Tau Day Conversation with Pinkie and Maud · 8:28pm Jun 28th, 2016
Hello everyone! It’s Pinkie Pie here for another of my crazy Math Is Fun blog posts. This one is to mark Tau (τ) Day, as it’s June 28 or 6.28 which is the first two digits of the irrational circle constant tau (τ=6.283185307179586...). And as tau is two pi (τ = 2π) then for this post I get to bring along my sister Maud!
It’s nice to be with you Pinkie.
Maud and I have a super special connection with the number tau which is connected with the super special connection between us, which goes back to when we were both fillies living on the rock farm busy farming rocks, which is explained in the story: The Tau of Two Pie. It’s kinda nice to have a special mathematical connection with my sister, but of course I would have a super special connection with Maud even if the number tau didn’t exist, just as I have a super special connection with my friends and with Gummy and with Mom and Dad and all my sisters. Come to think of it, is it even possible that the number tau could not exist? I guess not since that would only be possible if circles didn’t exist, and that doesn’t make sense as you can always draw a circle if you have a crayon and piece of paper. But what if crayons didn’t exist? Well I guess we could still imagine a circle even if we couldn’t draw it, so tau would still exist, just as my love for Maud will always exist. But I digress. This post is supposed to be all about tau.
Tau is an irrational mathematical constant defined as the ratio of the circumference of a circle to its radius.
It’s irrational in the mathematical sense, which is not the same as the normal sense. If a pony is irrational then it means they’re being all crazy and not making any sense and stuff, but when a number is irrational it means that if you write it in the normal way it just goes on and on and on and on and on and on and on and on and on and on and on and on and on… It has an infinite number of digits, which is pretty amazing, but it means you can’t write it in the normal way as that would need an infinitely big piece of paper, an everlasting crayon, and way more patience than I have… And you can’t cheat and write it as one number divided by another. That doesn’t work either. So we just have to call it τ. By the way I wrote a lot more about irrational numbers in my blog post: The Pinkie Pie Guide to Irrational and Transcendental Numbers. Now you may be wondering why not just use pi which is the much better known circle constant defined as the ratio of a circle circumference to the diameter and has just as many digits and also works just fine for calculating circles, but there is a whole community of tau-ists who insist that pi is wrong.
Tau is a more natural circle constant. As there are exactly tau radians in a full circle. A right angle is a quarter of a circle or τ/4 radians. Using pi, a right angle is π/2, which is not so intuitive.
But the nice thing about pi is we get to celebrate pi day on March 14 (3.14) and eat lots of pie.
The Tauists celebrate tau day today, and say you can eat twice as much pie.
Or better still, celebrate both pi day and tau day with three times as much pie, and use whichever circle constant you like. You know it kinda seems like this discussion is just going round in circles.
... But where's the Greater Good?
Or celebrate Grad day twice a week?
¡TauDay is the bestest day of the year evar!
By the way, you are supposed to have 404 followers, but they are not found.
You made a mistake! But you were the chosen one!
Okat that was cute. Now if only that last part had ended with Lomestone saying something to the eating three-times more pies part. We need more Limestone in general. If she runs the rock farm... does that mean as the oldest sister she somehow paid for Maud's schooling? Does Equestria even make you pay to go to universities anyways? Why are my thoughts always derailing? I'm glad I wasn't in charge of making railways tracks...
I like pi better, because I prefer to use pi*r^2 over 1/4*tau^2, and Euler's equation doesn't look nearly as nice when it's e^(.5*i*tau)=-1. I'll admit tau is more convenient for trig and polar coordinates by a longshot, though.
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nd:
((d-1)/d)τr^d=interior
2d:
τ/2r^2=a
3d:
⅔τr^3=v
4d:
¾τr^4=interior
It is much easier to compute the interior of n-spheres with Tau. As for Euler's Equation:
e^iτ=1
¿What is so hard about that? Indeed, in Tau, it relates rotations:
* e^i(0/4)τ=+1,0
* e^i(τ/4)=0,+i
* e^i(t/2)=-1,0
* e^i¾τ=0,-i
* e^iτ=1,0
When we go all of the way around the circle, we are back to positive +1,0, where we started.
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My main issue with using tau for Euler's Equation is that a full circle doesn't adequately convey how much interesting stuff's going on: why should a person encountering it for the first time assume it's notably different from exponentiating by zero, which gives the exact same result? Getting a complex and/or negative value by exponentiating a non-complex positive number makes it immediately obvious. And sure, you can use a fraction of tau to get around that, but there's something to be said for elegance - "e to the i pi" looks and sounds a lot better than "e to the i tau over two" and its ilk, and I'd definitely call it more memorable.
(Forgot Pineta already said that position too in the Euler's Equation post until I was halfway through that last sentence, but oh well - if the benefits of tau are worth bringing up continually, repeating the tradeoffs shouldn't be a problem either.)
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The bad form of Euler's Identity only gets one halfway around the circle. Try showing students that e^0 equals 1 (as does everything to the 0th power). Have students play with integer values of i. After a while, the students soon discover that they rotate around the complex plane. They discover that the constant of 1 rotation is a bit more than 6. After that, they discover that the constant of rotation is τ and rotations are just fractions and multiples of τ.
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Well yeah, in a school setting letting the students figure out what's going on is obviously superior. But the thing is, in my experience most people encounter Euler's Equation "in the wild", and investigate it because they're curious. Going full circle cuts off that curiosity hook, and people just take a quick glance and move on rather than seeking to learn more about it.
Also, I must admit I'm having trouble seeing how any form of the equation can be considered "bad", except insofar as it has negative ripple effects like failing to interest people in math. Personally, if it weren't for the elegance issue (important because something simple and euphonic is more likely to catch peoples' attention and be remembered), I'd probably go with something like "e to the five-eighths i tau" to make things even more interesting.