# Fourfold Mandala

Polar · r = f(θ, t)

`r = exp(sin(θ + t)) − 2 · cos(4(θ + t)) + sin((2θ − π)/24)⁵`

[Open in the app](https://www.wavelace.com/app#p=23) · [This page](https://www.wavelace.com/presets/fourfold-mandala)

### 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.

### Read more

- [Butterfly curve (transcendental)](https://en.wikipedia.org/wiki/Butterfly_curve_(transcendental))
- [Polar coordinate system](https://en.wikipedia.org/wiki/Polar_coordinate_system)
- [Rose (mathematics)](https://en.wikipedia.org/wiki/Rose_(mathematics))
- [Rotational symmetry](https://en.wikipedia.org/wiki/Rotational_symmetry)