# Polar Mandala

Polar · r = f(θ, t)

`r = sin(6θ + t) + 0.4 · sin(17θ − 2t)`

[Open in the app](https://www.wavelace.com/app#p=22) · [This page](https://www.wavelace.com/presets/polar-mandala)

### What it draws

`r` is a sum of two angular harmonics and nothing else: a strong sixfold one, and a fine seventeenfold one at `0.4` of its size. The value swings between `−1.4` and `1.4`. Half of the angles carry a negative `r`, and a negative radius is plotted in the opposite direction, through the origin.

That is what fills the disc: the pen runs out along one bearing, falls back through the centre, and comes out on the far side. On its own `sin(6θ)` is a twelve-petal rose. The seventeenfold ripple splits and unbalances those petals, and one turn of `θ` now has 24 outward tips.

### Why no two arms match

Turn the picture by one sixth, `2π/6`. The first term does not notice, but the second shifts by `17 · 2π/6`, which is `5π/3` and not a whole turn. So the fine ripple sits at a different phase on every arm. Six is not a factor of seventeen, so only a whole turn brings everything back, and the figure has no rotational symmetry at all.

In time the two terms drift against each other. `6θ + t` stands still in a frame turning at `1/6 ≈ 0.17` radians a second at `Speed` 1. `17θ − 2t` stands still in one turning the other way at `2/17 ≈ 0.12`. Both terms return to a symmetry of their own after `2π ≈ 6.28` seconds, so the whole picture repeats then. In `θ` the sum has period `2π`, so a second turn lays the same closed curve down again.

### Try

- Drop `Turns of θ` to 1. The picture is unchanged, and the pen under `trace` draws it in half the time.
- Remove the ripple, `sin(6θ + t)`: the plain twelve-petal rose it was built on.
- Make the fine term a harmonic, `sin(6θ + t) + 0.4·sin(18θ − 2t)`: eighteen is three sixes, so the sixfold symmetry snaps back.
- Give the ripple equal weight, `sin(6θ + t) + sin(17θ − 2t)`: the six arms stop leading and the tips scatter to every length.

### Read more

- [Rose (mathematics)](https://en.wikipedia.org/wiki/Rose_(mathematics))
- [Polar coordinate system](https://en.wikipedia.org/wiki/Polar_coordinate_system)
- [Fourier series](https://en.wikipedia.org/wiki/Fourier_series)
- [Rotational symmetry](https://en.wikipedia.org/wiki/Rotational_symmetry)