Why Tau Should Replace Pi

Podcast Transcript

For the last 250 years, there has been a problem in the world of mathematics.

There is a good argument to be made that we have been doing it wrong.

We have been teaching it in a way that has been unnecessarily confusing and complicated, and we have been causing generations of students unnecessary headaches.

Learn more about Tau and why it is better to use than pi on this episode of Everything Everywhere Daily.

To understand what this episode is about and why there is a problem, we have to start with one of the simplest concepts in geometry, the circle.

A circle is defined to be the set of all points equal in distance to a single point.

Physically, you can make a circle with a compass by holding one end down on a single point and then swinging the rest of it around to make a circle.

Likewise, you could take a piece of rope or string, fix it to a point, and then move it around to make a circle as well.

The point being, the defining characteristic of a circle is its radius—the distance from the center to the edge. No other two-dimensional shape has a radius.

To hammer the point home, the radius is what makes a circle a circle.

Every circle has a special ratio that is built in, regardless of how big or how small the circle is. Most of you know this ratio is the number pi.

I’ve previously done an entire episode on pi, but just to refresh your memory, pi is the ratio of the circumference of a circle to its diameter.

….and therein lies the problem.

There is almost nothing in mathematics that we use the diameter of a circle for. Why is it that when we define the most important ratio found inside every circle, we all of a sudden define it using the diameter rather than the radius?

You might be thinking that this really isn’t that big of a deal. The diameter of a circle is just twice the radius. Just as every circle has a radius, so too does every circle have a diameter.

That is true, but there is a problem.

While a circle is the only two-dimensional shape with a radius, it is not the only two-dimensional shape with a constant diameter. There is an entire class of shapes known as Reuleaux triangles that also have the same diameter in every direction going through a center point.

A Reuleaux triangle looks like a triangle but with rounded sides.

So, if something has a fixed radius, it must be a circle. If something has a fixed diameter, however, it is not necessarily a circle.

So, again, why do we calculate ? using the diameter of a circle rather than the radius, which would seem to make much more sense?

The calculation of the circle ratio goes back to antiquity. In many histories of pi, you will hear of ancient mathematicians from Egypt, India, and China who all calculated, with various levels of accuracy, ?.

However, they were not always calculating the circumference divided by the diameter. Sometimes they were using the radius.

The modern usage of the Greek letter ? to represent the ratio of the circumference to the diameter is attributed to the Swiss mathematician Leonhard Euler, one of the greatest mathematicians of all time.

At the time he was writing in the first half of the 18th century, there was no standard symbol to represent the circle constant.

In 1727, in his paper Essay Explaining the Properties of Air, he used the Greek letter pi to represent the ratio of the circumference of the circle to its radius. In other words, how many times can the radius of a circle go around the circumference?

The number he associated with the Greek letter ? in that paper was 6.283185307….

Or in other words, his original definition of pi was twice what we now consider pi because he was using the radius instead of the diameter.

Had he ended here, I wouldn’t be doing this episode.

Instead, he kept on writing and changed his definition of ?. In 1736 he started defining ? to be half the value he gave it originally, and what really cemented into the world of mathematics was his 1748 book Introductio in analysin infinitorum where he said, “For the sake of brevity, we will write this number as ?; thus ? is equal to half the circumference of a circle of radius 1”.

Euler wasn’t dumb. He explicitly stated that ? was the constant that represented half a circle.

At this point, you might be thinking, “Who cares?”

When I first took trigonometry in high school, way back in the day, we were introduced to the concept of radians.

Radians are a way of measuring angles. A radian is just the angle measured by the arc of a circle with an arc length equal to the circle’s radius.

If you go all the way around a circle, in other words, 360 degrees, the number of radius lengths you’ve gone around the circumference is equal to 2?.

I always found this to be extremely confusing. Why was one circle equal to 2?? Why is a half a circle ?, and why is a quarter circle ½??

If you have a sine wave, it reaches its peak at a value of ½?, it’s back to zero at ?, reaches its lowest point at 3/2?, and then finally completes the wave at 2?. The cosine function is similar.

In fact, you find 2? popping up all over the place in mathematics and physics.

In quantum physics, the Planck constant is calculated using 2?. In statistics, normal distribution is calculated using 2?. Fourier transformations, the Riemann zeta function, Cauchy’s integral formula, and a host of advanced mathematical formulas all use 2?.

So, if 2? is everywhere, shouldn’t that be the constant that we use? Afterall, 2? is just the common sense ratio of the circumference to the radius.

Well, there are mathematicians who think exactly that.

In 2001, Robert Palais, a mathematician from the University of Utah, published a paper titled “Pi Is Wrong!”

In it, he argued for changing how we think of circles and angles. Saying that a full circle is 2? is like saying a complete trip somewhere is twice half the distance. Technically correct, but confusing.

It would be much easier to say that a circle is one turn. That is common language that everyone can understand. A turn would be equal to 360 degrees.

Half a circle is half a turn. ¾ a circle is ¾ of a turn.

He suggested replacing the pi symbol with a new symbol that had three legs instead of two.

His idea for a new symbol didn’t catch on, but the idea of using a single constant to represent 2? did.

In 2010, the physicist Michael Hartl wrote a blog post where he proposed a new symbol for the constant: the Greek letter Tau.

The Greek letter ? visually sort of looks like pi, only with one leg. It is closer to the letter T in the Latin alphabet.

It would also represent the concept of a ‘turn.’ One turn would equal one tau number of radii.

Tau would simplify many mathematical formulas and would make certain things like angles and trigonometry much easier to understand.

In theory, tau is a pretty good idea, and, personally, I think it makes sense. Using the radius instead of the diameter of a circle is more intuitive, and if we were to be sent some mathematical message by aliens, I’d be willing to bet they would be more likely to use tau than pi.

Since 2010 there has been a movement for the adoption of Tau amongst mathematicians.

Probably the biggest thing has been the adoption of Tau Day.

In the American system of dates, March 14 is written 3-14, which is the first several digits of pi. Hence, March 14 is Pi Day.

As tau is just two times pi, Tau enthusiasts have adopted June 28, 6-28, as International Tau Day. It is an excuse to talk about tau and do crazy things like record podcasts on the subject.

While there has been a vocal community of tau advocates, getting people to switch from using pi has proven extremely difficult.

There have been some programming languages which have adopted tau as a value, which is pretty easy to do considering it is just two times pi. These include python, java, rust, and .net.

The online education platform Khan Academy now accepts answers using tau instead of pi.

There has been at least one academic paper that used tau instead of pi.

However, that isn’t a whole lot to point to since Robert Palais write the original “Pi Is Wrong!” article over two decades ago.

There are over 250 years worth of mathematical papers and textbooks out there that use pi. Everyone who studied mathematics, even at a cursory level, knows about pi and has become accustomed to using it. There are literally buttons hardwired into most calculators just for pi.

Change will be difficult, if not impossible.

If the United States can’t convert to the metric system, getting the entire world to change from pi to tau might be a bridge too far.

Nonetheless, the tau advocates make a very good point. Mathematics would be simpler and easier to understand if tau had been adopted as the circle constant rather than pi.

We owe it all to an 18th century mathematical genius who changed his mind about what he wanted the symbol pi to represent.

The Executive Producer of Everything Everywhere Daily is Charles Daniel.

The associate producers are Thor Thomsen and Peter Bennett.

Today’s review comes from listener MaacoDale on Apple Podcasts in the United States. They write:

Extremely interesting podcast.

I love finding out about things I never even knew existed. This podcast really brings a lot of interesting topics to light. I’ve always said “it’s not what you don’t know that’s important. It’s what you don’t know, that you don’t know”. Thanks to Everything Everywhere Daily, there are far less things I don’t know. Keep up the good work!

Thanks, MaacoDale! As I’ve mentioned before, even if you can change unknown unknowns to known unknowns, in other words going from being totally ignorant about a subject to at least being aware you don’t know something about a subject, you’ve made substantial progress.

Remember, if you leave a review or send me a boostagram, you too can have it read on the show.