Fixed it all!

http://www.filedropper.com/tau_3

Bravo! I do not think I will repeal my statement! (I think it's quote 126)

More bug reports:

Within a program, pretty much anything related to displaying text seems to disable the functionality of tau for the rest of the program.

The following display the incorrect value of tau after the second line, for example:

I am completely happy with this program.


I also emailed it to Michael Hartl, in case he might be interested.

Well, http://www.filedropper.com/tau_7 is all I got, feel free to use it, and don't forget to ignore theta

For all you pi lovers out there, I'd just like to point out that this is in the OEIS, and there is no such sequence for Tau.

To go on the same topic, the sum of the first 144 digits of pi adds up to 666.

No such digit sum for tau!

Yes. But that is merely a coincidence. Pi has an even distribution of numbers, so any average of them should be around 5, which it is. However, small variation in the first hundred digits of so can lead to an average slightly below that. (666/144=4.625, close to 5)

[edit] still unbiased, just disproving a bit of numerology.
[double_edit] I have a particular love for this subject (even distribution of the numbers in pi and tau) that I spent a few days (ok, it was more like a few years) inputting 20,000 digits of pi and 20,000 digits of tau into a spreadsheet to test this... Let's just say I should have trusted my sources

I don't like using numerology as actual evidence, either. But it's a curiosity.

Yeah.


(JWinslow knows all this, below, see disclaimer at bottom)
Well, duh, and you are wrong. Due to the distribution of the digits of pi, each n-length string of numbers is EQUALLY LIKELY!

If you multiply 6 nines by 2, you get 1999998. If there is a four in front of the nines, it turns into: 9999998. If there is a n 8 in the back, you get 99999996.


Code:

[edit] There are spaces to make everything nice and neat!

DISCLAIMER:
Also (double edit) I know he's just pointing out a cool fact. I like that, and that's kinda cool. Now someone find one for pi! (to make it equal)

Using only 4 digits of tau in sequence, this program puts 2017 in Ans:


Code:


This beats Wolfram|Alpha's attempt with pi, which has 10 digits:


Code:

EDIT EDIT: He changed it (like 5 times), but I like that one, too ^^
Very clever.

Also: I would like to say that there is one for pi, but it's less "Elegant", if you ask me. But, it doesn't use cheaty trig functions.

EDIT: Crud. I got ~ninja'd...

Tau describes a full circle equaling 2pi. Pi is 180 degrees. Strobogrammatic numbers are those that are equal to themselves when rotated 180 degrees. The most recent strobogrammatic year was "1961", which contains the square number "961", so we must take the square root to have √(1961). "Pi" is a homonym of "pie". You make pies by smashing pumpkins. Smashing Pumpkins released a song titled "1979", which is another year, so in keeping our units of measurement straight, we add them up to get √(1961)+1979. It took us 2017 years to come to this revelation, so we subtract 2017 out to get √(1961)+1979-2017, which brings us all the way back to tau, which as you might know is about coming full circle.

I didn't read the thread, but this is the most important point inside of it.

Was laughing the entire time while reading that. Nice "revelations", Weregoose

Code:

Agreed. But to keep it fair, perhaps use this code instead:

Code:


Also, (This is the reason I made this post) I found some interesting graphics.

↑ Tau ↓ Pi

By myself, I tried to make a comparison of the formulae for τ and π (and, just for fun, I defined a constant η = π/2 = τ/4 and made formulas based off of that, too).

First off, some definitions, in case some aren't familiar.

n! is the factorial of n (meaning, 1*2*3*...*(n-1)*n).
n!! is the double factorial of n (meaning, 1*3*5*...*(n-2)*n for odd n, and 2*4*6*...*(n-2)*n for even n).
⌊n⌋ is the floor of n (the greatest integer ≤ n)

Also, a sidenote: the π and r formula is actually traditionally written in terms of the gamma function, which has a weird definition as:

This was supposed to describe circles and spheres with a few more dimensions, so what the heck is that doing in there? For this analysis, I will look only at formulae that don't require knowledge of calculus to evaluate.

Now for the formulas.





(there are closed-form formulae for each, but I went with the simplest formulae for a result in simplest form)

As you can see, the formulae with D for diameter look very compact, and are pretty easy to actually calculate, with τ and π being pretty much equal in this metric, but η coming out on top (by design). So if you're an engineer, or for some other reason you think diameter is more fundamental than radius, the "simplest" circle constant to use in this formula is η (though, because it is diameter, it'd be more compatible in the math to use π).

The formulae with r, however, show wildly varied complexities for each. Let's pick apart each one in detail.

In the π formula, we can see that for even numbers, the coefficient is the reciprocal of successive factorials, and for odd numbers, it is a power of 2 (which happens to be 1 greater than the power of π...this might suggest something) over successive odd double factorials. This description seems nice and easy...until you think of the following identity for even n:

Because the 2 here has the same power as the π in the volume formula, this seems to suggest we could unify the even and odd cases with τ. Indeed, this turns out to be the case.

In the τ formula, we can see that the coefficient is either 1 or 2 (depending on the parity of n) over n!!. This is even simpler to calculate, and it even demonstrates the recurrence formula more transparently, V(n) = τ*r²*V(n-2)/n . So if you, like almost all other mathematicians, think radius is more fundamental than diameter, τ is the best constant to use (and τ and the radius are compatibly defined, too).

In the η formula...trying to put it in simplest form is way too complicated. Sure, you could write both cases as 2^n / n!!, but this barely even matters, as the τ formula has no 2^n to worry about anyways, so the τ formula is simpler.

Bottom line: the simplest formula for volume of an n-sphere uses either tau and the radius, or eta and the diameter. However, because tau is defined as C/r, and eta is not defined in terms of D, tau is the best constant to use in this context.

Wow this topic is still going on! I like tau and all but I could never keep a topst going on this long. Good work !

ETA?
Eta?

I like it.

And nice point. I like the idea, and e^eta*i (which results in imaginary i) is cool!

I really like your mathematical arguments, that's the whole point of this thread!

Oh, I forgot, thanks to Cemetech pi still pops up as tau on my calculator.

While e^eta*i = i is kinda cool, I personally don't agree with the whole eta thing. The guy who made the original video promoting eta made the main arguments of eta being the ratio of the distance of a semicircular path between two points to the shortest distance between those same two points, the ratio of a circle's area to the area of the largest containing square, and the measurement of a right angle.

Of course, the first two are silly, but the last argument is really its only crowning achievement: right angles are common, and eta is a right angle.

While I don't have any argument directly refuting the point that right angles are common and useful, I'd like to compare it to pi's claim to fame in the eyes of many: the area of a unit circle is pi. The flaw there is that there is a natural factor of 1/2 when it comes to area measurement, and no such factor for angular measurement.

Hm, I think I should research some more about that eta number. If it really has an argument, I'd love to pick it apart

Also, an addendum to the volume analysis:


I think it's game, set, and match here for tau.

Lemme restate my solution:

Use them when they are more useful.

You (JWinslow) described numerous formulae with 2π and, yes, you admitted that although not all formulae use it, most do.

I think tau should be taught/used where it is more efficient/makes more sense logically, and pi used where is would be more efficient/makes more sense.