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Pi’s Evil Twin Goes for Infinity - Most people know about the numerical constant pi (or π, if you prefer). But did yo... - https://hackaday.com/2024/12/24/pis-evil-twin-goes-for-infinity/ #mischacks #infinity #math #pi
Hackaday · Pi’s Evil Twin Goes For InfinityMost 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 fun…
@itnewsbot - that blog article is mainly a link to my post here on Mastodon:
![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://schleuss.online/system/cache/preview_cards/images/001/153/398/original/efe82402ac69b9ab.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:
\[ \frac2\pi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12\sqrt{\frac12 + \frac12\sqrt\frac12}} \cdots \]
\[
\frac2\varpi = \sqrt\frac12 \cdot \sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12} \cdot \sqrt{\frac12 + \frac12 \Bigg/ \!\sqrt{\frac12 + \frac12 \bigg/ \!\sqrt\frac12}} \cdots
\]")
MathstodonJohn Carlos Baez (@johncarlosbaez@mathstodon.xyz)Attached: 1 image
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'.)
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