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The tau-manifesto (with a misleading symbol of the yin-yang on the background) is asking us to switch from “pi-π” to “tau-τ”. Today, I want to talk about one good thing about pi. We all know the area of a circle is $\pi r^2$ which lead to the ellipse area formula to be $\pi a b$ given a, the semi-major axis and b the semi-minor axis.

(Proponents of “τ” can argue that the circle area formula isn’t very special as $\frac14 \tau r^2$ is beautiful just like

$\frac12 \int_{\alpha}^{\beta} r^2 d \theta$

is a beautiful formula of area in polar coordinates and the old pi lovers claimed that $2 \pi r$ is “beautiful” much to the annoyance of “τ”-lovers.)

You see, your honour,$\pi a b$ extends to$\frac43 \pi a b c$ in 3 D and that is pleasing to the eye.

Objection! 4/3 is a constant very much like  $\frac14 \tau r^2$ is !

If we define the the Π(z) function (sorry for using pi’s capital letter) as the shifted gamma function

$\Pi(z) = \Gamma(z+1)$

then $\Pi(n) = n!$ and the volume of a generalized ellipsoid is a nice function of π and Π as

$V(r_1, r_2, \ldots,r_n) = \frac{\pi^{n/2}}{\Pi(\frac{n}{2})} r_1 r_2 \cdots r_n$

And doing it in “τ”(tau) will spoil the simplicity.

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