John Carlos Baez on Nostr: Take 2 points. Draw all the curves where the product of the distances from these 2 ...

Take 2 points. Draw all the curves where the product of the distances from these 2 points is some constant or other. These are called the 'ovals of Cassini'.

There's one that's special, shaped like ∞. This is the 'lemniscate'. This is the one connected to pi's evil twin.

Here's a formula for the lemniscate in polar coordinates:

r² = cos2θ

Just as the perimeter of a circle is 2π, the perimeter of this curve is 2ϖ where

ϖ ≈ 2.62205755...

People have computed this number to over a trillion digits.

Just as we can use the circle to define the trig functions sin and cos, we can use this curve to define functions called sl and cl. Most of the usual trig identities have mutant versions that work for sl and cl. For example just as we have

sin²θ + cos²θ = 1

we have

sl²θ + cl²θ + sl²θ cl²θ = 1

To see a graph of sl and cl, go here:

https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions

But I want to show you some formulas for π and ϖ!

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