Video Transcript
I’m sure that you’re already familiar with the whole 𝜋 versus 𝜏 debate. A lot of people say that the fundamental circle constant we hold up should be the
ratio of a circle’s circumference to its radius, which is around 6.28. Not the ratio to its diameter, the more familiar 3.14. These days, we often call that larger constant 𝜏, popularized by Michael Hartl’s tau
manifesto. Although, personally, I’m quite partial to Robert Palais’s proposed notation of a 𝜋
with three legs.
In either of these manifestos and on many, many other places of the Internet, you can
read to no end about how many formulas look a lot cleaner using 𝜏. Largely because the number of radians describing a given fraction of a circle is
actually that fraction of 𝜏. That dead horse is beat. I’m not here to make the case further.
Instead, I’d like to talk about the seminal moment in history when 𝜋 as we know it
became the standard. For this, one fruitful place to look is at the old notes and letters by one of
history’s most influential mathematicians, Leonhard Euler. Luckily, we now have an official Three Blue One Brown Switzerland correspondent, Ben
Hambrecht. Who was able to go to the library in Euler’s hometown and get his hands on some of
the original documents.
And in looking through some of those, it might surprise you to see Euler write. Let 𝜋 be the circumference of a circle whose radius is one. That is, the 6.28 constant that we would now call 𝜏. And it’s likely he was likely using the Greek letter 𝜋 as a “p” for “perimeter.” So was it the case that Euler, genius of the day, was more notationally enlightened
than the rest of the world, fighting the good fight for 6.28? And if so, who’s the villain of our story, pushing the 3.1415 constant shoved in
front of most students today?
Well, the work that really established 𝜋 as we now know it as the commonly
recognized circle constant was an early calculus book from 1748. At the start of Chapter 8, in describing the semicircumference of a circle with
radius one. And after expanding out a full 128 digits of this number, one of them wrong by the
way. The author adds, “which for the sake of brevity I may write 𝜋.”
Now there were other texts and letters here and there with varying conventions for
the notation of various circle constants. But this book, and this section in particular, was really the one to spread the
notation throughout Europe, and eventually the world. So what monster wrote this book with such an unprincipled take towards circle
constants?
Well, Euler again. In fact, if you look further, you can find instances of Euler using the symbol 𝜋 to
represent a quarter turn of the circle, what we would call today 𝜋 halves or 𝜏
fourths. In fact, Euler’s use of the letter 𝜋 seems to be much more analogous to our use of
the Greek letter 𝜃. It’s typical for us to let it represent an angle, but no one angle in particular. Sometimes it’s 30 degrees. Maybe other times it’s 135. And most times, it’s just a variable for a general statement. It depends on the problem and the context before us.
Likewise, Euler let 𝜋 represent whatever circle constant best suited the problem
before him. Though it’s worth pointing out that he typically framed things in terms of unit
circles, with radius one. So the 3.1415 constant would almost always have been thought of as the ratio of a
circle’s semicircumference to its radius. None of this circumference to its diameter nonsense. And I think Euler’s use of this symbol carries with it a general lesson about how we
should approach math.
The thing you have to understand about Euler is that this man solved problems, a lot
of problems. I mean, day in, day out, breakfast, lunch, and dinner, he was just churning out
puzzles and formulas and having insights and creating entire new fields, left and
right. Over the course of his life, he wrote over 500 books and papers, which amounted to a
rate of 800 pages per year. And these are dense math pages. And then, after his death, another 400 publications surfaced.
It’s often joked that formulas in math have to be named after the second person to
prove them, cause the first person is always gonna be Euler. His mind was not focused on what circle constant we should take as fundamental. It was on solving the task sitting in front of him in a particular moment and writing
a letter to the Bernoullis to boast about doing so afterwards.
For some problems, the quarter circle constant was most natural to think about. For others, the full circle constant. And for others still, say at the start of Chapter 8 of his famous calculus book,
maybe the half circle constant was most natural to think about. Too often in math education, the focus is on which of multiple competing views about
a topic is “right.” Is it correct to say that the sum of all positive integers is negative
one-twelfth? Or is it correct to say that it diverges to infinity? Can the infinitesimal values of calculus be taken literally? Or is it only correct to speak in terms of limits? Are you allowed to divide a number by zero?
These questions in isolation just don’t matter. Our focus should be on specific problems and puzzles, both those of practical
application and those of idle pondering for knowledge’s own sake. Then when questions of standards arise, you can answer them with respect to a given
context. And inevitably, different contexts will lend themselves to different answers of what
seems most natural. But that’s okay.
Outputting 800 pages a year of dense transformative insights seems to be more
correlated with a flexibility towards conventions than it does with focusing on
which standards are objectively right. So on this 𝜋 day, the next time someone tells you that, you know, we should really
be celebrating math on June 28th. See how quickly you can change the topic to one where you’re actually talking about a
piece of math.
