# Fermat Spiral

Polar · r = f(θ, t)

`r = sqrt(θ) · (1 + 0.2 · sin(t))`

[Open in the app](https://www.wavelace.com/app#p=28) · [This page](https://www.wavelace.com/presets/fermat-spiral)

### What it draws

`√θ` is the whole shape. Squared it reads `r² = θ`, which is Fermat's spiral. The pen leaves the origin and is out at `√(2π) ≈ 2.51` after one turn. It reaches `√(12π) ≈ 6.14` at the end of the sixth turn, where `Turns of θ` opens. The second factor, `1 + 0.2·sin(t)`, multiplies every radius alike, so the spiral keeps its shape and only swells and shrinks by a fifth either way, once every `2π ≈ 6.28` seconds at `Speed` 1.

### Why the turns crowd

A spiral that grows as a square root is the one that grows in area rather than in radius. The disc out to the curve at angle `θ` has area `πr² = πθ`, so every radian adds the same `π` of area, however far out the pen already is. Radius has to pay for that. The gap between one crossing of a bearing and the next falls off as `2.51, 1.04, 0.80, 0.67, 0.59, 0.54`, and the arms grow tighter without ever touching.

That is the law a sunflower head follows. Helmut Vogel's 1979 model puts the `n`th seed at radius `c·√n` and turns it by the golden angle, `137.5°`, from the one before. The square root gives every seed the same area, and the angle keeps any two of them from lining up. The screen shows only the `θ ≥ 0` half of the curve; the full Fermat spiral has a mirror branch at `r = −√θ`.

### History

Pierre de Fermat wrote the spiral down in 1636, working out the area it sweeps as an exercise in his method for quadratures.

### Try

- Raise `Turns of θ` to 20 and the arms crowd into a disc, the sunflower packing.
- Add the other branch by hand: `−√(θ) · (1 + 0.2·sin(t))` draws the missing half, turned through half a turn.
- Change the root to a plain `θ`, `θ · (1 + 0.2·sin(t))`: an Archimedean spiral, whose arms keep a constant gap instead of tightening.
- Deepen the breath to `√(θ) · (1 + 0.9·sin(t))`: the spiral now shrinks to a tenth of its reach and swells back out to nearly twice it.

### Read more

- [Fermat's spiral](https://en.wikipedia.org/wiki/Fermat%27s_spiral)
- [Archimedean spiral](https://en.wikipedia.org/wiki/Archimedean_spiral)
- [Phyllotaxis](https://en.wikipedia.org/wiki/Phyllotaxis)
- [Golden angle](https://en.wikipedia.org/wiki/Golden_angle)