I’m sure you’re well-familiar with the whole pi vs. tau debate.
Many people say that the fundamental circle constant we hold up should be the ratio of
a circle’s circumference to its radius, around 6.28, not the ratio to its diameter,
the more familiar 3.14.
These days we call this larger constant “tau”, popularized by Michael Hartl’s “Tau manifesto”,
although personally, I’m quite partial to Robert Palais’ proposed notation of a pi
with three legs.
In this manifesto and many other places on the internet, you can read to no end about
how many formulas look at lot cleaner using tau, largely because the number of radians
describing a given fraction of a circle is actually that fraction of tau.
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 pi 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 3blue1brown Switzerland correspondent Ben Hambrecht who
was able to go to the library in Euler’s hometown and get his hands on some of these
original documents.
It might surprise you see Euler write (in French), “Let pi be the circumference of
a circle whose radius = 1”.
That is, the 6.28 constant which we call tau today, where he was likely using pi as the
greek letter “p” for “perimeter”.
So was it the case that Euler was more notationally enlightened than the rest of the world?
Fighting the good fight for 6.28?
If so, who’s the villain of our story pushing the 3.14 constant that students are shown
today?
The work that really established 3.14... as the commonly recognized circle constant was
an early calculus book from 1748.
At the start of chapter 8, in describing the semi-circumference of a circle with radius
1, and after expanding out 128 digits of the this number, the author writes “which for
the sake of brevity I may write π”.
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 through Europe, and eventually the world.
So who wrote this text with such an unprincipled take towards circle constants?
Well...Euler again.
In fact, we can also find instances of Euler using the symbol pi to represent a quarter
turn of a circle, what we would today call “pi/2”.
In fact, Euler’s use of the letter pi seems to be much more analogous to our use of the
greek letter theta.
It’s typical for us to let it represent an angle, but no particular angle.
Sometimes it’s 30o, other times 135o, most times just a variable for a general statement.
It depends on the problem and context before us.
Likewise, Euler just let pi represent whatever circle constant best suited the problem before
him.
Though it’s worth pointing out he typically framed things in terms of unit circles, with
radius 1, so the 3.14 constant would have been thought of as the ratio of a circle’s
semi-circumference 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 to math.
What you have to understand about Euler is that he solved problems.
A lot of problems.
I mean, day in day out breakfast lunch and dinner this man was thinking about puzzles,
formulas, having insights and creating entire new fields left and right.
He wrote over 500 books and papers during his lifetime, what amounted to about 800 pages
per year, with another 400 publications appearing posthumously.
It’s often joked that formulas in math have to be named after the second person to prove
them, since the first person will always be Euler.
His mind was not focused which circle constant should be taken as fundamental; it was how
do I solve the task sitting in front of him and writing a letter to the Bernoulli's boasting
about doing so.
For some problems, the quarter-circle-constant was most natural to think about.
For others, the full circle, and for others still, the half circle.
Too often in math education the focus is on which of multiple competing views of a topic
is “right”.
Is it correct to say the sum of all positive integers -1/12, or is it correct to say it
diverges to infinity?
Can the infinitesimals 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 0?
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.
When questions of standards arise, then, you can answer them with respect to a given context.
Inevitably, different contexts will lend themselves to different answers of what seems most natural,
but that’s okay.
Outputting 800 pages a year of 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 pi day, the next time someone tells you we should really be celebrating math on
June 28th, see how quickly you can change the topic to one where you’re talking about
an actual piece of math.