Pi vs Tau, #shorts #science #maths

[OfficialYoyleberryS]

Tau. Literally just pi multiplied by 2

[isavenewspapers]

Alright, τ supporter here. Let's break this down. I'll be grouping similar points together for convenience.

Radius (τ): C/r
Perimeter (π): L/w

Agreed. We speak of the radius far more often than we speak of the diameter, so using it to define the circle constant makes sense.

Now, I am curious: is the π point meant to be referencing curves of constant width, or shapes with the same width in every direction? Rollers, if you prefer? If so, I think that's pretty cool. It's something I admire a lot about π. However, it makes it clear that π's relationship with the circle is not special, so if the point of the video is to compare how well these constants serve as circle constants, then that's kind of a ding against π.

Area of a disk (τ): A = ∫τr dr
Area of a disk (π): A = πr^2

The first one is superior; A = ∫τr dr = (1/2)τr^2 actually shows you where the formula comes from, and it shares a form with other formulas derived via integration, such as y = (1/2)gt^2, U = (1/2)kx^2, and K = (1/2)mv^2. Writing A = πr^2 is like writing the formula for distance fallen as y = ht^2, where h is half of the gravity on Earth.

Euler's identity (τ): e^(iτ) = 1
Euler's identity (π): e^(iπ) + 1 = 0

I just want to point out that you wrote the π version of Euler's identity incorrectly, stating "e^{iπ}-1=0", which is false.

Anyway, think about what these identities are telling us. e^(iθ) represents starting at 1 in the complex plane and then traveling θ units counterclockwise along the unit circle centered at the origin. e^(iτ) = 1 tells us, "Walk τ units along the circle, you come full circle back to 1." e^(iπ) + 1 = 0 tells us, "Walk π units along the circle, walk 1 unit east, and you're at 0." At that point, just write e^(iπ) = -1; there's no use trying to hide the minus sign just because people think it's ugly. In any case, I'd prefer going full circle rather than going halfway.

Radians (τ)

Agreed.

Direct measurements (π)

Okay, sure, it's easier to measure the diameter than the radius of a round thing. But if you want it to rotate, and you're in charge of figuring out how it'd work, you'll probably end up having to know the radius anyway.

More known (π)
Used more (π)
Historical relevance (π)

This kind of makes me wonder if the video is even about comparing merits, because popularity is obviously not an indication of quality. I have no idea why the first two are listed as separate points; the third point being separate is marginally more understandable, but it's still in the same vein, which is why I put it here.

Period of pendulums (τ)
Kepler's third law (τ)
Rhodonea curves (τ)

All agreed, though these are really just consequences of the radian thing from earlier. I guess it just goes to show how important radians are.

Easier to write (τ)

I mean, slightly, I guess? Not very important.

Looks more based (τ)

If that's your opinion, sure.

Normal distribution (π)

I don't know what in particular you are referring to. The probability density function for the normal distribution is f(x) = (1/(σ√(2π)))e^((-1/2)((x-μ)/σ)^2), where 2π can naturally be replaced with τ, so I don't think it's that.

Surface area of a torus (π)

Given a torus with major radius R and minor radius r, the formula for surface area is A = 4π^2 * Rr. This comes from (2πr)(2πR) = 4π^2 * Rr, which is really just (τr)(τR) = τ^2 * Rr. However, if you have p and q as the distances from the center to an outermost and innermost point of the torus, respectively, then R = (p + q) / 2 (the average of p and q), and r = (p - q) / 2 (half the minor diameter of the torus). some algebraic shuffling yields A = π^2 * (p^2 - q^2) or A = (1/4)τ^2 * (p^2 - q^2), which certainly looks like a good case for π if you ignore how you obtained it.