Pi’s Evil Twin Goes For Infinity

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.

Then again, if you are bored calculating more digits of pi, here’s something new to try. Not that you need that many digits.

10 thoughts on “Pi’s Evil Twin Goes For Infinity

          1. I never understood why it’s not π2r or 2rπ to reflect relation between diameter and π rather than double π (τ) and radius.

            I just notice, τ is twice as big as π but if you cut π symbol vertically you will get something like double τ. Look: ττ vs π. Symbols are all wrong! ;-)

            Merry Christmas.

  1. “Like pi, the lemniscate constant probably never ends, but it is roughly 2.622057.”

    … Uh, both numbers have been proven to be transcendental. It is absolutely certain that neither one ends or repeats.

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