# Trefoil Bloom

Polar · r = f(θ, t)

`r = 1 + 0.25 · sin(3θ + t) + 0.15 · cos(6θ − t)`

[Open in the app](https://www.wavelace.com/app#p=21) · [This page](https://www.wavelace.com/presets/trefoil-bloom)

### What it draws

The leading `1` is a circle of radius one, and the two waves after it are ripples laid on that circle. `0.25·sin(3θ + t)` pushes the rim out and in three times a turn. `0.15·cos(6θ − t)` does the same six times a turn, at a smaller depth. Together `r` stays between `0.6` and `1.4`, so it never reaches zero: the curve is a closed ring that bulges, and it never doubles back through the middle.

### Why it turns

Each term is in a frame of its own. `3θ + t` holds its value when `θ = −t/3`, so the three-lobed ripple slides round at `1/3` of a radian a second at `Speed` 1. `6θ − t` holds when `θ = t/6`, so the six-lobed one slides the other way at `1/6`. The bloom is the two drifting past each other. It returns to its opening shape after `2π ≈ 6.28` seconds, when the first has turned by `2π/3` and the second by `π/3`, a whole lobe each.

The threefold look is exact and never breaks. Turning the plot by `2π/3` leaves `3θ` unchanged and moves `6θ` by `4π`, which is also nothing. So the picture has the same three arms at every moment. That is what the second harmonic buys: it sharpens and blunts the lobes without adding a count of its own.

### Try

- Deepen the ripple to `1 + 0.6·sin(3θ + t) + 0.15·cos(6θ − t)`: the dips fall to `0.25` and the ring turns into three fat petals.
- Change the `3` to a `4`. Now `6` is no harmonic of it, and the threefold symmetry drops to a twofold one.
- Raise the fine count to nine, `1 + 0.25·sin(3θ + t) + 0.15·cos(9θ − t)`: nine is three threes, so the three arms survive.
- Send both terms the same way, `1 + 0.25·sin(3θ + t) + 0.15·cos(6θ + t)`. Nothing drifts now, and the whole ring turns rigidly.

### Read more

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