Bad arguments in defence of π
Yet another useless comment on τ vs π
I recently read through The Pi Manifesto and The Proper π Manifesto and I want to point out a few bad arguments:
To get it out of the way first: π is fine. It’s not ugly, it’s not “wrong” (that’s a meme), and it isn’t going anywhere. My claim is narrower. τ is the more consistent choice and the far better one to teach, and the manifestos defending π mostly do it with bad arguments. And when I say inconsistent I never mean that the math doesn’t work out - it obviously does. I mean that the choice is inconsistent with every other convention around circles, which all use the radius.
Pi Manifesto
Most of the wording is pretty loaded. That’s not a bad argument itself, but it shows that the author leans into defamation instead of just providing good arguments.
They celebrate Tau Day (June 28th), wear τ-shirts and spread pro-tau propoganda.
[…] and those who have, simply dismiss tauists as cranks.
a lot of mathematicians simply shrug off the Tau Movement as being silly
The benefits of τ only appear when viewing π from a narrow minded two dimensional geometrical point of view
Wording like propaganda, cranks, silly, narrow-minded, etc. is not from direct quotes but either the author’s own words or intentionally misinterpreted quotes.
In this article we attempt to give a serious rebuttal to τ in the defence of π.
With “serious” the author probably meant a rebuttal against the silly, narrow-minded τ-crank propaganda 😉
But pi is far from being ugly and mathematicians are certainly not going to replace the circle constant any time soon.
As said above, π is not ugly, it’s just a constant, and they are right, nobody is going to “replace” it. But claiming that π has benefits over τ is like claiming that defining the conventional direction of current in the exact opposite direction of electron flow has some obscure benefits; it hasn’t. It was a historical accident (no hard feelings, that happens). Historical accidents are everywhere; what differs is how fixable they are. Base 10 instead of 12, or 12 months instead of 13, are baked in far too deep for anyone to seriously try. Some of those “accidents” are stuck because there is no gradual path to fix them. You’d have to flip every textbook, datasheet and schematic at once. τ is the easy case, it can arrive one constant, one course, one library at a time. So it’s not a “replacement” but a transition. One that might take a few decades, but would mostly happen naturally.
The fact is, most mathematicians have never heard of the Tau Movement, and those who have, simply dismiss tauists as cranks.
There is no source for that claim. I’m pretty sure (especially now in 2026) there aren’t many mathematicians who haven’t heard of the alternative circle constant τ.
Tauists argue that by using the constant τ=2π a lot of formulas become simpler. Unfortunately, the Tao Manifesto is full of selective bias in order to convince readers of the benefits of τ over π. They pinpoint formulas that contain 2π while ignoring other formulas that do not. We demonstrate below that when making the change to τ, there are lots of formulas that either become worse or have no clear advantage of using τ over π.
It’s true that there are situations where the 2 in 2π cancels out, but that always hides some key insight. $A = \pi r^2$ is exactly such a case. Written as $A = \frac{1}{2}\tau r^2$ it rejoins the family of $\frac{1}{2}kx^2$, $\frac{1}{2}mv^2$ and $\frac{1}{2}at^2$, and it becomes visibly $\int_0^r \tau x,dx$, the integral of the circumference. The Tau Manifesto page goes into this and many more at length, though I’m not sure if that was already the case back when the Pi Manifesto was written.
Tauists also claim that their version of Euler’s formula is better than the original, but we will see that it is in fact weaker.
Euler’s formula is neither an argument for nor against τ or π. There is no “better” in that sense; it’s like discussing whether people with black hair look better than people with blonde hair. Even if the τ version of the identity looked like a complete mess and the π version like an angel, it still wouldn’t be an argument, because τ vs π is not about which produces the most beautiful formulas, but about which one is the more self-consistent and intuitive/logical choice.
In particular, since a circle is defined as the set of points a fixed distance (i.e., the radius) from a given point, a more natural definition for the circle constant uses r in place of D.
So why did mathematicians define it using the diameter? Likely because it is easier to measure the diameter of a circular object than it is to measure its radius.
Mathematical definitions don’t care about how “easy” something is to measure 🤣. It’s about how simple (Occam’s razor) and consistent a definition is. Defining a circle through its radius is way simpler, because shapes of constant width are a thing.
Defining π through the circle’s diameter was just a historical accident. It’s not like a huge research team of mathematicians discussed for a long time which definition would be better in the long run. In fact, for actual mathematical work done by professionals it doesn’t matter. It’s just a factor of two you have to add or not. But τ vs π is not about mathematicians (at least not mainly, because they probably couldn’t care less). It’s about being consistent, intuitive and being the obvious logical choice. As a nice side-effect it’s also way more intuitive to teach.
And that teaching part is where it hurts the most:
| turn | with τ | with π |
|---|---|---|
| quarter | $\frac{1}{4}\tau$ | $\frac{1}{2}\pi$ |
| half | $\frac{1}{2}\tau$ | $\pi$ |
| three quarters | $\frac{3}{4}\tau$ | $\frac{3}{2}\pi$ |
| full | $\tau$ | $2\pi$ |
In the τ column the fraction is the answer. In the π column every single value needs a silent division first. That’s exactly where students lose the thread.
To be consistent and intuitive there are basically two options:
- Option 1: Switch from π to τ.
- Option 2: Switch the unit circle from having radius 1 to having diameter 1, and change the definition of all trig functions (sin, cos, tan, sinh, cosh, etc.). Then reword all circle definitions from radius to diameter (give it a try without accidentally including things that are not circles, like constant width shapes).
I know which option I’d choose. Realistically there is a third option, the one that most people who favor π (knowingly or unknowingly) implicitly chose, and that’s keeping the inconsistency and living with the 2π that turns up everywhere.
Another definition for π is to define it to be twice the smallest positive x for which cos(x)=0 [4], or the smallest positive x for which sin(x)=0. With this definition neither π nor τ is simpler than the other.
Sure - and note what is being conceded here: by their own admission this one is a tie, so it isn’t an argument for π either. It just hands you a periodic function and lets you pick which special value you want to name. Pick “the smallest positive $t$ with $e^{it} = 1$” and out comes τ. “twice the smallest positive x for which sin(x)=0” and “the smallest positive x for which cos(x)=1” also evaluate to τ.
Another common geometric definition for π is in terms of areas rather than lengths. Take r to be the radius of a circle. Define π to be the ratio of the circle’s area to the area of a square whose side length is equal to r.
“common” … sure. That one is another consequence turned backwards and called a “definition”. Nobody defines a circle constant like that. I’d love to see this “definition” in real literature, papers and textbooks 🤣.
In practice, the only way to measure the radius of a circle is to first measure the diameter and divide by 2.
The short answer: “so what?”. But seriously, there are two big flaws in this argument. First: mathematics is a framework of Platonic ideals and never cared about how you measure something “in practice”. Concepts are not defined by how they are measured in real life. The second flaw: there probably isn’t a single mathematically perfect circle (or sphere, for that matter) in the universe, and even granting some leeway on accuracy, without taking multiple measurements you wouldn’t even be able to truthfully claim that something is a circle. And because of constant width shapes, even taking multiple measurements of the “diameter” (for example with a caliper) could mislead you.
Why look at a ratio where you go all the way around the circle yet only HALF way across it? It just doesn’t seem natural.
That sentence is actually pretty ironic given the fact that π radians is only half the circle. As said before, everyone I know who prefers τ over π does so because of consistency. All mathematical definitions around circles and trigonometry use radius=1 (look up “unit circle”), and from that follow the definitions of all trig functions and many more things; π is the only exception here.
The Proper π Manifesto
Radians are a unit for measuring angles. An angle in radians is the length of the arc along a circle with radius 1. The circumference of such circle is 2π, and this results in 2π being the length of a full turn. With this definition, a single turn becomes twice the circle constant. This is absurd, and evidently this factor of 2 haunts us throughout mathematics. To fix this, some suggested that we should redefine π to be 2π, but this is not the solution.
Quoted that for context.
π is the all important circle constant. It has historical and cultural significance, and it is here to stay. Changing it is unreasonable, and most importantly unnecessary
I can’t tell if that’s sarcasm or not. Sounds a bit like someone is way too much into specific circle constants.
Radians are the problem, and fixing radians is the solution.
The proper way to define angles is to use the arc of a unit circle with a diameter of length 1.
To be fair, what the author is suggesting is at least self-consistent and basically Option 2 from before. Even though this would “fix” some things, it would make the definition of a circle way harder, and the switch would be next to impossible in practice.
Let’s take software as an example. Transitioning from π to τ can be done gradually and is pretty simple: just add TAU as a new constant to the standard library of the programming language. That’s already happening for major languages like C#, Java, Rust, Zig and Python - they all support TAU out of the box.
Doing what the manifesto suggests would imply duplicating the whole trigonometry API surface: old radians, to not break backwards compatibility, and new radians based on the new unit circle definition with diameter=1.
I can tell you what’s possible (and already happening), and what’s definitely not going to happen 🤣.
Conclusion
Given all that, τ is clearly more consistent (I’ve probably said that too many times already), simpler, more elegant and more intuitive. Given the chance to change it with a time machine (excluding other weird side effects of time travel), I’m sure almost all serious mathematicians, physicists and engineers would do it. And I’d take a huge bet that if the circle constant had historically been defined through the radius, there wouldn’t be a debate today about whether we should redefine it through the diameter.