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MACurlicue curves are fascinating geometric paths created by repeatedly adding tiny complex-number steps, one after another. In this animation, the curve is generated by an iterative rule of the form
z(n+1) = z(n) + step(n, t),
where each new point depends on the previous one. The step itself is a rotating complex number whose direction is controlled by an exponential with an imaginary argument, and whose size slowly changes with n.
The angle of each step is not random. It is driven by trigonometric functions involving both the iteration index n and a continuous time parameter t. As t changes, the direction of the steps oscillates, producing smooth curls, loops, and dense spiral structures. This is why the curve looks organic and fractal-like, even though it is fully deterministic.
Because the curve is built from thousands of small additions, local patterns repeat at different scales, a key idea behind fractals. At the same time, the gradual decay in step size keeps the curve bounded and visually coherent. Curlicue curves like this connect complex numbers, iteration, trigonometry, and geometry in a single visual process, showing how rich structure can emerge from a simple mathematical rule.
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