On the Tao of Tau

Tau day is coming up toward the end of the month of June - 6/28 to be precise. More than a few of you may have celebrated pi day back in March, where you could (with a certain amount of time twisting) come up with several digits of Pi (3.14159). And while on the subject of memorable days, the 4th of May came under special attention given the upcoming Star Wars movie.(May the fourth be with ... oh, never mind, you get the basic idea.

Tau day, on the other hand, is actually kind of special, and not just for its numerological significance in the calendar. Back in 2010, mathematics teacher Michael Hartl published a paper on the Internet that is beginning to make real waves - the Tau Manifesto (https://tauday.com/tau-manifesto). 

In it, Hartl makes an interesting assertion - that we have mathematics all wrong, most specifically our use of the most famous mathematical constant in the world: π. The Greek equivalent of our "p", π was originally proposed as far back as Euclid's time as the ratio between the length of a circle'd diameter and its circumference (the distance around that circle, famously expressed as

C=πD

Now, the problem with this is that the diameter of a circle is actually not all that interesting. Mathematically, a circle is the locus (i.e., set) of all points that are equidistant from a single point, where the distance between the point and the circle's edge is it's radius. Now, the radius is actually very important in mathematics - The radius is the distance from a point where something interesting is happening - a body of mass, or charge, or the length of a spring, and so on - to the point where you want to measure the effect of that thing.

Of course, given the old Euclidean formula this means that the length of the circumference in terms of the radius is actually double that of C/D, or put another way:

C = 2π r

So, we've now begun carrying this factor two around with us, and it's made mathematics really far more complicated than it needed to be. So, Hartl picked up an idea first proposed by Bob Palais. Create a new constant that is double pi. Palais' idea was brilliant, but he faltered on the symbol (left).

What Hartl proposed was, rather than give typographers a heart attack, the symbol for 2π be represented by the Greek letter τ:

τ = 2π

Another way of thinking of this is that τ represents the unit circle - that is to say, the circle when r = 1, and as such it has one leg. π, on the other hand, has two legs below a line, which is much like saying "half the unit circle". 

This simple change, has some mind-blowing implications. For instance, anyone who has ever graphed a sine wave knows that when the wave goes from 0 to π radians (0 to 180 degrees), it has only completed half of its arc - the one where the function is positive. The entire arc, on the other hand, has to go through the negative y axis before it gets back to where it started. Put another way, π will take you only halfway there, but τ will get you back to where you started, completing the cycle:

Moreover, with pi, one quadrant is described as π/2, one half of the circle is π, three fourths of the circle is 3π/2, and the full circle is then given as 2π. These are not intuitve relationships.

On the other hand, a quadrant using τ is given as τ/4 radians, a half a circle is τ/2 radians, 3/4 of a circle is 3τ/4, and a full circle is τ. This fits in much more with our expectations about the nature of circles - when the ratio of circumference to radius (the much more important quantity) is τ, then you've completely navigated the circle.

Now, this does spoil one of the greatest math jokes in fourth grade:

First kid: Pi R square
Second kid. No they're not. Pie are round!

In more mathematical terms, this can be stated as:

A = π r^2

Now, with τ , the area formula is no longer quite as simple:

A = τ r^2 /2

or, written slightly differently:

A = ( 1 / 2) τ r^2

(where ^ is a way of saying "to the power of" - r^2 is the same as r squared)

While this may look like you're adding some complexity here, people who have studied even basic physics may have picked up something by now. Area is an integral over a circle. Other integral relationships may look suspiciously similar:

E = ( 1 / 2) m v^2, (relating energy, mass and velocity)

E = ( 1 / 2) k x^2, (relating a spring constant, the extension or compression of the spring, and energy)

y = ( 1 / 2) g t^2, (relating distance fallen in the presence of a gravitational field over time),

and dozens of other quadratic forms. In other words, it's A = π r^2 that's the real aberration.

The kicker is one of the most famous (and profound) statements in mathematics - Euler's equation:

e^(i θ)=cos θ + i sin θ.

This equation establishes the (deep) relationship between complex numbers and polar coordinates - coordinates over angles. Here i is the square root of -1, and is typically called the imaginary unit number.

Now, when the angle θ is set to  π, then you get the truly bizarre statement:

e^ (π i) = cos π +i sin π = -1 + i * 0. = -1

However, with τ, you go from weird to sublime:

e^ (τ i) = cos τ +i sin τ = 1 + i * 0. = 1

In this case, it means that the base of the natural logarithms, raised to tau times the square root of -1, will always be equal to 1. Of course 

e^ (0 i) = e^ (0) = e^ (k τ i) = 1, where k = -2, -1, 0, 1, 2 etc.

as well. This is so simple that it's obvious, but with pi the relationship becomes much less self-evident:

e^(π i) = e^( - π i) = e^((k * 2 -1) π i) = -1, where k = -2, -1, 0, 1, 2 etc.

Given that the role of mathematics is to help identify patterns, replacing 2π with τ provides an incredible degree of service.

With symbols like e, π , τ and θ floating around, all this may seem more than just a bit academic, but there's actually quite a lot of good that can come out of pushing τ into the mainstream. Most students study a year of analytic geometry (better known as trigonometry) their second or third year of high school. I've long thought that teaching classical geometry before analytic geometry has been a mistake, for a very simple reason. Classic geometry teaches a form of mathematics from first principle, and requires that you effectively context switch from a functional form of mathematics (analysis) to one where you are in effect building up a logical system from first principles. Then, just as you are finally beginning to master the latter, you context switch again, only this time, in the other direction.

You can teach analytic geometry without having to derive a single proof, without having to worry about spending hundreds of hours tryingaster the why of mathematics. It's not that such logic isn't important - it's vital - only that by teaching mathematics at an analytical level first do you build up the understanding necessary to master the mechanics of building proofs.

τ is a big step in that direction. A τ is a revolution or a cycle - middle C is 440 τ per second, a car's pistons will move at 2500-3000 taus per minute. A pi is a form of baked flour dish with filling. With tau, the connection between trigonometry, geometry and physics becomes clear. A spring oscillates in a sine wave because it completes a full cycle, as does a pendulum, as does a satellite travelling around the Earth. It makes the analogies between these things stronger and more obvious, and also makes us ask the quintessential science question - "why?". Why should springs and pendulums and planets all share the same superficial formulas if there isn't something that's not in fact at all superficial about their relationships. Metaphors give way into insight, and insight into learning. 

Tau is making its way into the math curriculum (it dramatically simplifies trigonometry, for instance), and many virtual calculators are now sporting tau buttons in addition to pi buttons. This may seem like a silly change, but sometimes real innovations come simply by reframing what had been a hard problem and making it into an easier one, opening up insight into why the change does so as a means of exploring reality more deeply. Pi may be cool, but tau is everything.

Kurt Cagle is principal evangelist for Avalon Consulting, LLC, and describes himself as a contemporary Tauist.