Tau vs Pi

Pi (π) is an amazing number that defines the ratio of the circumference of a circle to its diameter.  It has important historical significance, because many civilizations knew that it existed, but until recently, none had been able to discover what it was to any degree of precision.  In the modern world, we use π all over the place.  It turns out though, that there are some places where π just does not make sense.  For example, the unit circle, used in trig and other more advanced mathematics, ends up with a lot of complicated and unintuitive fractions, and a single revolution around the circle is equal to 2π in distance.  This is confusing to new students and makes a lot of the math more complicated.  In fact, it turns out that 2π is used all over the place, and it may even be used significantly more often than π by itself.

A group of people have started advocating the use of an alternative to π.  Tau (τ) is equal to 2π.  It could be defined as the ratio of the circumference of a circle to its radius.  A unit circle using τ makes more sense, because a single revolution is equal to τ.  A quarter revolution is τ/4 (using π, it is π/2), a half a revolution is τ/2 (π), and 3/4s of a revolution is 3τ/4 (3π/2).  With τ, the fraction of the circle is the fraction of τ, but with π, it is twice the fraction of the circle, which makes it more difficult to understand and complicates the math.  So, this group supporting τ says we should switch from using π to using τ, because it simplifies the math.

It turns out that this is only part of the story though, and I am writing this, because I have not seen any evidence that anyone else fully understands the issue.  The problem is not that τ always makes more sense than π.  The problem is that we are using π wrong.  To fully understand this, we need to define both π and τ, without reference to the other.  In most debates on the subject, τ is defined as 2π, which is technically true but also misleading.   τ is not merely 2π.  Both τ and π are ratios relating the width of a circle to its circumference.  Mathematically, τ = c/r, while π = c/d, where c = circumference, r = radius, and d = diameter.  If you look closely, you might see why we end up using 2π all over the place.  Look at it this way: π = c/2r -> 2π = c/r.  It should be obvious by now.  π is a ratio of the circumference to the diameter, but we are doing all of our math using the radius.  Of course we have to multiply π by 2 all the time, because we are implicitly dividing the diameter by 2 nearly everywhere we use it!

The most obvious solution is to replace 2π with τ.  This is certainly a valid solution, but it is not the only solution, and it is not necessarily the best or most sensible solution either.  The other solution is to keep using π, but use d instead of r.  This would even fix the unit circle, as radians are defined as the distance from 0 multiplied by r.  π radians is not actually π.  It is the distance πr, but the r is not written, because it is implied.  If we replaced r with d, a full revolution would be exactly π (the d is implied this time).  The only problem is that we could no longer call them radians, because the name comes from "radius," and we would be using the diameter.

So, why do we use r instead of d?  This question stumped me when I took geometry in highschool.  It did not make much sense.  The reason is simple: When we find the area or perimeter of a square or rectangle, we use width (w) and height (h).  We don't use w/2 or h/2.  So why, with circles, are we always using d/2?  It just does not make much sense.  Even when we are calculating the area of a triangle, we use (w * h)/2, not w/2 * h or w * h/2.  It does not make sense to use half the width of the circle when finding perimeter (circumference) or area, when don't use half lengths anywhere else.  I recently realized why we use r instead of d, and the answer is rather disappointing.  The definition of a circle is the following equation: r^2 = x^2 + y^2.  This is the only place I can find where it makes significantly more sense to use half width over whole width.  So why is it that we are using this one equation to define the normal case instead of using all of the others and defining this one as a special case?  Honestly, I can see no reason for doing it this way, except perhaps that this is how it was done in the past.  I don't happen to subscribe to the theory that tradition trumps logic and reason.  If tradition does not make sense, it is time to trade it for something that does.

It turns out that this entire argument is almost pointless though.  Using 2π all over the place works, and it is pretty entrenched in our mathematics.  Outside of education, math and science are not going to suddenly change because someone decides it is better to do things a bit differently.  Within education is where the "almost" comes in though.  I am not the only person who noticed that using r instead of d does not make sense in the context of all of the rest of geometry.  I am certainly not the only person who noticed that the unit circle does not make sense using π as we do.  We could just replace 2π with τ, but that would only fix the unit circle and part of the math.  It does not fix the underlying problem with using only half of the distance for circles, while we use the entire distance for everything else.  If we really want to make education easier, we should keep using π, but switch to using d instead of r.