Fourfold Mandala
r = exp(sin(θ + t)) − 2 · cos(4(θ + t)) + sin((2θ − π)/24)⁵
Open in the app The dials and keys named below are the app's.
What it draws
The formula is Temple Fay's butterfly curve with the clock added to the angle in its first two terms, and those two carry all the shape. exp(sin(θ + t)) runs between 1/e ≈ 0.37 and e ≈ 2.72 once a turn, tipping the figure to one side. −2·cos(4(θ + t)) swings between −2 and 2 four times a turn. Their sum has four peaks a turn, two tall at 4.06 and two short at 2.50.
The third term, sin((2θ − π)/24)^5, is the slow one. Its period in θ is 24π, which is exactly the twelve turns Turns of θ opens at, and it adds at most 1. Over the whole sweep r runs from −2.52 to 5.06. Where it is negative the point is plotted through the origin, on the far side.
Why the layers
The first two terms depend on θ + t alone, so they repeat every turn and a lagged copy of themselves is just a rotation. The slow fifth power does not repeat: it is a little further along on each of the twelve turns. So the twelve loops are near copies of one four-lobed wing, each pulled in or pushed out by a different amount, and the mandala is the stack of them seen from above.
Time only rotates the fast part, at one radian a second at Speed 1, while the slow term stays fixed to the plate. After 2π ≈ 6.28 seconds the fast part has come full circle and the figure is exactly as it started.
Try
- Top in the deck: the twelve loops overlaid, which is the mandala proper.
- Set Turns of θ to 1: one wing, the four-lobed loop everything else is built from.
- Take the clock out of the second term, exp(sin(θ + t)) − 2·cos(4θ) + sin((2θ − π)/24)^5: the exponential and the cosine now slide against each other instead of turning together.
- Slow the drift to a crawl with exp(sin(θ + t/8)) − 2·cos(4(θ + t/8)) + sin((2θ − π)/24)^5, a full turn in 16π ≈ 50 seconds.