User:Waldyrious/Tau/Right angle

Some have suggested that pi/2 (tau/4) is the true circle constant.

The Pi Manifesto and Harremoes's "Al-Kashi’s constant τ" mention a proposal by Albert Eagle, quoted by Murray S. Klamkin and Charles W. McCutchen in this letter to The Mathematical Intelligencer:

“ There are one or two other important innovations I have made which have nothing to do with elliptic functions and the first of which may cause an outcry at first; but its immense convenience must surely soon be realized. Most mathematicians must have often wished that there was a single symbol that could be written instead of ${\tfrac {1}{2}}\pi$ or $\textstyle \pi /2$. When one thinks that for the elliptic functions we have the convenient symbols of $K$ and $K'$ for the two quarter periods, it is really too absurd that for the circular functions, which are employed millions of times as often as the elliptic functions, we have no symbol for the quarter period at all, and have to express it as "half the half period."

Aside on the name of the constant:

The letter $\pi$ should have been used to denote ${\tfrac {1}{2}}\pi$. but the meaning of $\pi$ can obviously not be changed now. But the Greek letter $\tau$, with its one leg instead of two, so closely resembles $\pi$ cut in two that I have appropriated that letter exclusively for ${\tfrac {1}{2}}\pi$. 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 ${\tfrac {1}{2}}\pi$! And how convenient it is to say "the integral of $\sin ^{n}\theta d\theta$ 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 $e^{-{\frac {1}{2}}\pi }$ or $e^{-\pi /2}$ printed instead of $e^{-\tau }$? Or who won't much prefer to write $\tau$ than ${\tfrac {1}{2}}\pi$ 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 $\tau$ to stand for ${\tfrac {1}{2}}\pi$ was a necessity that was forced upon me. How immensely nicer books on Fourier's Theorem would look with it!

” — Albert Eagle. Elliptic Functions as They Should Be. Galloway and Porter, Cambridge, 1958.

The Tau Manifesto mentions an observation by Jeffrey Cornell:

“ Tau Manifesto reader Jeff Cornell pointed out, to my utter astonishment, that the formulas [for surface area and volume of a hypersphere] can be simplified further using the measure of a right angle, which he called λ (lambda): $\lambda =\tau /4.$

The biggest advantage of $\lambda$ 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

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

The simplification in these formulas appears to come at the cost of a factor of $2^{n}$, but even this has a clear geometric meaning: a sphere in $n$ dimensions divides naturally into $2^{n}$ congruent pieces, corresponding to the $2^{n}$ families of solutions to $\textstyle \sum _{i=1}^{n}x_{i}^{2}=r^{2}$ (one for each choice of $\pm x_{i}$). 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 $\lambda$ 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 $x_{i}>0$ for every $i$—and then find the full value by multiplying by $2^{n}$. This suggests that the fundamental constant uniting the geometry of n-spheres is the measure of a right angle. (I liken the difference between $\tau$ and $\lambda$ to the difference between the electron charge $e$ and the charge on a down quark $q_{d}=e/3$: the latter is the true quantum of charge, but using $q_{d}$ in place of $e$ would introduce inconvenient factors of 3 throughout physics and chemistry.)

” — The Tau Manifesto, section 5: Getting to the bottom of pi and tau

Harremoes's text also makes three additional observations (numbering and footnote mine):

“ (1) Recently, David Butler has proposed to use the symbol $\eta$ to denote Eagle’s constant.

(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.

” — Peter Harremoës, Al-Kashi’s constant τ

---

Conclusions: Overall, my take-away is that there might be a benefit to having a dedicated symbol for a quarter-turn (just like there there is one for a half-turn — $\pi$), but there's no convincing argument as to why it should take the place of $\tau$ as the primary circle constant.

Such a quarter-turn constant could be represented as the ∟ symbol (U+221F, ∟) or one of its variants ⊾ (U+22BE, ⊾) / ⦜ (U+299C, ⦜), and such a choice would be quite intuitive and straightforward (and unlikely to be overloaded with unrelated meanings, like $\eta$ or $\lambda$). Using ⦜, in particular, may help reinforce the notion that it is a right angle and not merely a rotated ∢ or ∡.

Another option could be repurposing Robert Palais' original three-legged pi, $\pi \!\;\!\!\!\pi$, which introduces a nice sequence of bisection between the circle constants (tau → tau/2 → tau/4, or 2*pi → pi → pi/2).

That said, it would be more intuitive to have the legs of the symbol directly represent how many parts the circle is divided into, so that the symbols themselves would resemble a fraction, with the denominator using a unary representation (aka tally marks). In this case, the symbol for the quarter-turn would need to have four legs:

• $\tau ={\frac {\color {lightgrey}\bigcirc }{|}}$
• $\pi ={\frac {\color {lightgrey}\bigcirc }{||}}$
• $\pi \!\;\!\!\pi ={\frac {\color {lightgrey}\bigcirc }{\|\!\;\!\|}}$

(Of course, it would be typeset to look like an actual stand-alone symbol, instead of two pi's stuck together. For example, something similar to the "pfft" joke symbol shared by user plasmafrag on Tumblr on Tau Day 2023.)

Either way, this would be a nice way to keep Palai's legacy in popularizing the circle constant.