Butterfly Curve

The butterfly curve, introduced by Temple H. Fay in 1989, is a transcendental parametric curve in the plane that combines exponential, polynomial, and trigonometric terms. It belongs to a small family of "named" curves whose chief appeal is aesthetic: the resulting shape evokes a butterfly with detailed scalloped wings despite arising from a single compact closed-form formula.

The parameterization combines \sin(t), \cos(t), \exp(\cos(t)), \cos(4t), and \sin(t/12), evaluated over the long interval 0 \le t \le 12\pi. The \sin(t/12) factor is the key: its very slow period relative to the other terms ensures the curve does not close on itself for many wing-strokes, producing the dense overlapping detail. A single purple stroke traces the entire path, so the bilateral symmetry and the small nested loops along each wing read as a continuous trajectory.