eiπ + 1 = 0, written by itself. 420 rotating arrows, each one ckeikωt (Euler's formula in motion), draw the identity as a chrome wire. The coloured curves are the same drawing made with only 2, 3, 4, 6, 9… arrows, coming into focus. Move to bring a compass needle · click to flip it.
Euler’s identity
eiπ + 1 = 0 links the five most important numbers in mathematics. Here it is not typed but drawn: by rotating arrows, in metal, inside its own magnetic field.
How it works
Rotating arrows. Any closed drawing can be written as a sum of arrows that spin at whole-number speeds, each one ckeikωt with its own length and starting angle. Chained tip to tail, 420 of them trace the identity stroke by stroke: a Fourier series made visible, with Euler’s formula eiθ = cos θ + i sin θ doing the turning. With fewer arrows the pen can no longer follow the wire; the Fourier drafts show the drawing made with only 2, 3, 4, 6, 9… arrows.
A magnetised wire. Each stroke becomes a magnet as soon as it is drawn. Its field comes from the magnetic charges that appear wherever the magnetisation changes, and the iron filings are small needles that turn to follow it. The cursor carries a compass needle of its own.
Draw your own. Sketch anything, in one or several strokes, and the same machinery finds its spectrum, writes it in metal and magnetises it.