Butterfly Curve
r = exp(sin(θ)) − 2 · cos(4θ) + sin((2θ − π)/24)⁵
Open in the app The dials and keys named below are the app's.
What it draws
Three terms make the wings, and none of them holds the clock, so the figure stands still. exp(sin(θ)) runs between 1/e ≈ 0.37 and e ≈ 2.72, one long bulge a turn, which is the body. −2·cos(4θ) swings between −2 and 2 four times a turn, and it is what gives each turn four lobes: two long ones reaching 4.06 and two short ones at 2.50.
The last term, sin((2θ − π)/24)^5, is the slow one. Its period in θ is 24π, twelve whole turns, which is why Turns of θ opens at 12: only there does the curve close on itself. Over that sweep r runs from −2.47 to 5.05, and the stretches where it is negative are plotted through the origin, on the far side.
Why twelve turns
The fifth power is a slow dial on the size of each loop. Turn by turn it moves through its own cycle, so the twelve loops the pen lays down are the same wing at twelve different reaches, and their overlap is the shading in the wings. Raised to the fifth, the sine spends most of its range near zero and only briefly near ±1. So most turns sit close together and one or two swing wide.
History
Temple H. Fay published the curve at the University of Southern Mississippi in 1989, in a two-page note in the American Mathematical Monthly, volume 96, pages 442 to 443. It has been a standard demonstration of what a polar plotter can do ever since.
Try
- Top in the deck: the butterfly as it is usually printed.
- Set Turns of θ to 1, then to 6: one wing, then half the insect.
- Drop the slow term, exp(sin(θ)) − 2·cos(4θ): every turn is now identical, and twelve turns draw one loop twelve times over.
- Count the lobes differently with exp(sin(θ)) − 2·cos(6θ) + sin((2θ − π)/24)^5: six a turn instead of four.
- Set it spinning by writing θ + t for θ in the first two terms, which is the preset next door, Fourfold Mandala.