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pi is not so wrong

August 4, 2015

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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