# Square Petal

Polar · r = f(θ, t)

`r = Σ_(k=1)^(1 + floor(mod(t, 10))) sin((2k−1) · 5θ)/(2k−1)`

[Open in the app](https://www.wavelace.com/app#p=116) · [This page](https://www.wavelace.com/presets/square-petal)

### What it draws

The radius is a Fourier partial sum, built from odd harmonics of `5θ`. Take only the first term and it is `sin(5θ)`, the ordinary five-petalled rose. Every extra term adds the next odd multiple, `sin(15θ)/3`, then `sin(25θ)/5`, each one weaker than the last.

That series is the square wave. As the terms accumulate the radius stops easing smoothly between its extremes and starts holding near them, so the petals lose their round tips and gain flat ends and square shoulders.

### Why five petals

Every term is an odd multiple of `5θ`, so the whole sum repeats ten times in a turn. Five of those are positive and five negative, and a negative radius plots on the opposite side, which lands each negative lobe on top of a positive one. Five petals is what remains. This is the usual rule for `sin(nθ)`: an odd `n` gives `n` petals, an even one gives `2n`.

### The overshoot that never leaves

The square wave this converges to has amplitude `π/4 ≈ 0.785`, but the partial sums reach `0.93` and stay there. Adding terms narrows the bump without lowering it: the excess is about `9%` of the jump, and it survives every partial sum however long. That is the Gibbs phenomenon, and here it is the small lip at the corner of each petal.

### Try

- Watch one cycle. The term count climbs from 1 to 10 over ten seconds at `Speed` 1, then drops back to a plain rose and starts again.
- Drop `Stack` to 1. Only the current outline stays, so each term count is seen clean rather than layered over its own past.
- Change the 5 to a 4: `sum(sin((2k−1) · 4θ)/(2k−1), k, 1, 1 + floor(mod(t, 10)))`. An even multiple gives eight petals instead of five.
- Hold the count still at three terms with `sum(sin((2k−1) · 5θ)/(2k−1), k, 1, 3)`. The shape stops moving, halfway between a rose and a square.

### Read more

- [Square wave](https://en.wikipedia.org/wiki/Square_wave)
- [Gibbs phenomenon](https://en.wikipedia.org/wiki/Gibbs_phenomenon)
- [Rose (mathematics)](https://en.wikipedia.org/wiki/Rose_(mathematics))
- [Fourier series](https://en.wikipedia.org/wiki/Fourier_series)