Most people know about the numerical constant pi (or π, if you prefer). But did you know that pi has an evil twin represented by the symbol ϖ? As [John Carlos Baez] explains, it and its related functions are related to the lemniscate as pi relates to circles. What’s a lemniscate? That’s the proper name for the infinity sign (∞).
[John] shows how many of the same formulas for pi also work for the lemniscate constant (the name for ϖ). Some (as John calls them) “mutant” trig functions use the pi-like constant.
Mathematically, a circle is a point (the center) with a curve that describes x2+y2=r2. The lemniscate is a particular instance of a Cassini oval where r2=cos2θ. We all know the circumference of a circle—basically, the perimeter—is 2π; the perimeter of the lemniscate is 2ϖ.
Why does any of this matter? Well, [John] shows how it connects to elliptic curves and the Gauss constant.
Like pi, the lemniscate constant probably never ends, but it is roughly 2.622057. Will this be useful in your next project? Probably not. Will it help you win some bar bets? Maybe.
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'.)
![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://poliverso.org/photo/preview/640/29300721 "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:
\[ \frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots \]
\[
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12 \Bigg/ \!\sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12}} \cdots
\]")