The number pi has an evil twin

This discussion helped me discover my new favorite map. <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Peirce_Quincuncial_Projection_1879.jpg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Peirce_Quincuncial_Projec...</a>

And to protect you from it, you can use the following lucky clover charm (polar plot r=cos(2<i>theta) ): <a href="https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input?i=+plot+r%3Dcos%282theta%29" rel="nofollow">https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input?i=+plot+r%3Dcos%282theta%...</a> whose perimeter can also define a constant 4*E(-3) ~ 4 * 2.4221<p><a href="https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input?i=plot+r%3Dcos%282theta%29+from+theta+%3D+-pi%2F4+to+pi%2F4\" rel="nofollow">https:&#x2F;&#x2F;www.wolframalpha.com&#x2F;input?i=plot+r%3Dcos%282theta%2...</a></i>

&gt; <i>” This ∞-shaped curve is called a &#x27;leminscate&#x27;, and ϖ is called the &#x27;lemniscate constant&#x27;. I&#x27;ll show you the leminiscate in my next post.”</i><p>This got me confused, so I went to check. Apparently <i>”lemniscate”</i> is the correct spelling.

π is derived from the circle, which is defined by distance from a single point.<p>ϖ is derived from the lemniscate of Bernoulli, which is defined by distances from two points.<p>Is there an analogous constant that is derived from a shape defined by distances from three points?

&gt; I&#x27;m not enough of a cultural relativist to believe there&#x27;s a civilization that cares more about the shape ∞ than the shape ◯.<p>Maybe these are &quot;logarithmic&quot; beings, as opposed to us &quot;linear&quot; beings? The lemniscate is based on geometric mean, which is basically multiplicative mean and&#x2F;or mean in log-space -- as opposed to the additive mean in linear space.<p>If we assume we are linear beings good at intuitive addition but somewhat bad at intuitive multiplication, there could exist beings which live in log-space and whose minds are based on multiplication. Their circle would be the lemniscate.

aside: As the Professor points out, the ratio of pi to its evil twin is ~1.198, the arithmetic-geometric mean of sqrt(2) and 1. The geometric part involves a square root, and square roots are expensive. So I was like, well, if the AM converges to GM, then due to AM-GM-HM inequality, it must converge to the harmonic mean as well. And the HM does not need an expensive square root!<p><a href="https:&#x2F;&#x2F;imgur.com&#x2F;a&#x2F;UkxkPzW" rel="nofollow">https:&#x2F;&#x2F;imgur.com&#x2F;a&#x2F;UkxkPzW</a><p>Its quite wild that the AM GM convergence is almost immediate - in just 2 steps, whereas to get a decent convergence for the Gauss&#x27;s constant via HM, you need like 15 steps.You can dispense with expensive operators like square root but you end up paying for it with numerous iterations.

Other notable constants and where they show up:<p>Euler–Mascheroni Constant (integrals and sums involving the harmonic series, Gamma functions)<p>Catalan’s Constant (certain trigonometric series, lattice Green’s function)<p>Feigenbaum Constants (logistic map, chaos in dynamical systems)<p>Khinchin’s Constant (partial quotients in simple continued fractions)<p>Glaisher–Kinkelin Constant (asymptotic expansions of the Barnes G-function, combinatorial limits and certain product expansions)<p>Ramanujan’s Constant (complex multiplication of elliptic curves)<p>Omega Constant (Omega times e to the power of Omega = 1, Lambert W function, x^x^x^... = 2)

It seems obvious that these are not twins. We can only say that π and ϖ are two among the infinite multitude of siblings ϖₙ.

Hmm. Why only 2? Why not 3 points? Can you find an interesting curve produced by a constant product of distances from N points? Maybe even in higher dimensions, for 1 point, you have a sphere. What is the shape for 2 points? Is it more like an hourglass-like double droplet?

&gt; This ∞-shaped curve is called a &#x27;<i>leminscate</i>&#x27;, and ϖ is called the &#x27;<i>lemniscate</i> constant&#x27;. I&#x27;ll show you the <i>leminiscate</i> in my next post.<p>Two of these...do not belong?

Having that shape become more important to a civilisation than the circle because it has something to do with the geometry of hyperspace seems like it could be an interesting conceit for a sci-fi setting.

If I saw ϖ in the wild I would have assumed it was an omega (ω) with a macron over it. Makes me wonder how many more varient Greek letters are out there.

The Fourier transform is composed of trigonometric sines and cosines.<p>There must be an analogous transform composed of lemniscate sines and cosines?

Infinity symbol with Lissajous curve:<p>x = A<i>sin(at + delta)<p>y = B</i>sin(bt)<p><a href="https:&#x2F;&#x2F;ericfortis.github.io&#x2F;lissajous&#x2F;?preset=Infinity" rel="nofollow">https:&#x2F;&#x2F;ericfortis.github.io&#x2F;lissajous&#x2F;?preset=Infinity</a>

The post mentions that ϖ is called “varpi”; I just wanted to add that this is actually short for “variant of pi”, also known as an “archaic form of pi” from old Greek writing.

&quot;figure of eight&quot; curves .... perhaps the simplest is the lemniscate of Gerono, which has the parametrization:<p>x = cos(t); y = sin(2<i>t) &#x2F; 2; and looks like this:<p>Lemniscate of Gerono animation <a href="https:&#x2F;&#x2F;i.sstatic.net&#x2F;VKBgs.gif" rel="nofollow">https:&#x2F;&#x2F;i.sstatic.net&#x2F;VKBgs.gif</a><p>However, the lemniscate of Bernoulli may be visually more pleasing; it has a parametrization very similar to the lemniscate of Gerono, except that both axes are scaled by a factor of 1&#x2F;(sin(t)^2 + 1) = 2&#x2F;(3 - cos(2</i>t)):<p>scale = 2 &#x2F; (3 - cos(2<i>t)); x = scale </i> cos(t); y = scale * sin(2*t) &#x2F; 2; It looks like this:<p>Lemniscate of Bernoulli animation <a href="https:&#x2F;&#x2F;i.sstatic.net&#x2F;nOPMx.gif" rel="nofollow">https:&#x2F;&#x2F;i.sstatic.net&#x2F;nOPMx.gif</a><p>per: <a href="https:&#x2F;&#x2F;gamedev.stackexchange.com&#x2F;questions&#x2F;43691&#x2F;how-can-i-move-an-object-in-an-infinity-or-figure-8-trajectory" rel="nofollow">https:&#x2F;&#x2F;gamedev.stackexchange.com&#x2F;questions&#x2F;43691&#x2F;how-can-i-...</a>

You just blew my mind. I&#x27;m taking a dive on this.

Wish people wouldn’t inject weird social jabs into stuff like this

Change pi to ϖ in this setup.<p>2022 - Non-Euclidean Doom: What happens to a game when pi is not 3.14159… <a href="https:&#x2F;&#x2F;youtu.be&#x2F;_ZSFRWJCUY4?t=406" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;_ZSFRWJCUY4?t=406</a>

Is there an evil twin to the set of prime numbers?

&gt; This ∞-shaped curve is called a &#x27;leminscate&#x27;, and ϖ is called the &#x27;lemniscate constant&#x27;. I&#x27;ll show you the leminiscate in my next post.<p>I think others have commented, but this three-way spelling certainly got a chuckle from me.

So are there an infinite amount of constants like this? In terms of pi, e and this number?<p>Just wondering, there are an infinite number of shapes I suppose? But does that mean there is an infinite amount of constants?

Is there an abstraction of a leminscate&#x2F;consonant with 3+ center points?

The lemniscate really looks like a homoclinic orbit in a 2d dynamics problem

I thought it might be e. e is often used to model unbounded growth, so it&#x27;s chaotic, while pi is harmonic.<p>Plus, evil starts with &#x27;e&#x27;, so why not.<p>&quot;Laugh with me Jocko!&quot; &quot;Eeeeeeeeeeeeee!&quot;

<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Bizarro" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Bizarro</a>

Side by side, there is a clear parallel to monopolar and bipolar fields. Is this found in any version of Maxwell’s equations?

&gt;On our planet, it was Bernoulli, Euler and Gauss who discovered this math.<p>You don&#x27;t say. Newton must have been sick that day.

I thought this was going to be about tau, which is not pi&#x27;s evil twin, but rather, the One True Circle Constant.<p><a href="https:&#x2F;&#x2F;tauday.com&#x2F;tau-manifesto" rel="nofollow">https:&#x2F;&#x2F;tauday.com&#x2F;tau-manifesto</a>

Is there something like ThreadReaderApp for Mastodon?

Am I the only one who expected the evil twin to be 3 ?

Wow, pomega is such a terrible name for it!

Woah it even has a w for wa-pi-rio.

mupi (mutant pi) or piet (pi evil twin) would be better names

Seems like a fine number, but I bet there&#x27;s quite a few more irrational computable numbers out there.

how close is ϖ to e?

&gt; I&#x27;m not enough of a cultural relativist to believe there&#x27;s a civilization that cares more about the shape ∞ than the shape ◯.<p>Rumor has it there is one civilization of lizard-people out there. One is in fact running a company here on Earth with this shape as a logo!<p>&#x2F;s

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I follow John on Mastodon. He never fails to disappoint.