The Tau Manifesto | Hacker News Reader

The Tau Manifesto

tauday.com

82 pointsacjohnson5533 comments

Both tau and pi are inconvenient because you often need fractional multiples like 1/6 or 3/4.

We should instead take the fundamental unit of angle measurement to be pi divided by a highly composite number, say 2 * 2 * 3 * 3 * 5 = 180. Most common angles will be then be integer multiples of this unit. Let's give this unit a name, say "degree".

One full rotation = 360 "degrees"

Half a rotation = 180 "degrees"

1/10 of a rotation = 36 "degrees"

etc.

I can't believe I'm the first person to think of this. It's so simple that even the ancient Greeks could have figured it out. I should write a full internet manifesto and try to convert the world.

Stuff like signal processing is horrible if you try and use degrees.

I'm not even sure if calculating the area of a circle is in anyway better using degrees.

What happens when you have 1/7 of a rotation? There's no one multiplier that will solve all cases.

In any case, personally I think that fractions are way better, because they tell you exactly what's what.

It's bugged me as well. It feels like whenever you have a unit where all your measurements contain a multiplication by a constant factor, you should just pick a more convenient unit. This applies to radian measure regardless of whether one uses Pi or Tau. It's odd that the unit of radian measure is a quantity which one almost never encounters, and that the quantities one most often encounters are all transcendental (except 0), even in the most basic of situations.

But instead of picking an arbitrary number like 360 or 1337 or whatever, how about we pick 1? Let's give this unit a name, say "turn".

One full rotation = 1 "turn"

Half a rotation = 1/2 "turn"

1/10 of a rotation = 1/10 "turn"

etc.

What do you think?

I always liked this as well, and in this case, tau is basically that unit. In fact, maybe we could think of tau as being short for "turn".

One full rotation = tau

Half a rotation = 1/2 tau

1/10 a rotation = 1/10 tau

I especially like that this conversation went full circle and back into tau.

And as a bonus, this would make sin and cos library functions easier to implement on computers. Typically, to compute sin or cos, one has approximation functions like Taylor series which work for small values, and then one reduces the input down to that small range by doing a modulus, since sin and cos are periodic. This modulus is very complex to get right in practice, and can be very slow, especially for large values. But, if we switched from radians to turns, it would be as trivial and fast as just taking the fractional part of the input, and dividing by 4 if needed.

And as a further bonus, working in turns would mean that many more common quantities can be represented exactly, rather than requiring rounding as most multiples of Pi or Tau do.

Tau is turns. They're the same thing.

Recently went through this a few times, found it here on HN:

http://jackschaedler.github.io/circles-sines-signals/

Made me realize how useless degrees are, why pi is used, etc.

I think this whole argument is silly. I really do not think one is fundamentally better than the other. Factor of 2 constants will exist no matter which one you choose. Might as well go with pau http://xkcd.com/1292/

I don't like to think about it this way. I think it's silly to go all out and insist one is always objectively "better" than the other (it's a tradeoff), but to have this discussion is illuminating. Look through the comments at how many people gained a better understanding of geometry as a whole by reading.

I distinctly remember when I was learning geometry that there were some things that never really made sense. Why was pi defined in terms of the diameter when literally everything else we learned about circles used the radius? Why did radian angles feel off by a factor of 2? When I later stumbled upon the Tau Manifesto, it felt like a lot of things fell into place. And by that time, I had also studied calculus and had a familiarity with the kinds of things that happen in formulas which relate lengths and areas, so the discussion of the circle area formula resonated as well.

Despite all the cheap dismissals one sees, this feeling of "woah, that would have actually made sense!" is a big part of what makes the Tau Manifesto popular.

I didn't really understand why tau was "better" than pi until I understood the relationship to radians. Figure 8 [1] in the Tau Manifesto was the eye-opener for me. With tau, instead of pi, I now have a more intuitive sense of how to think in radians when doing trigonometry.

[1]: http://www.tauday.com/tau-manifesto#fig-tau_angles

http://www.thepimanifesto.com/

I like Terence's suggestion of using 2pii as the fundamental constant. Sqrt(pi) could also be useful given how often it appears.

Thank you.

Numberphile also has a really fun debate on Pi v. Tau: https://www.youtube.com/watch?v=ZPv1UV0rD8U

But changing our number system to base 12 would destroy pi day!

Yeah, but as Vihart pointed out, Pi = a half rotation, so Pi day is in June. :-)

I also recommend Phil Moriarty's numberphile video about Tau, where he covers the key point about how Tau is better for education, regardless of how it is used by everybody else in "regular use".

https://www.youtube.com/watch?v=83ofi_L6eAo

Of course, there is always Vi Hart's absolutely amazing discussion of the subject:

https://www.youtube.com/watch?v=jG7vhMMXagQ

Do we really need to do this every year? Come on.

This is probably the most pointless math argument. Apparently, some people consider it too cumbersome or confusing to write two glyphs instead of one, and would prefer to replace the whole thing with a single glyph. The chosen glyph happens to be one of the worst possible options, because it conflicts with torque (an angular force, which frequently appears in the same calculations as pi). This despite there being dozens of completely unused glyphs in the non-English non-Greek alphabets (Hebrew, Russian, etc).

It seems almost like a parody.

I think most of your arguments are valid, especially about the choice of glyph, tau is already used a ton. However, I think there is something to be said for the fact that, by virtue of being the ratio of C/R, there are exactly tau radians in a circle. This really does simplify the math, and gives more meaning to the constants on tau vs. pi.

There's more meaning from '3 * tau / 4' vs. '3 * pi / 2' because the constant tells you that you have exactly '3 / 4' of a circle. With pi, this is less obvious because there are '2 * pi' radians in a circle, but the 2 frequently disappears (like in my example), which leaves you with '3 / 2 of a half a circle', which isn't obvious how much that actually is. pi definitely does have it's uses when you're talking about the diameter, but when you're talking about something like radians, it makes more sense to use the ratio of circumference to radius rather then circumference to diameter. If we were using diameterians then it would make sense to use pi, since there would be exactly pi diameterians in a circle. Having them mismatched like we do creates a mess.

> which leaves you with '3 / 2 of a half a circle', which isn't obvious how much that actually is

I like pi. I think it's quite obvious too, but maybe you need to stop thinking about circles and start thinking about planes or lines instead. Pi is simple, straight line, or equivalently the whole half-plane above the x axis. Pi/2 is half the turning needed to get back to the straight line, i.e. right square. And so on...

I get what you're saying, but you can say the exact same thing using tau and it's simpler:

Tau is a simple, straight line, and stretches the entire length of a circle, starting from the x-axis in the positive x direction, and ending at the x-axis from the negative x direction. So Tau / 2 is the amount of turning needed to go half-way around the circle. Tau / 4 is the amount of turning needed to go a forth of the way around the circle, IE. a right square.

Pi might seem easier or obvious to you because you've already been dealing with it for years, but it still creates a situation that is more complex then it needs to be. Tau creates a simpler unit-circle, because Tau uses the radius, and we're talking about radians. Using something that's calculated using the diameter, when you're talking about a unit that's measured in radius's is asking for a mess.

It's not just a drop-in glyph replacement. The 2 is a number that can, and frequently does, disappear.

e.g. You would never write the area of a circle as (2pi)r^2/2

Are you really saying that he should have chosen a letter from the Hebrew alphabet as a glyph? Sure, it has no collisions, but it's completely bizarre and unlikely to gain widespread adoption, which is the whole point.

e^(τi/2) = -1

Didn't think so.

Yes, which means "a half turn around the unit circle in the complex plane is -1". Try explaining that in words without saying "half" or something equivalent to it.

Opposite of 1 on the unit circle in the complex plane is -1. Pi just means "enough of a turn to get back to the straight line".

That's ambiguous. Which straight line? Forwards or backwards? It's a half-turn.

You mean a vector in the opposite direction. Clear as mud.

...

e^(τi) = 1

so still better.

Do not want.

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