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Matt Henderson @matthen.com · 09/09/2026

Drawing a picture from the continued fraction of a number. The continued fraction terms drive a Fourier series.

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fry69 @fry69.dev · 09/09/2026

For the uninitiated (like me, also thanks Gemini) ->

$This equation represents a complex-valued parametric function $z(t)$, which traces out a 2D path in the complex plane (often used to construct Fourier transform animations, complex epicycles, or artistic curve-drawing algorithms like those popularized by 3Blue1Brown). $$z(t) = \sum_{k} \frac{e^{i q_k t}}{q_k}$$$ $Breakdown of Components $z(t)$: A point in the complex plane at time $t$, where the real part corresponds to the $x$-coordinate and the imaginary part corresponds to the $y$-coordinate. $\sum_{k}$: A summation over a discrete set of integer frequencies or indices, $k$. $e^{i q_k t}$: A rotating complex exponential representing a vector in the complex plane with a constant length of $1$. As $t$ progresses, this vector rotates counterclockwise (or clockwise if $q_k < 0$) at an angular frequency of $q_k$. $q_k$: The frequency (or wave number) associated with the $k$-th mode. $\frac{1}{q_k}$: The amplitude and scaling factor for the $k$-th term. Higher frequencies ($\vert{}q_k\vert{} \gg 1$) have smaller radii ($\frac{1}{\vert{}q_k\vert{}}$), while lower frequencies dominate the main shape of the path. The presence of the imaginary unit factor or sign within $q_k$ also introduces a phase shift/rotation to each term.$ $Physical & Geometric Interpretation 1. Epicycles (Circles on Circles): Each term $\frac{e^{i q_k t}}{q_k}$ acts as a rotating circle vector attached to the tip of the previous vector. As $t$ increases, the tip of the final combined vector draws out a smooth, intricate path $z(t)$. 2. Fourier Representation: This is a specific form of a complex Fourier series expansion where the coefficient amplitudes are inversely proportional to their respective frequencies $q_k$. 3. Integration Context: Since $\int e^{i q_k t} \, dt = \frac{e^{i q_k t}}{i q_k} + C$, expressions of this form frequently arise when integrating periodic complex signals or solving differential equations governing oscillatory motion.$

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Matt Henderson @matthen.com · 09/09/2026

haha thank you

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