A drafting table for signals · ferro-prussiate proof
Every message is a sum f circles.
Give the table one stroke of handwriting and it computes the discrete Fourier transform — then rebuilds your line out of nothing but rotation: one circle per frequency, each turning at its own rate, the pen riding the end of the chain. Slide the budget from 1 term to 128 and watch fidelity return, measured to the hundredth.
Terms64
RMS error—%
Samples—
Trace0%
Draw on the plate with a mouse or stylus — on touch, arm DRAW first so the page still scrolls. G overlays the ghost of the original stroke.
I · The analysis
Grenoble, December 1807. Fourier claims any curve can be written as a sum of sines.
He is reading a memoir on the propagation of heat to the Institut de France. Lagrange objects from the floor — a jagged line, built from smooth waves? The committee withholds the prize. Fourier is right, and it takes mathematics fifteen years to admit it.
In the plane, the theorem grows a body. Treat the stroke as complex numbers, z = x + iy, and its transform hands back a list of circles: for each frequency, an amplitude, a phase, a rate of turn. Chain them tip to tail, largest first, and the free end of the chain retraces the handwriting. Nothing else is stored. The circles are the message.
Amplitude
How much of frequency k the stroke contains — drawn here as the circle's radius, and ranked as a brass tick in the spectrum strip along the foot of the plate.
Frequency
Full turns per traverse of the path. Positive spins one way, negative the other. The broad gesture lives in the low terms; the wildest flicks of the pen live out in the high k.
Phase
Where each spoke points at t = 0. Lose the phases and the word dissolves — amplitudes alone are anagram soup. Timing, not volume, is what spells.
II · Lossy on purpose
Compression is choosing what to forget.
Rank the circles by radius and keep only the largest. At 4 terms the word is a wobble with intent; at 16 it grows bones; at 64 it is handwriting. The residue is measured live — root-mean-square distance between original and reconstruction, as a share of the plate's diagonal — and it falls, provably, with every term you restore.
JPEG does this to your photographs. MP3 does it to your ears. Draw your own stroke above and these three proofs re-develop with your line in them.
III · Provenance
The circles never left.
c. 150
Ptolemy's Almagest carries the planets on epicycles — circles riding circles riding a deferent. The cosmology was wrong. The mathematics was a Fourier series, eighteen centuries early.
1807
Fourier reads his memoir on heat to the Institut de France. The claim that arbitrary curves decompose into sines scandalises Lagrange; the Théorie analytique de la chaleur finally appears in 1822.
1842
John Herschel sensitises paper with iron salts and prints in Prussian blue — the cyanotype. Engineers adopt it for line drawings, and every plan becomes a blueprint. This plate borrows the chemistry.
1965
Cooley and Tukey publish the fast Fourier transform: N² work collapses to N log N and the transform becomes infrastructure — inside every phone call, every image, every second of compressed music. The circles never left; they just got fast.