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If you're a serious mathematician who has been suffering through all this pop math fluff, wondering what's *really* going on here, let me finally come out and say:

The reason ϖ and its mutant trig functions are important is that they're connected to one of the most symmetrical elliptic curves of all, the Gaussian elliptic curve!

You can get this by taking the complex plane and modding out by a square lattice. Any lattice in the complex plane gives an elliptic curve and elliptic functions. But this particular case is especially nice, because a square is more symmetrical than other parallelograms, so it was discovered early.

Gauss discovered that this elliptic curve is connected to the 'arithmetic-geometric mean', defined below. And the arithmetic-geometric mean of 1 and √2 is the ratio π/ϖ. This number is called Gauss' constant.

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

I believe the higher numbers ϖₙ are similarly related to certain specially nice hyperelliptic functions. Hyperelliptic functions come from hyperelliptic curves, which are defined to be branched double covers of the Riemann sphere. I think the numbers ϖₙ are connected to some very symmetrical hyperelliptic curves, but I haven't checked.

Happy holidays!

(4/n, n = 4)