I wasn't sure where to post this, but I saw a few math-related things here, so I'll try my hand at this.
π is wrong. (Not as a number, but as a concept.)
The ratio of circumference to radius, defined as τ=2π, is much more fundamental.
Seriously, if you haven't read the Tau Manifesto by Michael Hartl, read it now. Someone also wrote the Pi Manifesto - read this too.
So many things become simpler with tau:
There are τ radians in a circle:
EVERY trig function is periodic with respect to τ.
The Euler identity: some have decried it, saying that e1/2τi = -1 is less beautiful than eπi =
-1.
I'll give you this: eτi = 1. This means that a rotation by τ brings you back to unity.
The nth roots of unity derive from this identity as well, and it can be easily shown that eπi = -1 is simply the second root of unity.
And nothing more.
Then there's the area of a circle: 1/2τr2. This seems like it's a better case for π, but it's actually
supposed to have the factor of one half. It can be shown that the area of a circle is equivalent to that of a triangle with a base corresponding to
the radius r and an altitude corresponding to the circumference C. This relation doesn't exist with the diameter, which raises the question:
WHAT ARE YOU DOING WITH A CIRCLE CONSTANT DEFINED BY THE DIAMETER WHEN THE CIRCLE AND ALL OF ITS MEASUREMENTS ARE DEFINED BY THE RADIUS?!
I told some friends about this. Their reactions ranged from "who cares?" to "Oh my god, I've been lied to my entire life!"Etaoin Shrdlu - 13-4-2014 at 10:14
I like them both. Is there some reason we can't have both?
EDIT: Your poll is not sufficient for my purposes.
[Edited on 4-13-2014 by Etaoin Shrdlu]smaerd - 13-4-2014 at 16:19
I can think about Tau more easily. Pi trips me up if I am doing too much mental math.
I'm not in the camp that says "who cares", but I'm also not in the "I've been lied too" camp either. Both constants work.
Edit: one plus side to pi is it doesn't look like a lower-case t, or a + sign if I am writing too fast. I could see myself looking at a parametric
equation and getting very lost as to what was a t, +, or τ.
[Edited on 14-4-2014 by smaerd]ScienceHideout - 13-4-2014 at 17:03
I respect your tau-loving conclusion and I am happy it works (and makes sense) for you... BUT... Honestly, we are all chemists... higher order math is
not very useful and Algebra 2 is as far as most chemists really need to go (except for those few that can use a bit of calc). Therefore, I like it
simple and I will stick to the way I was taught since 3rd grade. smaerd - 13-4-2014 at 18:46
If you want to do anything semi-serious with kinetics expect to use calculus and differential equations. Or anything in physical chemistry. Honestly,
I've found a good deal of the 'higher math' helped make sense of a lot of the gobbldy guck I've learned as an undergraduate chemistry major. For
example rather then looking up zero, first and second order rate equations I can come up with them pretty easily with some mental math. From there the
respective half-life equations too.
Sure, when you're mapping out a synthesis you're not solving ODE's or anything but being fluent in math can give you a good idea as to whether an SN1
or SN2 will be favored on a reactant. Even things in biochemistry such as structure solving require some rudimentary level of calculus
(electrostatics, etc). Yes computers and soft-ware handle a lot of this now, but it's not all inclusive, and someone has to tell the computer what to
do. Sure most of quantum mechanics after pchem can be forgotten for a lot of people, but in spectroscopy knowing the basic principles of simple
harmonic oscillators can say tell you where if a dueterium atom was substituted from a hydrogen atom on a molecule where the peak 'should' be in an
FT-IR spectra. Come to think of it, FT-IR, Raman spec, and NMR are all expressions of higher mathematics.
I wouldn't be so quick to chuck 'higher level undergraduate' math out the window if you're interested in 'graduate level' chemistry, is what I'm
trying to say I guess.
Edit- jeese come to think of it I even had to use calculus on a bomb calorimetry experiment I did the other semester. Not sure how I would have done
it accurately without it and polynomial curve fitting.
Edit again - even the most simple reactor models definitely requires the understanding of differential equations and chemical thermodynamics.
[Edited on 14-4-2014 by smaerd]Töilet Plünger - 13-4-2014 at 19:00
The rationale for tau is that it doesn't needlessly confuse children learning about circles. Other than that, I don't really care. Let people confuse
themselves.
In all seriousness, I only use tau in any calculations I do.Twospoons - 13-4-2014 at 20:16
If you find a factor of 2 confusing, then maths is not for you.
One rationale for pi is that if you are holding a circular object, measuring the diameter is easier than measuring the radius (if the center isn't
marked). (Imagine yourself 2000 years ago trying to work out the relationships in a circle.)
Yes, its a weak rationale, but if you have to pick some basis for a definition you want a simple one.
e.g 24 hour day because 12 (and 24) has lots of factors, making it easy to divide up time using fractions.
[Edited on 14-4-2014 by Twospoons]Töilet Plünger - 13-4-2014 at 21:55
I forgot to include something: η (eta). τ = 4η
η appears to be the n-sphere constant. It's shown to be an integral part of the surface area and volume formulae for an n-sphere. It's mentioned
in the Tau Manifesto, and there's a good video on it. Though η likely will not catch on because it's not really common in basic mathematics,
unlike τ.
I'm surprised that so many people want to keep using π! When I learned about τ I stopped using π completely. It's almost foreign now.
There's also a video on the matter by ViHart and Michael Hartl. I'm also going to try to promote it at Spaceweather.com (they had a thing for π
Day).Twospoons - 13-4-2014 at 22:08
Also τ is used in electronics to denote the time constant of a circuit, and pi is used in various other places (relating to frequency), so using
τ to replace 2*pi would cause a lot of confusion. I'll be sticking with pi thanks.12AX7 - 15-4-2014 at 00:07
Also τ is used in electronics to denote the time constant of a circuit, and pi is used in various other places (relating to frequency), so using
τ to replace 2*pi would cause a lot of confusion. I'll be sticking with pi thanks.
What, you think
tau = R*C
and
F = 1 / (tau * tau)
is confusing?! Bah! Brain&Force - 15-4-2014 at 12:25
Well, if the notation doesn't work, change the notation. Why not χ for the time constant (from Greek χρόνος? Or N for torque (another common complaint)?
Remember, physicists have to deal with the formula ψ(r) = Ne-me2r/ħ2. And the expression for N has an e in
it (the exact e intended is left to the reader as an exercise). Notational conflicts can easily be resolved.
π is used somewhere else in chemistry, I forgot exactly what it's used for...
[Edited on 15.4.2014 by Brain&Force]The Volatile Chemist - 16-4-2014 at 13:22
Tau is a fun concept, but it shouldn't change pi or normal math.Eddygp - 1-5-2014 at 03:28
But... wait... the area of a circle would be an equally valid statement! In fact, the perimeter of the circle is the derivative of the area!
LONG LIVE PIThe Volatile Chemist - 1-5-2014 at 09:37
But... wait... the area of a circle would be an equally valid statement! In fact, the perimeter of the circle is the derivative of the area!
LONG LIVE PI
My second wonderful contribution to this form:
LONG LIVE PI!Brain&Force - 1-5-2014 at 13:08
Not really.
A circle is defined by the set of all points a fixed distance (the radius) away from the center. These points form an arc which is τ times the
length of the radius. The area is not constructed: it is a DERIVED definition. The area relationships have a natural factor of 1/2 that arises from
integrating the proportionality τr with respect to r.
The area is also defined by breaking the circle into an infinite number of skinny triangles of area 1/2 × r × dC and summing them up through
integration to get 1/2 × r × C. C = τr, so A= 1/2 × τ × r2.
tl;dr: Τau is the τruth; π is for eating.
[Edited on 1.5.2014 by Brain&Force]
[Edited on 1.5.2014 by Brain&Force]Mildronate - 1-5-2014 at 22:07
Actualy who cares.Brain&Force - 10-5-2014 at 10:46
I care!
The reason being that π, being ill-defined, makes math harder for students. I'm not saying we should rewrite our scientific papers and replace
every instance of π with τ - that's unnecessary and pointless. Scientists are fine with 2π. It's the kids that are affected by this.
I would be OK with using both π and τ, but there's a problem with π that extends into the geometry of n-spheres (this is explained in
the Tau Manifesto; I can't really explain it myself). I am OK with using η in the context of n-spheres because it is the fundamental unifying
constant of their geometries.Mildronate - 10-5-2014 at 11:14
Its only constant, you dont need 2 constants. Only problem with pi ts transcendental nummber. jock88 - 10-5-2014 at 13:16
Sorry, couldn't resist!
Brain&Force - 28-6-2014 at 08:41
Happy Tau Day! (Well, today is 6/28 for us in the US - but it doesn't matter for those of you in Europe, either. 6 and 28 are perfect numbers, so
today is a PERFECT DAY!)Texium - 25-10-2014 at 17:10
Just wanted to say this: When I first read this thread back when it was started, it didn't really mean much to me at all and I thought "who cares?"
Now, I've been in Precal for a few months, and ever since we started getting into trig functions using the unit circle and stuff, I've realized how
much easier it would be to use τ than π. It seems to me like using τ/2 where π is needed would be simpler and more efficient than using 2π in the
multitude of situations where τ fits. No disrespect is meant to π, since it has had such a great and productive history, however history alone is
not reason enough to continue using it.
