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The number pi has an evil twin! It's a number called ϖ with many properties similar to π. There are even mutant trig functions connected to this number, called sl and cl.
So maybe while you were studying trig in high school, some kid in another galaxy was having to memorize all the identities for these other functions.
I doubt it. Just as pi and trig functions are connected to the circle, this number ϖ and its mutant trig functions are connected to a curve shaped like the symbol for infinity, ∞. But this curve is just less important than the circle. I'm not enough of a cultural relativist to believe there's a civilization that cares more about the shape ∞ than the shape ◯.
This ∞-shaped curve is called a 'lemniscate', and ϖ is called the 'lemniscate constant'. I'll show you the lemniscate in my next post.
A civilization will probably only get interested in ϖ when it gets interested in the lemniscate.... or the deeper math it's connected to. On our planet, it was Bernoulli, Euler and Gauss who discovered this math.
(Why does unicode even have the symbol ϖ? Here's why: it's a script version of the Greek letter pi, sometimes called 'varpi' or 'pomega'.)
(1/n)
![The lemniscate constant $\varpi$ is like a mutant version of the number $\pi$:
\[ \pi = \int_{-1}^1 \frac{dx}{\sqrt{1 - x^2}} \approx 3.14159 \]
\[ \varpi = \int_{-1}^1 \frac{dx}{\sqrt{1 - x^4}} \approx 2.622057 \]
It obeys a lot of similar formulas. For example:
\[ \frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots \]
\[
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12 \Bigg/ \!\sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12}} \cdots
\]](https://media.mathstodon.xyz/media_attachments/files/113/703/358/405/037/392/original/12262b712b868b80.jpg "The lemniscate constant $\varpi$ is like a mutant version of the number $\pi$:
\[ \pi = \int_{-1}^1 \frac{dx}{\sqrt{1 - x^2}} \approx 3.14159 \]
\[ \varpi = \int_{-1}^1 \frac{dx}{\sqrt{1 - x^4}} \approx 2.622057 \]
It obeys a lot of similar formulas. For example:
\[ \frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots \]
\[
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12 \Bigg/ \!\sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12}} \cdots
\]")
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 the unit 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!
(2/n)
Lots of nice formulas for π have partners for the number ϖ!
Back before Twitter became a Nazi bar, I issued a challenge there: find a whole series of numbers like pi, each with its own bunch of formulas. @duetosymmetry took me up on this and invented the numbers ϖₙ:
https://math.ucr.edu/home/baez//diary/june_2022.html#june_12
π is ϖ₂ and ϖ is ϖ₄. So between them there's another interesting mutant version of pi!
(3/n)
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 where there are n branch points, one at each of the nth roots of unity.
Happy holidays!
(4/n, n = 4)
@johncarlosbaez can you remind me — did we go over the Wallis-style product for the lemniscate constant? I don't know how it would be derived, and why it's similar to the Wallis product for pi, and if there's a similar product for all the generalizations.
@duetosymmetry - no, we did not get into that stuff! I think a bit of it is on the Wikipedia page 'Lemniscate constant':
https://en.m.wikipedia.org/wiki/Lemniscate_constant
Btw, notifications from you were filtered on Mathstodon because you're on an instance that puts out a lot of spam. 
@duetosymmetry - oh, this paper seems to do Wallis-style products in a systematic way:
https://arxiv.org/abs/1212.4178
We'd seen this paper before.
@johncarlosbaez hah! That's why it was in my subconscious... You linked it and mentioned the title in your post!
@johncarlosbaez I've been browsing "Squigonometry: The Study of Imperfect Circles" by Poodiack and Wood this weekend, which talks a lot about this. I know this is all intimately connected to the modular group and my Aggregate Theory of Concrete Mathematics, but I've not figured out many of the details yet.
...It's been a long time since I've really thought about calculus.
@leon_p_smith - wow, this could be cool if it gets into enough detail.
@johncarlosbaez
The introduction gives three equivalent definitions of sine and cosine. The first is based on a differential equation, specifically a Coupled Initial Value Problem. The second is based on the unit circle, and the third is an analytic definition based on an integral. That sorta seems to set the theme of the book.
Chapter 12 mentions continued fractions, and their application for approximating Pi and variants of Pi. It doesn't really go into any detail here, and just mentions without proof a few facts about continued fractions.
Then Chapter 16-17 seems to be the bulk of the material most directly related to elliptic integrals and the lemniscate.
So I'd like to understand better understand the connections between continued fractions, the modular group, and elliptic curves, which this book doesn't really get into.
Even so, it seems like this book should offer some intriguing hints into the analytic side of Langlands, especially given it's emphasis on connecting analysis and geometry.
https://old.maa.org/press/maa-reviews/squigonometry-the-study-of-imperfect-circles
@johncarlosbaez "the arithmetic-geometric mean of 1 and √2 is the ratio π/ϖ" math is so cool
@f4grx - Yeah! And Gauss was amazing!
@johncarlosbaez @f4grx I only know that arithmo-geometric mean can be used to compute pi, and ϖ plays a role.
@johncarlosbaez As someone who's really into math but suffers from poor attention span due to Brain Problems, I appreciate the pop math fluff for helping ease into a topic. This is so cool, thanks for posting about it
@november - thanks! Believe me, even supposedly good mathematicians like myself find math easier to understand with a bit of an "on ramp" to help us ease into it. I'm glad you enjoyed this.
@johncarlosbaez "the arithmetic-geometric mean of 1 and √2 is the ratio π/ϖ"
I was wondering if I could dispense with square roots by replacing the geometric mean with its harmonic counterpart. Turns out convergence is a bit slow, but it works out beautifully.
@johncarlosbaez Thanks for reminding me to get back on with a little project of mine, https://observablehq.com/@liuyao12/real-numbers-with-bigint , which lets you compute digits in your browser. Not there yet to do the AGM algorithm, but it occurs to me that you can design a whole course on calculus just on the various ways to compute pi. (I'm never a fan of the pop math side of it, but apart from the digits, pi is indeed awesome.)
@liuyao - here's a way to compute pi using AGM - not the most practical, but conceptually simple:
@johncarlosbaez yes, I remember your posting about it on the other site.