Lemniscate
r = sqrt(abs(cos(2θ + t)))
Open in the app The dials and keys named below are the app's.
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.