User:Waldyrious/Tau/Right angle

${\displaystyle {\tfrac {1}{2}}\pi }$

${\displaystyle \textstyle \pi /2}$

${\displaystyle K}$

${\displaystyle K'}$

Aside on the name of the constant:

The letter ${\displaystyle \pi }$ should have been used to denote ${\displaystyle {\tfrac {1}{2}}\pi }$. but the meaning of ${\displaystyle \pi }$ can obviously not be changed now. But the Greek letter ${\displaystyle \tau }$, with its one leg instead of two, so closely resembles ${\displaystyle \pi }$ cut in two that I have appropriated that letter exclusively for ${\displaystyle {\tfrac {1}{2}}\pi }$. Consequently I ask my readers every time they see it to say to themselves "half-pi" or "pi by two" instead of "tau" or "taw". If a shorter name is desired for it what could be more appropriate to imply half-pi than "hi"? How convenient is it to write τ as the upper limit of a trig integral instead of ${\displaystyle {\tfrac {1}{2}}\pi }$! And how convenient it is to say "the integral of ${\displaystyle \sin ^{n}\theta d\theta }$ from nought to hi" since the integrand is zero at the lower limit, and the upper limit is its high point!

It is natural that the practical man, measuring the diameter and the circumference of a cylinder, should want a symbol for the ratio of the two lengths. But a pure mathematician, noting that a diameter of a circle divides the circumference into two halves, would think it more reasonable to introduce a symbol for the ratio of half the circumference to the diameter. And he, perhaps rather surprisingly, would be showing better common sense about the matter than the practical man did!

Seriously, who can want to have ${\displaystyle e^{-{\frac {1}{2}}\pi }}$ or ${\displaystyle e^{-\pi /2}}$ printed instead of ${\displaystyle e^{-\tau }}$? Or who won't much prefer to write ${\displaystyle \tau }$ than ${\displaystyle {\tfrac {1}{2}}\pi }$ for the upper limit of a trig integral?

Those who peruse the numerous formulae in my book will, I think, come to see that the finding of a single Greek letter ${\displaystyle \tau }$ to stand for ${\displaystyle {\tfrac {1}{2}}\pi }$ was a necessity that was forced upon me. How immensely nicer books on Fourier's Theorem would look with it!

${\displaystyle \lambda =\tau /4.}$

The biggest advantage of ${\displaystyle \lambda }$ is that it completely unifies the even and odd cases, [...] eliminat[ing] the explicit dependence on parity. Applying this to the formulas for surface area and volume yields

${\displaystyle S_{n}={\frac {2^{n}\,\lambda ^{\left\lfloor {\frac {n}{2}}\right\rfloor }\,r^{n-1}}{(n-2)!!}}\;\;\;}$ and ${\displaystyle \;\;\;V_{n}={\frac {2^{n}\,\lambda ^{\left\lfloor {\frac {n}{2}}\right\rfloor }\,r^{n}}{n!!}}.}$

The simplification in these formulas appears to come at the cost of a factor of ${\displaystyle 2^{n}}$, but even this has a clear geometric meaning: a sphere in ${\displaystyle n}$ dimensions divides naturally into ${\displaystyle 2^{n}}$ congruent pieces, corresponding to the ${\displaystyle 2^{n}}$ families of solutions to ${\displaystyle \textstyle \sum _{i=1}^{n}x_{i}^{2}=r^{2}}$ (one for each choice of ${\displaystyle \pm x_{i}}$). In two dimensions, these are the circular arcs in each of the four quadrants; in three dimensions, they are the sectors of the sphere in each octant; and so on in higher dimensions.

What the formulas in terms of ${\displaystyle \lambda }$ tell us is that we can exploit the symmetry of the sphere by calculating the surface area or volume of one piece—typically the principal part where ${\displaystyle x_{i}>0}$ for every ${\displaystyle i}$—and then find the full value by multiplying by ${\displaystyle 2^{n}}$. This suggests that the fundamental constant uniting the geometry of n-spheres is the measure of a right angle. (I liken the difference between ${\displaystyle \tau }$ and ${\displaystyle \lambda }$ to the difference between the electron charge ${\displaystyle e}$ and the charge on a down quark ${\displaystyle q_{d}=e/3}$: the latter is the true quantum of charge, but using ${\displaystyle q_{d}}$ in place of ${\displaystyle e}$ would introduce inconvenient factors of 3 throughout physics and chemistry.)

${\displaystyle \eta }$

(2) In geometry the idea of using a right angle as unit dates back to Euclid.
(3) In Germany and Switzerland the symbol ∟ has been used to denote a right angle and it has been officially recognized as a unit for angle measurements in the period 1970-1996.