Pi Day 2016

I recently wrote an article on the subject of whether we should use π (pi) or τ (tau) in circle related math.  As a refresher, π is the ratio of the circumference of a circle to its diameter.  It turns out that in a majority of equations that use π, it is multiplied by 2.  A growing group of mathematicians and scientists is pushing to replace π with τ, which is just 2π.  They argue that this would simplify a lot of math and make geometry and trig significantly easier to learn.  This is probably true, but it addresses the problem from the wrong side.  I argue that π is not the problem.  The problem is that we have this unreasonable attachment to radius.  The reason for having to apply a multiple of 2 everywhere is not that π is the wrong value.  The reason is that π is a ratio of the diameter of a circle, but we always pair π with the radius, which requires a multiple of 2 to correct this.

There are a lot of different arguments for why we should use τ instead of π, but in the end, they are based on the idea that the radius is the natural way to measure a circle.  This comes from the idea that a circle is composed of an infinite number of infinitely small triangles, but while that is a valid representation, it is not actually true.  A circle is composed of a continuous, regular curve, with no straight lines anywhere.  Just as we would not try to define any other shape as the distance from the center to any point along its perimeter, radius is not the natural way to measure a circle.  Outside of mathematics and science, where it has become tradition to use radius, measuring merely half the width of an object is not something that is often useful.

The answer is not to switch from π to τ.  The answer is to use the natural measure of the size of a circle: diameter.  Some "tauists" claim that the decision of ancient civilizations to use diameter instead of radius was arbitrary.  If that is true though, then why did every civilization that set out to measure the ratio of the circumference of a circle to its size choose to use diameter instead of radius?  The answer is not chance.  The answer is that the choice was not arbitrary.  Diameter is the natural way to measure a circle.  To someone that has not been taught to measure a circle by radius, diameter is the obvious measure of size.

The fact is, replacing radius with diameter in circle equations simplifies them just as much as replacing π with τ, with the added benefit that new students will not be confused as to why we suddenly only care about half of the size of the circle.  Besides that, how does the average person measure the radius of a circle?  They measure the diameter and divide by two, because half a width is not a natural measurement to try to take!

In celebration of Pi Day, I bring you the following:

This is the real unit circle.  You will notice that one full turn is not equal to 2π.  This is because we are measuring in diameters instead of radians.  There is no need to redefine the circle constant when using diameter, because diameter is what it was made from!  Unfortunately, "diameterians" does not sound as good as "radians," so I propose just calling them "diameters."  That works just fine, since the circumference of a circle is, by definition, π diameters.  Besides that, I think "diameters" would be far less confusing to new students, as it does not sound like some kind of new unit like "radians" does.  (Be honest, how many of you struggled with radians, because it was not initially clear that radians are literally just the distance around the circle measured in radii?  Now, think about π diameters.  Without the fancy sounding name, it is much more clear what it means.)

Now, with my nice new diameter based unit circle in mind, here are some basic circle and sphere equations using diameter instead of radius:

Circle

Definition: d² = 4(x² + y²)
Circumference: πd
Area: ¹⁄₄πd²

Sphere

Definition: d² = 4(x² + y² + z²)
Surface Area: πd²
Volume: ¹⁄₆πd³

I considered adding some trig to this, but there is so much, and most of it is already so complex, that it would have taken a lot of algebra to reduce things down.  In short, I gave up.  This is a good taste though.

Here are some things I notice with the above.  First, in the circle equations, you can replace π with diameters (see the unit circle above) to find the values for partial circles.  For example, ^π⁄₂d will give you the circumference of half a circle (^π⁄₂ diameters is halfway around the circle).  You can do something similar using τ with radius, but I just wanted to point out τ does not have an advantage here.  Now take a look at the circle circumference equation and the sphere surface area equation.  Notice the logical step from the first to the second?  Merely squaring the diameter promotes the equation to the analogous equation of the equivalent shape one dimension higher.  Area to volume is not so pretty, but sphere volume is simpler than its radius based version, and the additional factor of 4 on the circle area equation adds trivial complexity. (I made an algebra mistake in circle area.  It is fixed now.)  I also notice a similar progression with circle area to sphere volume, where the multiplier is 1/(2 * dimensions), and like circumference to surface area, the exponent is the number of dimensions of what we are measuring.  The sphere volume is simpler with diameter, and the circle area is only trivially more complex.  In fact, the only places that complexity is increased noticeably are the two standard form equations that are typically used as the definitions for circles and spheres, and outside of education, more general forms are typically used for these, which are so much more complex that the extra factor of 4 would not make any difference.

The big advantage τ has is that replacing a bunch of "2π"s in textbooks is much easier than doing the algebra required to simplify the equations when you swap r for ^d⁄₂.  If you aren't going to do it right though, what is the point of doing it at all?  It is true that teaching τ with radius will be easier than what we are doing now, but many students will still start off confused that we are measuring only half of the circle (and isn't one of the big "tauist" arguments that it is absurd that 1 π only gets us halfway around the traditional unit circle?).  Sticking with π, but using diameters instead of radians, means that we don't have to overthrow a constant that has been ingrained in mathematics over the course of many millenia.  We don't have to deal with teaching two constants just so students will be able to understand even recently written papers, not to mention all of the classical mathematics treatises.  It is a lot easier to teach students that r = d/2 than it is to get them to memorize two circle constants out to n digits just so they can work with math from different eras.  We also don't have to try to explain to students why we only care about half the width of a circle.  The only advantage τ has is convenience in fixing textbooks.  Using π with diameters just makes sense, even to those without a heavy mathematical background.

I hope you like my Pi Day celebration!  Give a few minutes to celebrate diameter today as well.