Temple Fay's butterfly: one long looping line that folds into wings. About a dozen circles give the outline; the fine wingtips take hundreds.
Measuring the shape…
x = sin t · (e^cos t − 2 cos 4t − sin⁵(t/12)), y = cos t · (same), 0 ≤ t ≤ 12π
Published by Temple H. Fay in 1989 as a curve for students to plot. The part in brackets sets the distance from the centre as the angle turns, and the slow sin⁵(t/12) term makes each lap a little different, which is why the wings look feathered rather than repeated.
Each path of the butterfly curve is traced by a chain of circles, each one spinning on the tip of the one before. A Fourier transform measures the path and works out every circle's size, speed and starting angle. The circles are sorted biggest first, so the first few give the rough shape and the rest add detail.
"Within 1%" means the drawing is, on average, less than 1% of its width away from the real outline. The bars show the circles of the main outline: height is size, position is how many times it turns per loop and in which direction. Click a bar to remove that circle and see what it was doing. Learn how it works.