The Pythagoras tree stands a right triangle on each square and a square on each of its legs, ten times over: 1023 squares. Its outline, with the triangular gaps joined in by thin bridges, is one closed curve and a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ.
The widest gap to it is 49.7 % of the tree’s height at N = 1, 5.04 % at 128 and 1.39 % at 512, and the circles shrink like k^−1.32.
Joseph Fourier’s 1823 engraving, traced as 57 ink shapes joined by thin bridges into one closed outline, is a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ, the cₖ from its discrete Fourier transform.
With the N largest circles the widest gap to the outline is 42.0 % of the portrait’s height at N = 1, 3.16 % at 128 and 0.92 % at 512, and the circles shrink like k^−1.47. The tip’s distance |z(t)| plays as pitch.
The treble clef 𝄞, its four contours joined by thin bridges and sampled at 4096 points, is a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ.
With the N largest circles the widest gap to the outline falls from 35.7 % of its height at N = 1 to 0.37 % at 256. The tip's distance |z(t)| is played as pitch, floored to C major.
One circle keeps |z| fixed: one note. With 64 the tip plays the outline's own note 93 % of the lap; with 256, 98 %.
Each coordinate of a closed outline has its own real Fourier series: x(t) = ∑ (aₖ cos kt + bₖ sin kt), and the same for y.
A cosine and sine of frequency k make one arm of length √(aₖ² + bₖ²) turning k times per lap. One chain sets x, one sets y, and the pen sits where their guides cross. With N harmonics per axis the knight is off by 30.2 % of its height at N = 1, 0.70 % at N = 50.
At N = 1 both coordinates are pure sinusoids: an ellipse.