
Happy Tau Day - huguesdk
http://tauday.com/
======
MrManatee
I have found the idea of tau useful even though I have never used it in
writing.

One argument in favor of tau is that in many formulas pi often has the
multiplier 2 in front of it. If these formulas are written in terms of tau,
they may become slightly easier to memorize and manipulate. Perhaps so, but I
don't really care about this. It’s not a big difference. Besides, there are
also lots of formulas that are easier to memorize and manipulate using pi
instead of tau. The probability density function of the standard Cauchy
distribution f(x) = 1/pi * 1/(1 + x^2) is one example.

However, to get a deep understanding of mathematics, I want to understand the
connections between different parts of mathematics. If I see a mathematical
formula with the constant pi in it, I ask myself: "How is this connected to
circles?" The idea of tau taught me that that 2pi is the natural state of
affairs. If I see pi by itself, I need to ask myself: "How is this connected
to half-circles? Or has the multiplier 2 been cancelled away?”

So, why does the pdf of the standard Cauchy distribution above contain pi
instead of 2pi? What is the standard Cauchy distribution anyway? Take a gun
that shoots particles in random directions, and place it one unit distance
away from an infinitely long wall. Standard Cauchy distribution is where the
particles will hit the wall. The particle will only hit the wall if it is shot
in a direction towards it. This corresponds to 180 degrees - and there you
have it: the connection to half-circles. Of course, you also have to work out
the technical details. But on an intuitive level, when I see pi in the pdf of
the standard Cauchy distribution I don’t think about how the missing
multiplier makes it easier to remember a bunch of symbols; I think of
particles hitting a wall.

~~~
jeffwass
Your Cauchy argument is actually misleading and runs right into my joking
example elsewhere in this thread about needing to define a new constant
Sigma=4pi to work better with steradians.

Pi doesn't always represent _only_ a pure circle. Eg, in solid angles there
are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma.

This extra factor of two when using Tau should be just as disconcerting to tau
enthusiasts for steradians as pi is for radians.

Your example is talking about shooting particles in all directions over 3D
space. This calls for a solid angle approach. Which means you should be
integrating over steradians just as you'd use radians for an angular system.
There are 4pi steradians over a sphere's full solid angle, the wall only
covers 2pi steradians. Meanwhile the extra factor of two comes from the
integration of decreasing infinitesimal wall cross sections over the azimuthal
angle.

The fact that it comes out to 1/2 tau is merely happy coincidence. Ie, the
geometry introduced an extra factor of two because it's just as easily 1/4
sigma, and for a solid angle system (shooting particles in all directions) you
should be using steradians. Hence sigma.

~~~
tgb
Surely the Cauchy distribution you're replying to doesn't have any steradians
going on, right? It's a 1D distribution and the gun described would point
towards a random point on a circle, not on a sphere.

~~~
jeffwass
Argh, you may be right.

I was actually thinking of a separate example from physics (years ago)
calculating the classical flow of particles through an aperture. Where you
assume particles have maxwell-Boltzmann distribution of speeds and consider
them travelling in all directions over 3D and also the effective size of the
aperture vs the azimuthal angle of travel. It's quite similar to this example
actually but given the speeds you also must account for which velocity range
[v,v+dv] as a function of angle will put a particle through the hole in time
dt.

------
skrebbel
If mathematics has a bikeshed, this is it.

I enjoy the ridiculousness of it all, but people who consider this anything
other than a well-executed joke really should get a hold of themselves.

~~~
jessriedel
You're right about the bikeshedness, but that doesn't mean there isn't a
clearly better bike shed design. I also find that these sorts of
inefficiencies/inelegances compound. One or two are trivial, but when you have
30 of them, suddenly the mental tax becomes noticeable.

Furthermore, the cost falls mostly on the students, while the experts have
already paid it and don't see the need to worry about it anymore.

~~~
lamontcg
Yeah, but this bikeshed got decided hundreds of years ago.

Its only appealing to internet hipsters who want to brag about their
organically grown free range tau.

Making the next generation of math students use "2 pi" in their formulas is
going to be vastly easier than literally rewriting all the books.

~~~
jessriedel
As a practicing physicist, I found it illuminating and non-trivially useful to
know this when it was first introduced to me years ago. This is true even
though I will never use in a paper, and even if I only rarely write it in my
notes. In my head, 2\pi is a single character, to the point that I often write
\frac{1}{2} 2\pi.

------
rectang
I feel very fortunate that the tau vs. pi argument was around when I went back
to study math in earnest. I found tau much more intuitive and it helped me to
visualize and learn the material more effectively.

I now use tau whenever I can. However, I don't find switching back and forth
to accommodate pi loyalists that taxing. You don't really have to choose one
exclusively.

------
libeclipse
It's just the same useless argument every year. Just use what you want. I've
been taught pi since secondary school. I understand it well and can use it
effectively. Never have I sat there and thought, "if only there was a shortcut
to multiplying this by 2". There's whole sites, videos and movements to get
tau popular. I just personally don't understand the point.

~~~
jobigoud
That's exactly the issue. You are _thinking_ in terms of pi, so it might be
hard to see from a different perspective.

It's like using a slightly off abstraction for a concept. At first you have to
make a small effort to hold it in your head, then at some point it's committed
and you can manipulate the concept directly.

~~~
MichaelBurge
Let x be a randomly chosen real number in [0,1]. What does it mean to
"manipulate x's concept" or "think in terms of x"?

Do these phrases attach themselves to the real number, or to the expression
language? If the latter, do you say two number-expressions are equivalent if
applying some normalization function yields two equal expressions? Do you
consider two number-expressions distinct if they evaluate to the same real
number, but cannot directly be related to each other?

For example, let:

    
    
        S = { (x, e^(ix) + 1) | x in R, x > 0 },
        T = { x | (x,y) in S, y = 0 },
        c1 = min(T),
        c2 =  6 * sqrt(sum(n^-2, n > 0))
    

If my memory's right, c1 and c2 evaluate to the same real number which happens
to be equal to Pi. What does it mean to manipulate c1's concept or think in
terms of it? Does c1 have the same concept as Pi?

------
thecopy
The choice between Pi vs. Tau is completely arbitrary and should be based on
minimizing the friction of use. Everyone knows Pi, few knows about Tau. I see
no point in spending any more energy than that on this kind of nonsense
decisions.

~~~
greydius
"Because that's the way we've always done is" is __never __a valid argument.

~~~
dri_ft
Often it is. It's a perfectly good answer to, for example, "why do we drive on
the side of the road that we do?"

~~~
jobigoud
If the choice really is arbitrary then yes. If there was a compelling argument
to driving one side or the other then it would cease to be a good answer IMHO.

[https://en.wikipedia.org/wiki/Dagen_H](https://en.wikipedia.org/wiki/Dagen_H)

~~~
sevenless
Given that most people are right handed and right eye dominant it's surprising
there isn't a widely accepted 'right answer' for the best side of the road to
drive on.

~~~
kazinator
Maybe it's because driving involves both eyes and both hands.

------
nabla9
While choice between tau and pi is arbitrary from mathematical point, things
like pedagogy, notational simplicity and aesthetics matter.

The reason why I think π is probably better choice is because small multiples
of constant are cognitively easier to process than fractions. 2π is easier to
write and see as single object than τ/2\. All we need to do is to make slight
cognitive adjustment and think and teach 2π as a number instead of 2×π.

~~~
BenFrantzDale
My favorite example of why τ is the true circle constant is actually the
equation A = 1/2 τ r^2. It fits the usual quadratic form and shows how
circumference and area relate: c = dA/dr = τ r and so A = ∫ c dr = ∫ τ r dr =
1/2 τ r^2. At first glance those 1/2s seem superfluous and like it would be
nicer to just have a 2 in the other terms, but the 1/2 shows that
differentiation and integration with respect to r are happening, where the
"1/2" and the "^2" anhialate each other on differentiation and then re-form on
integration.

~~~
agumonkey
Reminds me of Taylor series. The factorial terms eluded me for so long until I
realized it was a cancelling factor for accumulated derivation of polynomials.
Now there's this link in my mind between powers / derivation / factorials.

~~~
Steuard
What polynomials are being derived, and from what premises? Are you thinking
of how to derive series solutions to differential equations or something? I'm
unclear on what you're getting at here.

(Or, oh: did you mean "differentiation" when you said "derivation"? That would
fit what you've said. Sorry for the pedantic post; I'll just leave this here
in case anyone else is confused.)

~~~
agumonkey
Sorry for the fuzzy vocabulary. TBH I wasn't even convinced I understood
taylor series correctly. A quick search on wikipedia hint at that derivation
is used sometimes; derivative; differentiation etc etc.

About the point:

    
    
        d(n)(x^n.dx)
        = n d(n-1)(x^(n-1).dx)
        = (n * (n-1 * (...)))
        = n! * x
        hence the 1/n! term.

------
rplst8
I just wish instead of celebrating these days on arbitrary month/day
combinations, that we'd instead use some physics or math based use of them. Pi
day should be when we've gone half way around the Sun and Tau, New Years Day.

~~~
pizza
Multiply the current dates by 2 and to a very near approximation (~1% off)
that's what you get :P

------
singularity2001
As today tauday is tuesday,

the Angle programming language now knows tau:

⦠ assert tau / 2 = pi

True

 __[https://github.com/pannous/angle](https://github.com/pannous/angle)

------
kerkeslager
I'm in favor of keeping pi because of one simple equation:

    
    
       e^{i\pi} + 1 = 0
    

This is, IMHO, the most beautiful equation I've come across. It's concise and
it relates all the most basic constants in mathematics. It's also useful, for
example, for operating on the logarithm of a negative number:

    
    
        e^{i\pi} + 1 = 0
        e^{i\pi} = -1
        i\pi = \ln{-1}
        \ln{x} + i\pi = \ln{x} + \ln{-1}
        \ln{x} + i\pi = \ln{-x}
    

For more see Euler's Identity[1].

[1]
[https://en.wikipedia.org/wiki/Euler%27s_identity](https://en.wikipedia.org/wiki/Euler%27s_identity)

~~~
karmakaze
This simultaneously illustrates how e^(180') is halfway round a unit. Tau
(literally/numerically?) > Pi

------
lazyant
I'd rather multiply by 2 75% of the time than divide by 2 25% of the time
(just a wild-ass guess of how often one or the other appear in common
equations)

~~~
xigency
It's probably closer to 98% and 2% of the time for serious math, engineering,
and physics.

------
jordigh
On to more important matters, Gamma function or Pi function?

[http://mathoverflow.net/questions/20960/why-is-the-gamma-
fun...](http://mathoverflow.net/questions/20960/why-is-the-gamma-function-
shifted-from-the-factorial-by-1)

Obviously, the Pi function is the right one since Pi(n) = n! for all integers
n.

------
dajohnson89
Using tau would uglify euler's equation, wouldn't it?

~~~
kqr
You be the judge of that. Compare

    
    
        e^(iπ) = -1
    

to

    
    
        e^(iτ) = 1.
    

The whole business of re-writing the identity as

    
    
        e^(iπ) + 1 = 0
    

is nothing but a hack to get around the weirdness of π as a constant.

~~~
kej
Writing it equal to 0 isn't a hack, it's a common method of understanding a
function. You factor polynomials by setting them equal to 0, for example.

In the case of Euler's identity, what we're really asking is "what values of
_x_ make e^(i _x_ ) + 1 = 0 true?" and the answer is "every multiple of π".
Using τ instead hides half of the answers.

~~~
kazinator
You mean "every multiple of 2π, shifted by π". The value -1 only comes around
once per revolution of the unit circle!

The solutions to

    
    
       e^(ix) + 1 = 0
    

are

    
    
      { π, 3π, 5π, ... }
    
    

Whereas the solutions to

    
    
      e^(ix) - 1 = 0
    

are:

    
    
      { 0, 2π, 4π, 3π, ... }
    

i.e.

    
    
      { 0, τ, 2τ, 3τ, ... }
    

Also, note that when we set a polynomial to zero, the roots appear subtracted
on the opposite side from the independent variable:

    
    
      (x - r0)(x - r1)...(x - rn) = 0
    

In the Tau-oriented Euler formula written homogeneously, there is a vague
analogy to this since we're similarly subtracting that 1:

    
    
      e^(ix) - 1 = 0

~~~
kej
Poor phrasing and math-before-coffee on my part. The point I was going for is
that e^(i _x_ ) has real values for each multiple of π. That is mathematically
interesting, and is obscured if you use τ instead.

~~~
kazinator
Yes; similarly, that a sinusoidal crosses zero twice in each period is
obscured by a focus on the period itself.

------
kazinator
Angular frequency (ω = 2πf) is related to this:

[https://en.wikipedia.org/wiki/Angular_frequency](https://en.wikipedia.org/wiki/Angular_frequency)

Note how convenient ω is when squared, in the expression of angular
acceleration, eliminating a ridiculous 4 factor.

~~~
Bromskloss
This ties into "ħ" vs. "h" as well:

E = ħω

E = hf

I don't mean this as an argument for anything; I just felt like mentioning it.

------
mkane848
OT, but Michael Hartl's RoR tutorials are amazing and helped me learn so
quickly. Worth checking out[0] if you're looking to learn

[0]: [https://www.railstutorial.org/](https://www.railstutorial.org/)

------
NaNDude
i totally suck at math, however i think i know what is a circle, i can draw
one with a compas. then i think i know what is the diameter of this circle, i
can draw it by drawing two more circles and a line with a ruler. because a
mathematician told me that the product of this diameter by a number is the
circumference of the first circle i believe him, but if another mathematician
ask me to draw two more circles and another line to define the same
circumference... i will probably believe that the first one suck less than the
second one at maths! (sorry for my english!).

------
Angostura
Pi is better because 'a slice of pie' links nicely to radius and makes it easy
for kids to remember. QED.

~~~
chronial
But pi isn't a slice of pie – it's half a pie, right?

~~~
lgas
But half a pie is a slice of pie, right?

------
juped
Wow, they've really redesigned that site's look. Not sure I like it... the
figures especially look bad.

------
thanatropism
What's a nice infinite series summing to tau that isn't merely 2S(n) where
S(n) sums to pi?

~~~
thanatropism
Ha. I've said a lot of continental-philosophy stuff that gets downvoted by the
techie culture, but this is amazing.

~~~
Bromskloss
Just checking: Do you mean that your comment amounts to continental
philosophy?

~~~
thanatropism
I'm saying this comment should be far less contentious than my comments that
somehow cite Heidegger or Deleuze.

------
devishard
Okay, tau is a little better.

Meanwhile, millions of lines of code are written in languages with no type
system to speak of, millions of Americans use imperial measurements,
_billions_ worldwide speak languages that are inefficient and ambiguous, and
many many people aren't even educated enough to know about pi _or_ tau.

We have much more damaging problems than multiplying by 2. Given the gigantic
amount of effort it would take to fix this one, I think that effort could be
better spent.

~~~
wtbob
> millions of Americans use imperial measurements

There are two types of countries: those using 'metric' units, and those who
have been to the Moon.

There's absolutely nothing fundamentally wrong with standard units (indeed,
they are better for concrete manipulation). One can do science and engineering
just as well with grains as with grams, with cups as with litres, with inches
as with centimetres. They could do with a bit more regularisation (e.g. to
make a fluid ounce a cube exactly 1, 1¼, sqrt(2) or 2 inches — each has
advantages — and adjust the ounce accordingly), but they're each completely
acceptable for their purpose.

The advantage of the standard units is that they are easy to physically
convert: cutting a yard into feet or a foot into inches is each (⅓ in one
case; ⅓, ½, ½ in the other); dividing a gallon into cups is easy (½ × 4);
dividing a pound into ounces is easy (½ x 4).

The number of times a person converts between barleycorns and parsecs in a
lifetime is approximately zero, while the number of times he takes one
physical quantity of a unit and breaks it into smaller quantities is pretty
common; optimising for the former is IMHO kinda foolish.

~~~
hexane360
But I think the standard unit system is one of many reasons engineers rely on
formulas for 100% of their calculations. When you have that many conversion
factors just to get between energy, force, distance, power, etc., you lose out
on the universality of physics. At that point you find someone who has already
done all the unit conversions and isolated it in a nice factor out front.

Also, IMO stuff in decimal is almost always easier to compare than fractional.
When someone asks me for a size up from an 8mm wrench, I know to grab the 9mm.
When someone asks me for a size up from 5/16", it'll take me a bit to get to
11/32".

~~~
wtbob
> Also, IMO stuff in decimal is almost always easier to compare than
> fractional. When someone asks me for a size up from an 8mm wrench, I know to
> grab the 9mm. When someone asks me for a size up from 5/16", it'll take me a
> bit to get to 11/32".

A lot of that is just due to how we teach fractions — but decimal notation
really is nice. I wish that we used duodecimal instead: all the advantages of
decimal, but with a far better base.

