# Lemniscate

Polar · r = f(θ, t)

`r = sqrt(abs(cos(2θ + t)))`

[Open in the app](https://www.wavelace.com/app#p=27) · [This page](https://www.wavelace.com/presets/lemniscate)

### What it draws

Squared, the formula reads `r² = |cos(2θ + t)|`, and that is the lemniscate of Bernoulli written in polar form. The radius is `1` along the four bearings where the cosine is `±1`. It falls to `0` a further `π/4` on, where the curve passes through the origin and starts the next lobe. Because `r` is a square root, it leaves the origin steeply and rounds off at the tip: four lobes meeting at a point, each one `π/2` of angle wide.

### What the absolute value adds

The true lemniscate is `r² = cos(2θ)`, and it has no points at all where the cosine is negative. It is the two-lobed figure of eight along one axis, and the quarter turns either side of it are empty. Wrapping the cosine in `| |` fills those quarters with a second copy of the same curve, at right angles to the first. What is on screen is two crossed lemniscates, and the four-lobed clover is the honest reading of the formula.

The `t` inside the cosine turns the whole thing. `2θ + t` holds its value along `θ = −t/2`, so the figure rotates at half a radian a second at `Speed` 1. It has quarter-turn symmetry, so it looks the same again after `π ≈ 3.14` seconds.

### History

Jacob Bernoulli described the curve in Acta Eruditorum in 1694 and named it after the *lemniscus*, the ribbon hung from a victor's wreath. It is the special case of the Cassini ovals in which the two foci lie at the distance that pinches the oval into a figure of eight. Measuring its arc length led Fagnano and then Gauss to the lemniscatic elliptic functions.

### Try

- `Top` in the deck, and the clover reads as the two crossed figures of eight it is.
- Take the absolute value out, `√(cos(2θ + t))`: the two quarters where the root has no real value leave gaps, and the true two-lobed lemniscate is left.
- Change the `2` to a `3`, `√(abs(cos(3θ + t)))`: six lobes, and the rotation slows to `1/3` of a radian a second.
- Drop the root, `abs(cos(2θ + t))`: the same four lobes and the same tips, but each one now narrows to a thin wedge at the centre.
- Turn on `trace` and watch the pen close each lobe at the centre before it starts the next.

### Read more

- [Lemniscate of Bernoulli](https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli)
- [Cassini oval](https://en.wikipedia.org/wiki/Cassini_oval)
- [Lemniscate elliptic functions](https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions)
- [Polar coordinate system](https://en.wikipedia.org/wiki/Polar_coordinate_system)