Islamic mathematicians like Jamshīd al-Kāshī (c. 1380–1429) focused on the circle constant 6.283... although they were fully aware of the work of Archimedes focusing on the circle constant that is nowadays called π.
David Gregory used π/ρ to represent perimeter/radius.
William Jones first used π as it is used today to represent perimeter/diameter (Synopsis palmariorum matheseos (London, 1706), p.263.)
Leonhard Euler adopted the same definition as William Jones, which helped popularized it into the standard it is today.
Paul Matthieu Hermann Laurent, though never explaining why, treated 2π as if it were a single symbol in Traité D'Algebra by consistently not simplifying expressions like 2π/4 to π/2.
Fred Hoyle, in Astronomy, A history of man's investigation of the universe, proposed using centiturns (hundredths of a turn) and milliturns (thousandths of a turn) as units for angles.
2009: Mathematics (28 June 2009). "2pi Day". Facebook. Retrieved 2011-07-24. {{cite web}}: |author= has generic name (help); External link in |author= (help)CS1 maint: numeric names: authors list (link)
T. Colignatus. Trigonometry reconsidered. Measuring angles in unit meter around and using the unit radius functions Xur and Yur. T. Colignatus, 2008.
T. Colignatus. Conquest of the Plane. Using The Economics Pack Applications of Mathematica for a didactic primer on Analytic Geometry and Calculus. Consultancy & Econometrics, March 2011. ISBN978-90-804774-6-9.
6:28 is a more convenient time to start celebrating than 3:15 (besides being after the actual start of the day rather than midnight)
Feynman point better in τ: starts earlier (761 digits after the radix mark[7] rather than 762 in π), is longer (7 nines[8] rather than 6 nines in π), and thus more improbable (0.008% vs. 0.08% in π[9])
First of all, the pun is not that strong of an argument: it only works because π is mispronounced "pie" in English, rather than "pea" as in the original Greek and most other languages. Even if people decided to eat peas instead, the pun would still only work for English speakers, which doesn't play well with the universality of a mathematical constant.
Second, pi radians is half a circle, not a full circle as most pies are, which weakens the association. If this inconvenience is ignored, then this ends up actually backfiring into favoring Tau, since on Tau day you can eat two pies!
An intriguing comment by Terence Tao: "It may be that 2*pi*i is an even more fundamental constant than 2*pi or pi. It is, after all, the generator of log(1). The fact that so many formulae involving pi^n depend on the parity of n is another clue in this regard."[1]
3Blue1Brown's "Euler's formula with introductory group theory" shows the significance of as highlighting the equivalence between multiplicative actions (rotations) and additive actions (translations) in the complex plane.
Furthermore, from 20:08 onwards: "what makes the number special is that when the exponential maps vertical slides to rotations, a vertical slide of one unit, corresponding to , maps to a rotation of exactly one radian — a walk around the unit circle covering a distance of exactly one. (...) and a vertical slide of exactly units up, corresponding to the input maps to a rotation of exactly radians, half way around the circle; and that's the multiplicative action associated with the number negative one."
This seems to be a special case of Euler's rotation theorem, which states that any affine transformation (TODO: confirm) can be represented as a single rotation around a given "half-vector" (origin point + direction).
The tau symbol having one leg (compared to pi's two) may be interpreted as the diameter (horizontal stroke of the character) over the radius (vertical stroke), while pi is the diameter over twice the radius
There's a formal proof of tau = 2pi in Metamath here. It's surprisingly more extensive than I'd expect. I wonder if other formal math systems/libraries (e.g. Lean's Mathlib, Coq's Mathematical Components, etc.) could have something equivalent, and whether they would choose different approaches to prove the fact.
References
TODO
Gather more stuff from Harremoës' and Palais' pages.