, portable way to downregulate a nervous system: slow attention, repetitive motion, low stakes. settle is an attempt to give that back by deleting the part that scares people. The symmetry group makes every mark ornamental. The smoothing makes every wobble deliberate. And instead of a critic, the canvas responds the only way mathematics can: by describing precisely what you did, in figures and in sound.

How to use it, in full: press settle — the breath-shaped button — and draw anywhere; your line sings as you make it. Pause, and your last stroke is retraced by a chain of spinning circles while its melody plays. Stay still for sixteen seconds and the room takes the pen, tracing a rose curve to itself until you want it back. When you're done, release — the drawing pours away like sand. That's the whole interface.

What slow drawing does to a nervous system

The wellbeing claim isn't vibes; there's a small, consistent literature. In a frequently cited study, Kaimal, Ray & Muniz measured salivary cortisol before and after 45 minutes of unstructured art-making and found significant reductions in a majority of participants — independent of prior art experience, which is essentially this product's thesis in a petri dish: skill was never the active ingredient. Earlier, Curry & Kasser found that coloring structured patterns — mandalas, specifically — reduced anxiety more than free-form coloring, which is why Slow Fourier leads with symmetry rather than a blank page: structure is load-bearing. And Andrade's doodling study suggests idle mark-making isn't attentional junk food; doodlers in her experiment retained more of what they heard, not less.

The breathing ring is its own citation. It swells and settles on a ten-second cycle — 0.1 Hz, about six breaths per minute — the region breathing researchers call the resonance frequency, where heart-rate variability tends to peak and the baroreflex entrains (Lehrer & Gevirtz review the mechanism). You don't have to breathe with it. Most people just do.

Your stroke has a symmetry group

The kaleidoscope isn't a filter; it's a group action. In 8-fold mode the app takes the dihedral group

D₈ = ⟨ρ, σ | ρ⁸ = σ² = e,  σρσ = ρ⁻¹⟩ eight rotations · eight reflections · order 16

and applies every element of it to your gesture in real time. The image on screen is the orbit {g·S : g ∈ D₈} of your stroke S. What reads as "the app making me look good" is literally the group acting on your input — sixteen images of one gesture, none of them more yours than another. Switch to mirror and you've quotiented down to the reflection group of order 2, the same bilateral symmetry your face uses. The folds button is a group selector wearing a cardigan.

Every line is a sum of circles

When you pause, your stroke is treated as a closed curve in the complex plane, z : ℝ/ℤ → ℂ, and expanded as a Fourier series:

z(t) = Σ_k∈ℤ c_k e^2πikt,   c_k = ∫₀¹ z(t) e^−2πikt dt computed as a 128-point dft · terms kept by |c·k|

Each coefficient is a circle: radius |c_k|, spinning at integer frequency k, phase arg c_k. Chain them tip-to-tail and the end of the chain retraces your line — that's the animation you watch, a partial sum drawing itself. Fourier proposed the idea in his 1807 memoir on heat and scandalized the Académie with the claim that essentially arbitrary functions decompose this way. Two millennia earlier, Ptolemy was already doing it in the sky: deferents and epicycles are a Fourier series with the terms picked by hand. For the long version of the whole pipeline — and the unlikely life of Fourier himself — see the deep explainer.

Here's the part worth savoring: the number of circles you need is a smoothness meter. If a curve is m-times continuously differentiable, its coefficients decay like |k|^−m. A calm oval collapses into two or three fat, slow circles; a jittery scribble scatters energy into dozens of fast, tiny ones. Roughness is high-frequency content — in ink exactly as in sound. Which is the hinge of the whole instrument, because—

Timbre is a spectrum you can draw

What makes a violin and a flute different on the same note is the relative strength of their harmonics — the timbre is the Fourier spectrum. Helmholtz built the theory in On the Sensations of Tone (1863), and your cochlea agrees: the basilar membrane performs a running frequency analysis, resolving sound into components before your brain ever hears "a sound." So Slow Fourier does the only honest thing available. It takes your curve's harmonic amplitudes,

A_k = |c_k| + |c_−k|,  k = 1 … 16 handed to the synthesizer as a periodic wave at f₀ = 110 hz

and builds an oscillator whose waveform has exactly that spectrum. The same mathematical object is rendered twice — once to your eye as spinning circles, once to your ear as a chord of partials. Draw a centered circle and all the energy sits at |k| = 1: a circle is a sine wave, heard. Draw a spiky star and you've drawn brightness. The SPECTRUM panel in the corner isn't decoration; it's the actual array the synthesizer received.

The melody layer is different, and the difference is disclosed. Pitch-from-radius — sampling r(s) along your line and snapping it to the A-minor pentatonic lattice — is a chosen mapping, a sonification, not an isomorphism. We chose it because it's the one you can hear yourself in: spiral outward, the melody climbs. The pentatonic snap is what keeps randomness consonant — five tones whose harmonics overlap generously with the drone's root of A, so there are no semitone collisions to flinch at. Your scribble can't play a wrong note because the lattice contains no wrong notes.

The room hums in polar coordinates

Leave the canvas alone and it begins tracing a rhodonea — a rose curve, r = cos(kθ), catalogued by Guido Grandi in the 1720s. These curves carry a tidy little parity theorem: odd k gives k petals (the curve traces itself twice per revolution), even k gives 2k. The rest mode exists because the product's promise is passive mathematics — you should be allowed to watch θ sweep and r breathe through the readouts, hear the petals dip through the low notes as r crosses zero, and do absolutely nothing about it. It is the app doodling, and showing its work.

Descriptive, never evaluative

Every number on screen is computed from your actual gesture — the radians your hand swept, the meters of line, the harmonic count, the theorem you tripped over when you closed a loop (Jordan, 1887: a simple closed curve creates an inside and an outside; famously easy to state, famously miserable to prove). And none of it grades you. There is no accuracy, no streak, no score, no undo — stray marks are absorbed by the group like everything else. The mathematics here plays the role a good teacher plays: it pays close attention, and it keeps its opinions.

The slow Fourier transform

The fast Fourier transform — Cooley & Tukey, 1965, though Gauss had the trick in an unpublished manuscript around 1805 — is among the most-run algorithms on Earth; every codec, call, and synthesizer leans on it, always in service of speed. This is the other one. The same decomposition, taken at six breaths per minute, O(N²) and unbothered, computed not to compress anything but so that you can watch. Slow food, slow television, slow Fourier. Inside the instrument the verb survives: the breath-shaped button you press to begin says settle.

Slow Fourier is one HTML file: Canvas 2D for the ink, the Web Audio API for every sound — the drone, the bells, your line's voice are synthesized on your device, never streamed. The DFT is the naive O(N²) sum at N = 128, because 16,384 multiply-adds is a rounding error to your phone. Frequencies are twelve-tone equal temperament, so the just ratios are approximated (a fifth is 2^7/12 ≈ 1.4983, not 3/2 — apologies to Pythagoras). Nothing you draw or play leaves your device; there are no accounts and no analytics. This is not a medical device and makes no clinical claims — it's a quiet instrument with good manners. Slow Fourier is made by Macguffin Co.