Saturday, February 9, 2013

Pi vs Tau: Pi's Rebuttal

As you may know, I have been strongly intrigued by the tau movement. This movement is promoting the idea that we should not be using pi as the circle constant, but rather 2π, which has been renamed with the greek letter tau.

This movement was started by Bob Palais of University of Utah when he wrote the article Pi is Wrong which was published in the Mathematics Intelligencer. Physicist Michael Hartl proceeded to write The Tau Manifesto, and founded Tau Day on June 28th (instead of Pi Day on March 14th) as well as the website This gave dozens of reasons in geometry, trigonometry, physics, and statistics why tau is more practical and natural than pi. On a side note, I currently have the world record for tau memorization at 2012 digits, but that's nowhere near the world record for pi memorization at 67890 digits.

If you have some more time, The Tau Manifesto is a fascinating read. I'd also recommend watching this video done by Vi Hart for an overview of the tau movement.

You can also click here to see all of the blog posts I have done giving details on why tau is better than pi.

Just two days before writing this, I found that someone was inspired by Hartl's Tau Manifesto to write The Pi Manifesto, which presents an interesting rebuttal for tau. This wasn't recent news in mathematics, but it was definitely news for me!

I found that all of the reasons for either side can actually be debated. I think this is the first post where there is actually a debate over what is the right answer.

An argument for tau is that the radius is what a circle is measured by. defines circle as a closed plane curve consisting of all points at a given distance from a point within it called the center. Clearly, the radius is the main measurement here.

Pi supporters would then argue that the radius can only be found by taking the diameter and dividing it by two. When looking at a circle on paper, it is impossible to pinpoint the center and find the measurement out to the end.

Yet, constructing the circle requires knowing the radius. If you were to use a compass, you must put the point where you want the center to be and trace a line with a constant distance around it. This is the true way to form a circle. For practical uses, you would need circles to see what restaurants were within ten miles or something. In this case, ten miles is the radius, and you are constructing a circle with this measurement.

This example requires the choice between constructing the circle easier or deconstructing the circle easier. The construction requires the radius, while a circle already given uses the diameter.

An argument for pi is that the area formula, which is one of pi's most common uses, is messed up by this change.

A = πr^2  –>  A = 1/2τr^2

Yet, tau-ists argue that this is more natural. First, there are many other shapes that have 1/2 at the beginning of their area formula. Triangles, trapezoids, and n-sided polygons all use formulas with a 1/2 in it. Second, the area formula can be proven (I will in a future blog post), and this proof's last step is to multiply 2πr^2 and 1/2. It turns out to be more natural just to tack on the 1/2 rather than hiding the 2π with it.

This is again an argument where we need to choose between having a natural number or an efficient number. It is clear that tau is the constant that belongs, but we have an opportunity to simplify the equation.

Both manifestos list other reasons that take longer to explain, but are very interesting (I think a lot more interesting than the two above). These involve finding measurements of the unit circle, graphing trigonometric functions, and rewriting Euler's identity.

Because math is such a definitive discipline, it is rare for Cool Math Stuff posts to have comments. I strongly encourage you to comment on this post. After reading both manifestos, watching several YouTube videos, and seeing lots of press coverage, it is difficult to take a side. Comment what you think about the different arguments (I think the three I listed earlier are the closest arguments to call), and if you are a Tau-ist or not. This is one of the few chances where you can get into a debate about mathematics!


  1. I was just directed to a part of the Tau Manifesto where some Pi Manifesto points were addressed.

    Let's see how pi supporters respond...

  2. If "simplify" means get rid of any numbers out front, then no constant can simultaneously simplify all formulas. It's impossible. Even with just the two formulas for circumference and area of a circle, tau simplifies the circumference formula (τr), while pi simplifies the area formula (πr^2). And there are lots more formulas out there. The sphere volume formula is only simplified if we create a new constant β = 4π/3. Then we can just write volume = βr^3. The sphere surface area formula is only simplified if we create a new constant γ = 4π. Then we can just write surface area = γr^2.

    There is no universally "efficient" number. That's why it makes sense to use the universally "natural" number, which is tau.

    Joseph Lindenberg

    1. Tau isn't necessarily more natural; if you do a little research, you'll find that pi was originally defined as the ratio between the area of a circle and the square of its radius.

    2. Yes, but who cares about area? Pists usually say that it's unnatural to define a circle from using the center (because it's unmeasurable, which is clearly not the case for a bike wheel, for instance), but they then say they want to define a circle based on area? That is DEFIANTLY not better than the radius.

      It's kinda like barley corns. An inch is 3 barley corns (the long way). Who wants to define an important unit in barley corns? You may say that it's fundamental, since it relates to a simple physical object, but I object. Why not use something less arbitrary? Make it actually definable? How about some large power of 12 times the planck length? Why 12? Because we should use base 12. Throw out the old, bring in the new (I think the bosses will agree).

      We use a lot of deprecated numbers based on ancient physical objects and their traditions. We need to throw those out now that we know better.

  3. The very definition of a circle is all points on a plane equidistant from an origin. π is wrong. τ makes more since. The greatest victory of π is a terrible failure:

    a = πr^2

    We derive this as the limit of of stuffing an infinite number of isosceles triangles whose base is on the circle and whose acute ange is at the center. The formula for the area of a triangle is base times height divided by 2 (bh/2). Logically, the formula for the area of a circle after integrating those infinite triangles should be the radius squared times the circle constant divided by 2 ( a = (τr^2)/2 ).

    The point is that τ is the natural circle-constant. We should abandon π.

    I am a Tauist.

    1. Huzzah! Why do we need infinite triangles when we could have a properly defined circle?

  4. Why not advance to the new "out of the box" concept of rPi? It promotes a trigonometric understanding of the Pi ratio:

    The 62.402887364309.. degree radius is consistent in all of the designs: any circle can be squared once this angle is known (the trigonometry proves this).

    What is rPi? The geometric complement to Pi (all of the known digits of Pi can be substituted into the formula).

  5. The biggest reason in support of Tau lies in secondary education. Every time I work with HS students learning trigonometric functions, the biggest obstacle is the period of 2pi. Getting them to truly understand that 3/4 of the way through a period is 3pi/2 (Wow, I still have to stop and think about that each time.) adds so much confusion. These non-mathematicians could envision 3/4* tau so much more readily. The Tau manifesto is not so much for the mathematicians but for the 16 year olds and we who teach them.

    I once showed Vi Hart's video to a pre-calculus student. Her only response was, "I'm pissed."

    1. Yes. A good point. I've mentioned this before to Pi supporters, but all they say is that all the HS people are stupid. There is no bad student, there is only a bad teacher (to certain limits. It isn' completely true, but it is true that a good teacher is more important than a good student).

    2. I completely agree with this. Mathematicians have adapted to the number pi for centuries, and have clearly been able to advance as a result. Changing to tau would not ease things for them, but for the general public, and high school trigonometry students in particular. I very recently went through the unit in my precalculus class involving sine and cosine graphs, and I would convert every problem to tau just because I got confused every time I would do it with pi. I just presented a series of mental math workshops to fifth graders today, and I tested out a teaching bit that combined mental math with proofs and algebra. Hearing a fifth grader walk out saying "I'm really excited to learn algebra in high school" is about the best compliment I can get. By taking something that American society believes to be hard and boring like algebra and showing how beautiful it can be is a real inspiration to students. Tau is the gateway to do this with trigonometry and further mathematics, and that is certainly the biggest reason why this conversion needs to be made.

      I'm thrilled at the amount of commentary that this post has gotten. It's great to see that math is not strictly quantitative, but can have two (and even more) sides to each argument.