The pitch for Tau is as a replacement for Pi. That it is more logical than Pi.
Tau has an important lesson to tell: intuition is important and in many cases shifting things around so they're easier to create an intuitive understanding is often very trivial and boils down to aesthetic/representation choices.
Claiming that "pi is wrong" is picking a fight that doesn't need to be fought with a crowd that will not be convinced.
Instead, if I were a more full Tau advocate I would do more along the lines of Khalid from Betterexplained and seek to find other interesting intuitive takes on dry topics that are often taught by wrote.
Pedagogy is important. And pedagogy can get stuck.
"Electromagnetism" is a good example. We still learn the Heaviside-Hertz formalisms which are based around an "ether". Why?--Because they work and you can calculate a lot of simple systems by hand. This helps A) develop intuition and B) generate calculations which approximate more complex systems.
However, once you start trying to do motors and capacitors and stuff--things get weird (starting with "displacement current"). At that point, the fields formulations tend to work much better (See: "Collective Electrodynamics" by Carver Mead). However, your calculations take a jump in complexity that you didn't have to deal with before--your linear algebra and vector analysis better be pretty solid. However, dealing with motors and coupled fields and stuff gets much easier.
I mean Newtonian methods put is on the moon so they are good enough for a lot of cases.
Something as 'simple' as inertia looks very different once you step from Newtonian/special to general relativity.
There were others before that suggested other symbols like a Pi with three legs. The concept of Tau is not new. The issue at this point is just convincing people to use Tau and getting it into textbooks. And for this purpose, the loudness of the manifest by claiming Pi is wrong has worked wonderfully. The topic has been debated and reiterated plenty if times using this title.
In the same spirit, I think it is time to rethink the verb 'to be': the only consistent conjugation is obviously 'I/you are; he ares; we/you/they are'. And similarly the past tense should be 'ared' instead of 'was/were'. Generations of children have been getting themselves needlessly confused because of this terrible convention!
Would it be possible to correct this wrong? Or do you think the time for doing so has passed and, by now, it are what it are.
Childs perhaps then?
So in that spirit I think it might be better to only propose fixing the verb 'to be' and to leave the poor children out of this discussion.
Also, pi was not strictly pi by Euler, who used pi for both 3.14... and 6.28... in different papers.
Edit: did you know you can do without “be” at all? It purely synthetic.
Wouldn't it become rather difficult to do identity statements without "is"? E.g.
"The Tor network is an anonymous relay network overlaid on the internet".
You could perhaps say
"The Tor network works on top of the internet and performs anonymous routing",
but that's not quite the same thing as saying the anonymous routing is Tor's reason for existence (as opposed to just something it happens to be doing).
The E-Prime guys want the is-of-identity to be hard (if not impossible). But it's not obvious that everybody would agree with their sentiment.
Not really, depending on the language.
Nitpick: This isn't an identity statement by the way: You can't replace "is" with "equals" here because it's being used to classify. It denotes subset, not equality or equivalence.
"Thingy is blue". (nominative connection)
"Thingy is presenting". (helper verb)
"Thingy is." (existence)
Other languages separate them and so can have more logical agreement.
The unit of measurement you propose will be just a new radian - basically 2pi = 1, or pi (old radian)= 1/2 (new radian) except you want to call it a "revolution" because for the circle, this length is the circumference (screw learning other shapes, right?). Then the power series for sine, cosine, etc would be rescaled by this term. This means the derivative of the sine is no longer the cosine, there is this ugly term added. Thus neither is the derivative of the cosine the negative sign. Fourrier series formulas now change. The wave equation now changes as it's not true that taking the derivative twice and adding to the original function is zero -- trig functions no longer satisfy the wave equation! So all these other formulas now get this scaling term inserted into them as well, it just keeps multiplying.
Moreover students would discover when they turn to ellipses and other shapes, that for their formulas to generalize, a "revolution" stops being a "revolution" but merely becomes a distance along an arc and they have to go back and really understand what they were doing all this time was working in terms of arc length rather than revolutions, except that for a unit circle and this special scale, there was an illusion created that it was revolutions that was going into the formulas rather than arc length. That would then be even more confusing and is kinda the definition of a bad education.
So the advantage of rescaling the radian to this new scale and calling it a "revolution" is a net negative to anyone who actually learns the field - which is hopefully the student.
The key concept, the more generalizable and useful concept, is "distance along an arc", not "numbers of revolutions". These two concepts are not the same, and only one of them is at the heart of trigonometry and calculus. We have moved past the understanding the Babylonians had and now we have a deeper understanding of unifying concepts. It is these concepts that we should teach students, even if they are a bit more subtle than "revolution". As the saying goes, "make everything as simple as possible. But not simpler."
My point is that I with kore than a decade of working theoretical physics I have NEVER seen anyone, student, professor or otherwise not use revolutions as their unit of concern. However they do it by squeezing together 2Pi (or very few by using tau) as a pseudo-unit. Imagine if we stopped using meters and instead used a unit we called meterians which was 1/sqrt(2) meter, yet everyone always in every single equation or table listing wrote out “72 sqrt(2) meterians” would that not be stupid?
As for redefining sin and cos, there really is no need you almost always see equations using cos(x pi) and the “annoying factors” you get in derivations still pop out, just not from the function itself, but instead from the factor you’ve put in place so you can keep dealing with rotations or half rotations while pretending to use radians.
As for concerns about other shapes it’s literally the case that the opposite of what you proclaim is true. If you rotate any other shape than a circle then there is no connection between radians as a length along the perimeter and rotation. A unit square requires 8 perimeter length units to rotate. By any kid even before they learn to multiply will be able to show you exactly that rotating a square once completely is one rotation. The fact that you might personally say “no that’s not intuitive! We should say that it’s 1 times (8 radial perimeter units) just shows that you’ve disconnected the math from what’s intuitive.
And changing how we deal with trigonometric formulas to drop the constant reference to pi, even though we’re multiplying it in and deciding net it away, doesn’t change anything, the fact that it shows up everywhere in formulas is just a concealed even of mathematical robustness, if we decided instead to measure angles in units of 1/200*(4pi^2)/tau as is done in land measurements doesn’t make the math simpler or harder, it just changes the scaling factor. But there is the benefit that people immediately and intuitively understand the measurements written down without need of a calculator, though that would also be the case with simple fractions and direct percentages.
I think this is false. Everytime someone uses a trig identity or takes a derivate, they are using arc-length as their unit of concern. You may not have realized it, but that is what was happening.
To speak merely of "rotation" and be disconnected from any kind of length or position in coordinate space doesn't allow you to do anything useful, and you have a degree of freedom in choosing what is a full revolution and now you have to do bookkeeping with coordinate transformations that ugly up all your formulas and and add extra steps.
The radian is the natural unit to use: 1 unit of distance of arclength. You are not better off with a fake degree of freedom and then transforming back and forth whenever you need to work with any kind of function relating that angle to a length or rate of change.
Moroever when you start generalizing to other curves -- for example, hyperbola and working with hyperbolic sines and cosines, it no longer makes sense to speak of rotations per se as these need not be periodic at all, but it makes a lot of sense to speak of arclength. Arclength works on any curve. It is just distance, it is canonical. It is the natural unit that measures how far you have travelled along a curve (once you select an orientation), thus allowing for a consistent approach and unification of all these identities.
The reason why radians are the One True Way is the power series expansions for trig functions. Measuring the angle in radians makes
e^ix = cos x + i * sin x
work out where
e^t = 1 + t + t^2/2 + t^3/3! + t^4/4! + ...
(Exercise: derive the power series expansion for sin and cos from this)
This formula connects geometry of circles to algebra and calculus in a fundamental way - there is no geometry needed to calculate e^ix, sin, or cos with power series (a computer could do it!), while cos and sin are all about angles and circles.
Measuring angle in anything other than radians makes the formula horrendous (try it).
This is a law of the universe. It doesn't know anything about degrees or sexagesimal number systems.
You can't give this to students straight away because they don't have the background to deal with it. Hence the diversions into triangles, imaginary numbers, and radians.
It’s amazing people assume math will break somehow if you use a different unit for angles, when most should at least be familiar with using both 360 and ~6.283 and some using 200 as the scale for one rotation. And when trying to reason about it every single person always translates this into ratios of rotations or percentages of rotations.
I had thought myself sometimes why there was always 2PI everywhere and how PI was half a circle (instead of a whole).
I was always eventually convinced by the PIr^2 = area function that PI must be correct. Also a lot of people thought about it for a long time right?
The example that many other surface area functions have 1/2 in them convinces me that the function should be 1/2 TAU * r^2.
Good luck convincing the rest of us degenerates.
- e^(iτ) = 1, meaning that taking a full revolution around the complex plane brings you back to 1, where you started
- e^(iτ/2) = -1, meaning that taking a half revolution takes you to the opposite of where you started, -1.
If you insist on the "contains the additive identity" formulation (which IMO is gratuitous because e^(ix) is all about the complex plane, not regular addition and multiplication), you can do either e^(iτ) + 0 = 1, or e^(iτ) - 1 = 0.
Moreover, it seems to me that starting with tau is more logical in that a circle results from the rotation of the radius, still nothing will change as pi is so absolutely entrenched.
And we should drop the stupid Earth-centric metric system and go to some kind of Planck units.
Base twelve is better than thirty because you can represent 1/4 (0.3) and 3/4 (0.9) with a single digit.
And it proved itself useful on this Earth for probably longer than the decimal system.
We should not be calling it "tau". That's already used for too many things, and anyway the constant is important enough to deserve its own symbol.
Instead of cutting it down from two verticals to one, we should add a symbol that has three verticals. Which has the additional advantage of giving a real visual cue to what's going on.
We could arrange them in a square shape.
And somehow at the same time The Tau Manifesto is written by Michael Hartl, who also wrote the Ruby on Rails Tutorial .
None of these is 2 * pi (M_PI_2 is pi / 2, and M_2_PI is 2 / pi)
That is a shame because 2 * pi is commonly needed and having it in 2 * M_PI form can cause mistakes such as order of operations if it's a divisor.
Since names M_2_PI and M_PI_2 are already taken, M_TAU would be very sensible.