matharium @matharium

Each of these thirteen graphs is also a sound: the height y at the pen sets the pitch, f = 161 · 2^(2.15y) Hz with y in half heights of the axis, so a curve read left to right is a melody and a jump is a new note. Blocks of decaying exponentials such as 2.5e^(−4.35(x mod 2)) drop in pitch fast enough to land as drum hits, and the poles of 1.2 sin(x/2)/sin(1.2x) sweep out to the top and bottom of the range.

Temple Fay's butterfly curve is r = e^(cos t) − 2 cos 4t − sin⁵(t/12) with x = r sin t, y = r cos t, for t from 0 to 12π. As two Fourier series it lives almost entirely on t, 2t, 3t, 4t and 5t, since sin t·e^(cos t) = Σ 2n Iₙ(1) sin nt and −2 sin t cos 4t = sin 3t − sin 5t; only the slow sin⁵(t/12) adds small terms. With 24 harmonics per axis the gap is 20.2 % of the height, and the 30th closes it to 0.01 %.

The curves y = 1/x and y = −1/x are separated by a gap of 2/x at every x > 0: below 0.01 from x = 200 on, below 2·10⁻⁹ at x = 10⁹, and never 0. The map (x, y) ↦ (kx, y/k) sends each curve to itself, so stretching x by k while magnifying y by k leaves the picture unchanged at every scale. Both curves have the x axis as asymptote and never meet it or each other.

The curves y = x² and y = 1.1ˣ cross exactly three times: at x = −0.955, at x = 1.051 and at x = 95.717. Between the second and third crossings the square leads, by a lot: 50² = 2500 while 1.1⁵⁰ ≈ 117. The ratio 1.1ˣ/x² bottoms out at x = 2/ln 1.1 ≈ 21 and grows without bound after it, so the third crossing is final. By x = 200, 1.1ˣ ≈ 4 700 · x².

The point z(t) = e^(−γt) e^(iωt) turns at angular speed ω while its distance from the origin shrinks as e^(−γt). Its parts e^(−γt) cos ωt and e^(−γt) sin ωt are two damped waves, one on the floor and one on the wall. Each turn is q = e^(−2πγ/ω) times the one before, and the path meets every ray from the origin at the same angle, so it is a logarithmic spiral. It never reaches the origin, yet its length is finite: ∫₀^∞ |z′| dt = √(γ² + ω²)/γ.

The Collatz map sends n to n/2 when n is even and to 3n + 1 when n is odd. From 27 it climbs to 9232 and takes 111 steps to reach 1. Each trajectory is drawn backwards from 1, turning +11° at an even number and −19° at an odd one. Every n has exactly one successor, so paths that meet never separate, and the starts n ≤ 10 000 grow into one tree of 21 664 numbers. Every start below 2⁷¹ has been checked to reach 1. No proof is known.

Twelve curves, each traced by a pen while its height is its pitch: f = 440 · 2^(0.4y) Hz. A heartbeat, a chirp, a ramp with poles, a comb of spikes, a line with a ripple. The square wave y = sgn sin(19.7x) alternates between two notes, 580 Hz and 333 Hz. The chirp y = 0.195 sin(3.93x²) is a vibrato that speeds up toward the edges, because its phase grows like x².

The superellipse |x/a|ⁿ + |y/b|ⁿ = 1, with a = b = 1, for n = 0.3, 0.5, 0.7, 1, 1.5, 2, 3 and 5. For n < 1 the curve is concave, a four-pointed star; n = 1 gives the diamond |x| + |y| = 1 and n = 2 the circle. Each curve meets the diagonal at x = y = 2^(−1/n), which climbs from 0.099 at n = 0.3 to 0.87 at n = 5. As n → ∞, (|x|ⁿ + |y|ⁿ)^(1/n) → max(|x|, |y|), and the curve becomes the square.

A shear adds a multiple of one coordinate to another: Sₓ(f) sends (x, y, z) to (x + fy, y, z), the identity matrix with a single off-diagonal entry f. The plane y = 0 stays fixed, and every other point slides parallel to it by f times its distance from it. Square faces lean into parallelograms, yet det S = 1 for every f, so the cube keeps its volume. Shears along one direction compose by adding, S(f)S(g) = S(f + g), so S(−f) undoes S(f).

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.

An electric field gives the force per unit charge at every point, E = kq r̂/r², summed over all the charges present. With +q at (−3, 0), −q at (3, 0) and a test charge on the ellipse x²/2.3² + y²/2² = 1, the field it feels is kq r̂₁/d₁² − kq r̂₂/d₂², where d₁² − d₂² = 12x. At (2.3, 0) both pulls point the same way and add to 2.08 kq. At the top their vertical parts cancel, leaving 6kq/(13√13) ≈ 0.13 kq parallel to the axis.

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 sawtooth f(t) = t/π on (−π, π), repeated every 2π, is the limit of f_k(t) = (2/π) ∑ (−1)ⁿ⁺¹ sin(nt)/n for n ≤ k. Term n is a circle of radius 2/(nπ) turning n times as fast as the first, and the chain's tip sits at height f_k(t). Heard at 220 Hz, f₁ is a pure tone; f₅₀ carries every harmonic up to 11 kHz. Near each jump the sum overshoots: 1.159 at k = 50, tending to 1.179, about 9 % of the jump.

Each gear's pitch curve is the golden spiral r = a φ^(θ/π), cut off after three quarters of a turn. Two of them roll on each other without slipping exactly when r₁ + r₂ = D, and then ω₂/ω₁ = r₁/r₂. Turn the first gear 142° and the next turns 97°, then 59°, 39°, 25°: each link passes on about 0.64 of the motion, near 1/φ, so 22 gears on a golden spiral rock like a wave that dies out.

The Sierpinski triangle’s outline is one unbroken path: 729 steps of the arrowhead curve plus the 64-step base. Two chains of rotating circles draw it, one setting x and one setting y, where their dashed guides cross. One circle per axis is off by 34.5 % of the height, 16 by 7.91 %, and 128 still by 1.31 %. The fractal has dimension log 3 / log 2 = 1.585, so the path wiggles at every scale and the circles never settle.

Euler's formula, e^(iθ) = cos θ + i sin θ, puts e^(iθ) on the unit circle at angle θ: its real part is cos θ and its imaginary part sin θ, two waves of period 2π. The reason is one derivative, d/dθ e^(iθ) = i·e^(iθ). Multiplying by i turns a vector through 90°, so the velocity stays perpendicular to the radius with length 1, and the point runs round the circle at unit speed, covering arc θ by time θ.

The X mark, drawn as one 20-corner outline with a thin slit cut into its thick stroke, 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 49.9 % of its height at N = 1, 2.15 % at 64 and 0.49 % at 256. The tip's distance |z(t)| plays as pitch. Each corner adds one term to every circle, which caps them at |cₖ| ≤ 5.35/k² (k ≥ 1).

A Klein bottle is a closed surface with one side. Here it is a tube of radius r = 4 − 2cos u whose centre runs round (6cos u(1 + sin u), 16 sin u). For u ≤ π each circle stands in the vertical plane at angle u; for u > π it lies parallel to the xz plane. The ends meet as (2π, v) = (0, π − v), a reflection, so a normal carried once round returns reversed. No closed one-sided surface embeds in 3-space, so the neck must pierce its own wall.

The Moore curve of order 4 is a loop of 256 unit steps through every cell of a 16 × 16 grid. Each coordinate has its own real Fourier series, and one chain of arms per axis draws it. The second half of the loop is the first half turned 180°, so every even harmonic is zero, and its mirror leaves x only sines and y only cosines. A curve that wiggles everywhere converges slowly: 32.3 % of the height at N = 1, still 0.47 % at N = 255.