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:

The Tau Manifesto mentions an observation by Jeffrey Cornell:

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



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: (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.)
 * $$\tau = \frac{\color{lightgrey}\bigcirc}{|}$$
 * $$\pi = \frac{\color{lightgrey}\bigcirc}{||}$$
 * $$\pi\!\;\!\!\pi = \frac{\color{lightgrey}\bigcirc}{\|\!\;\!\|}$$

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