🌀 Fourier Epicycles

The Gibbs phenomenon is the overshoot that appears when a Fourier series approximates a signal with a jump discontinuity (such as a square wave). Near the discontinuity, the partial sum overshoots the true value by about 9% of the jump height, regardless of how many terms are included — the overshoot does not disappear, it just moves closer to the discontinuity. This is visible in the epicycle animation as a "ringing" at sharp corners. It can be reduced by using window functions (Lanczos sigma factor) or Fejér summation instead of partial sums.

Yes, with modifications. For a non-periodic path you can either treat a finite segment as one period (introducing a jump discontinuity at the endpoints, causing Gibbs ringing), or use the Discrete Cosine Transform (DCT) instead of the DFT — the DCT implicitly mirrors the path to create a periodic extension with no jump. Alternatively, the path can be parameterised to close it (e.g., by adding a return stroke), making it genuinely periodic. The JPEG image compression standard uses 8×8-pixel DCT blocks for exactly this reason.

The real Fourier series expresses a function as Σ (aₙ cos(nωt) + bₙ sin(nωt)), using pairs of sine and cosine terms for each frequency. The complex Fourier series uses complex exponentials Σ cₙ e^(inωt) for n = −∞ to ∞, where c₋ₙ = cₙ* (complex conjugate) for real signals. The complex form is more compact and maps directly to epicycles: positive n gives counter-clockwise circles, negative n gives clockwise circles. Most signal-processing derivations use the complex form for algebraic convenience.

The simulator sorts epicycles by their radius |cₙ| in decreasing order — largest first. This is the most visually informative ordering, as the largest circles (lowest-amplitude, coarsest-frequency components) set the overall shape while the smaller, faster circles add fine detail. Mathematically, any ordering produces the same final path; the choice only affects the animation's visual progression and the intermediate tip positions of the chain.

Keeping only odd harmonics (n = 1, 3, 5, 7, …) corresponds to the Fourier series of an odd-symmetric signal — one where f(t + T/2) = −f(t). Visually, the resulting path has 180° rotational symmetry: the second half of the drawing is the first half rotated by half a turn. Square waves, triangle waves, and sawtooth waves (in their half-wave symmetric form) consist entirely of odd harmonics, which is why their epicycle chains show this elegant 2-fold symmetry.

Yes — this has been a popular mathematical visualisation since 3Blue1Brown's 2019 video popularised it. The path of a handwritten letter or portrait outline is sampled, its DFT computed, and the resulting epicycles animate to trace the shape. A typical portrait outline sampled at 1,000 points needs roughly 200–300 significant epicycles to look convincing. The technique is essentially the same as SVG path compression: keeping only the dominant frequency components discards fine, imperceptible detail.