No, pi is wrong: The Tau Manifesto | Hacker News Reader

No, pi is wrong: The Tau Manifesto

tauday.com

128 pointsadunk60 comments

I like the idea of Tau but "pi is wrong" is wrong. Pi is correct.

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.

While the article goes for tongue-in-cheek, it brings up an important point.

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.

Not unusual, we still mostly use Newtonian physics even though we know it's only an approximate solution because the math comes out close enough until you get in the realm where you are talking decent fractions of C.

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.

"pi is wrong" It's just hyperbole to get everyone's attention. From that perspective it's pretty good I reckon.

It got my attention, and when I realized it's yet another article about tau, it lost my attention.

It is not yet another article about Tau, it is THE original article about Tau. As in, the one who proclaimed this is the symbol and name of the symbol that should replace Pi.

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.

But Pi is the wrong transcendental number. That's the point. We're really studying Tau/2. Can we at least stop beeping out Pi picking say e and stop being an embarrassment in the galaxy?

That was a surprisingly entertaining and well-researched manifesto.

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.

> Generations of children have been getting themselves needlessly confused because of this terrible convention!

Childs perhaps then?

Generations of childs have ared geting themselfs

On the contrary! The manifesto suggests one specific improvement to mathematical notation whilst ignoring all the other issues, for example that Euler's Gamma(n) equals (n-1)! instead of n! or that df/dx is not a fraction.

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.

Childs and mouses are very common words for non native speakers

But why “are” and not “be” (past. bed)?

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.

>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.

> Wouldn't it become rather difficult to do identity statements without "is"?

Not really, depending on the language.

https://en.wikipedia.org/wiki/Zero_copula

> "The Tor network is an anonymous relay network overlaid on the internet".

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.

The problem with "to be" is that we have multiple conflated meanings.

"Thingy is blue". (nominative connection)

"Thingy is presenting". (helper verb)

"Thingy is." (existence)

Other languages separate them and so can have more logical agreement.

I don't like "are" for that purpose. "Is" is phonetically clearer and less awkward. I is, you is, he/she/it is, they is. I'll admit the past tense "ised" (pronounced izz-d) is still not very nice. Perhaps it should be "be"... I be, you be, yesterday they beed.

I think there's a bit of a difference between a woolly grammar you pick up organically at age 1-2, and a precisely specified system of abstract reasoning, specifically constructed as a tool of mind, that has to be hammered into you over the course of a decade.

Why do people even measure rotations in radians? It’s a stupid unit that’s never ever written out but always expressed as either “pi radians” or “tau radians”, and just like with degrees, or the measure which has 200 to a circle you end up spending time teaching students how to convert to the sensible unit we all default to, which is just plain rotations. Everyone understands intuitively half a rotation. A full rotation or a quarter rotation. And when you tell a student 30 degrees, they all spend a bit of mental effort thinking before they go “Ah yes, 1/12 of a full rotation” and if you tell them 0.52 radians they’ll just look at you blankly stating they need a calculator before they know what are you are on about.

We don't name things based on what students who know nothing about the topic find most intuitive based on zero background knowledge. We name things based on what people working in a field find most convenient for their work and the job of the students is to learn the field.

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."

> We name things based on what people working in a field find most convenient for their work

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.

> 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.

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.

I think this is the real point of confusion. It is true, but far from obvious, that radians are the One True Way of measuring angles (for understanding the universe, not necessarily for communicating to human beings). Once you believe in radians, the full angle is tau=2pi, so that is clearly the more fundamental number than pi.

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).

e^(iπ)+1 = 0

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.

Hz, rpm, no radius nor pi involved.

because sin and cos have simple Taylor series, but only when measuring angles in radians. sin(x)/x at zero is one only in radians. Just multiplying by the angle is a good approximation of sin for low angles, again only in radians. I bet even an alien mathematician will describe angles in radians.

The equivalent if sin taking revolutions (let’s say sinr) is just sinr(x)=sin(x2pi) there is no fundamental mathematical relation broken by this, that would indicate math was broken at the core. And the Taylor expansion isn’t any less simple, you simply state the pi’s in the expansion rather than having them implied in your input variable.

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.

There's a bad science fiction story in here = humans broadcast pi to the stars in many different formats, but nearby aliens sail right past oblivious, mistaking the signal for random noise because their maths uses tau.

The aliens might conclude that we have some taboo against giving correct values and instead write halves of everything. So they would come expecting us to have two heads, four arms, and four legs.

Was it absurd tongue-in-cheek sci-fi comedy similar to Douglas Adams? I can't imagine a technologically advanced civilization missing a perfectly correlated signal.

Isn't this just a refactor? He should submit a PR, maybe with a few tests.

Convinced me that PI is in fact wrong.

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.

Figure 2 in that article, illustrating a constant-width figure with three sides (a Reuleaux triangle), reminded me of a science fiction story from long ago. I finally found the title: it's "The Three-Cornered Wheel" by Poul Anderson. A spaceship crew on an alien planet needs to transport something heavy but the natives worship circles and won't let them use cylindrical rollers. They eventually find a solution that satisfies everyone.

It reminded me of an earthly vehicle usage:

https://en.m.wikipedia.org/wiki/Wankel_engine

The problem here is that pi shows up in a lot of places that have nothing to do with circles, and in those places tau does not do better (it frequently does worse). It's just not worth the hassle to make it the default, and I say this as someone who celebrates tau day every year (but I call it "my birthday" instead :P)

But as explained in the manifesto, quite often when you see π on its own generally it is a case of 2 * 1/2 π. And by removing the 2 * 1/2 π terminology you're missing the significance of why the 2 * term is there! The best example of this in the article is the λ (nee τ) version of the volume of a n-sphere; the symbolic algebra form makes far more sense.

Eh, the example I had in mind when I was writing that comment was of the evaluations of the Riemann zeta function, where the actual "factor on the pi" is not all that important because there's other things being mixed in anyways.

I might be misunderstanding you, but if you're referring to things such as ζ(2)=π²/6, then that doesn't strike me as a good example of "nothing to do with circles" and "in those places tau does not do better". This value is a special case of a more general formula that specifically features (2π)^(2n), so on surface tau would seem more natural here [1]. Also, the most intuitive explanation that I have seen for this particular case (the value of ζ(2)) was in a 3Blue1Brown's video, and it did involve circles. [2]

[1] https://en.wikipedia.org/wiki/Particular_values_of_the_Riema...

[2] https://www.youtube.com/watch?v=d-o3eB9sfls

So in what way does tau do worse?

It kinda ruins Euler's identity.

Arguably it makes it more clear what's really going on. With tau, the identity can be formulated either:

- 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.

Good points, I was just thinking about it superficially.

Over the years I've seen a number of similar articles on pi and tau but I reckon this one is by far the most comprehensive. All have the same thread and it's a shame that tau and pi weren't reversed in meaning from the outset as clearly this would have simplified many things. For example (as the article mentions) the number of times I've see this ℏ=h/2π explained or alternatively its explanation omitted when it should have been explained is numerous. With tau, all that could have been omitted.

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.

While we're trying to change the status quo in mathematics around the world, let's also all switch to the World Calendar and start using Base 12.

You could start by calling it base twelve (because 10 would be too confusing.)

Or to really obfuscate called it base 10 (in base 12) :))

60 is a better base than 12. Even 30 would be better.

And we should drop the stupid Earth-centric metric system and go to some kind of Planck units.

Base sixty would be much worse for humans. There are too many digits to memorize, and the multiplication table is enormous. It does give you a factor of 5, but 5 is only so common because we use base ten.

Base twelve is better than thirty because you can represent 1/4 (0.3) and 3/4 (0.9) with a single digit.

We do, for physics. Very often c is treated as 1. The "meter" is really just a conveniently human-scaled unit, equivalent to the "degree" in trigonometry.

> 60 is a better base than 12.

And it proved itself useful on this Earth for probably longer than the decimal system.

Yes, we should be using a circle constant that has a value twice that of "pi".

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.

And maybe we could call it: 2 pi ?

Just saying

Logically, it should have twice the verticals of pi, i.e. four.

We could arrange them in a square shape.

I only found out Tau where there was a slightly heated discussion about adding it to Ruby Core[1], still unsure if this is needed or required.

And somehow at the same time The Tau Manifesto is written by Michael Hartl, who also wrote the Ruby on Rails Tutorial .

[1] https://bugs.ruby-lang.org/issues/17496

POSIX defines M_PI, M_PI_2 and M_2_PI

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.

Bikeshedding

I am sold. Tau is the law.

Is this title trying to riff on "No, it is the children who are wrong"?

Every year, I look forward to the comments here of that page. Tauday, like the SR-71 ground check story, I’ll happily read the comments every time it’s posted.

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