Small Clever Rooms
Pi Day Curmudgeonliness
Happy Arbitrary Number Of Convenient Yet Also Arbitrary Time Units Elapsed Since The Death of a Historical Figure— Whose Divinity, If You’re Reading This, You Probably Don’t Even Believe In— Happening to Match Our Physiologically-Based (But Also Arbitrary) Number System Representation Of An Actually Fundamental Universal Constant Day!
Once upon a time I was enthusiastic about Pi Day, but I’ve become a bit of a curmudgeon about it without even realizing it. Much of it is due to typical hipster nonsense, probably, the “I was into math before you posers,” kind of thing when I see like fifty pictures of pies pop up on my Facebook wall. But my first paragraph sums up my more palatable reason, which Vi Hart expounds on at even more length with somewhat more verve but about as much use of the word “arbitrary”.
And I think that’s what my grumpiness about Pi Day comes down to: for a day ostensibly about celebrating a mathematical constant that’s remarkable for its timelessness and transcendence (little math joke there) of human conventions, Pi Day sure is contingent on a lot of those conventions. Pi, as the ratio of a circle’s diameter to its circumference, cares not about what number system we use to express it, or what year it is, or (I’d claim, as a mathematical Platonist) whether humans or any intelligent life even exist to know about it and give it a name. It’s just there, and it will always be there. I don’t think I’m irreverent about Pi Day. I think it’s my reverence for pi that makes me feel Pi Day itself is irreverent.
Still, though… it is nice to see the oft-maligned discipline of mathematics get a day in the sun, however superficial I may find it. And as a lapsed mathematician it’s unbecoming for me to be too snooty about it. So despite the arbitrariness of the day, I wanted to blog about some of the paths that complaining about Pi Day led me down. This will be very, very digressive and stream-of-consciousness. But the point is that thinking about Pi Day, despite its blatant disrespect of Pi’s sublime universality, can lead to some interesting if only tangentially related (another little math joke there) facts and curiousities. And isn’t knowledge what Pi Day should rightly be about?
Arbitrary Time Units
I started thinking about the arbitrariness of Pi Day’s relation to the Judeo-Christian Gregorian calendar and what more universal and unambiguous metrics it’s based on. I don’t think there’s anything to be done about the (to me as a non-Christian) arbitrariness of Pi Day’s being relative to the date of the birth of Jesus Christ (does this mean that Pi Day should be far more meaningful to Christians, who attach a profound and universal significance to that date?), but what about the divisions of time?
Modern science, which deals in particles that exist for far less than one-billionth of one billionth of one second, would not be satisfied with the imprecision in the layperson’s definition of a second as 1/31,536,000 of the time it takes for the Earth to revolve around the sun (or 1/86,400 of the time it takes for it to rotate once). The difference between any such revolution and the next is surely long enough for countless Higgs bosons to be created and destroyed. But the second is a Système International base unit. How is it measured these days in a way that’s good enough for science on this level?
The answer is quite arcane:
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
but it’s interesting because of the lengths that people presumably had to go to in order to ground what we’d already defined as a second in something that doesn’t change every time a comet swings by and tugs something a bit. And even decades after that definition was cooked up, it had to be amended to specify that it refers to the caesium atom at a temperature of absolute zero, because the radiation implied by a nonzero ambient temperature can affect the above measurement. In practice, this means no real-world clock can measure a second directly according to the strict definition. They have to use whatever temperature they can hold stable and extrapolate based on that what the measurement would be at absolute zero.
How Universal Is Pi?
Looking at the contortions necessary to make a “second” something that’s useful for high-precision science got me wondering whether my understanding of pi is as “universal” as I reverently assume it is. In particular, our value of 3.14 and so on is based on a circle in Euclidean geometry, which is characterized by the axiom that given a line L and a point P not on that line, there is exactly one line through P parallel to L. That seems obvious, but in fact you you can instead declare that there are no lines through P that parallel L or that there are infinitely many such lines and get perfectly coherent and even wildly useful systems.
If you make the first of those two assumptions, you get something called spherical geometry, called that because things work as if they were all on a sphere. In spherical geometry, the angles of a triangle add up to more than 180 degrees (or π radians, if you prefer) and there’s no such thing as parallel lines; all lines intersect. Circles on a sphere look much like they do in “flat” Euclidean geometry, (although some circles in spherical geometry, known as “great circles”, are also lines) but if you measure distance along the surface of the sphere, their circumference-to-diameter ratio is not what we know as π! In fact that ratio is 2 for the great circles and gets closer to our π as the circles get smaller! Under the second assumption, in hyperbolic geometry, instead the ratio is greater than our π (though, again, it gets closer to π as the circles get smaller because both hyperbolic and spherical geometry look like Euclidean geometry if you zoom in real close).
So dang, I should reconsider my position on my high horse. The only reason I think of pi as “universal” is that I privilege Euclidean geometry over the spherical and hyperbolic kind! There’s no compelling objective reason that we should say pi (defined as the ratio of a circle’s circumference to its diameter) is approximately 3.14 at all; instead all we can say is that that’s the case in Euclidean geometry. Pure mathematics views the three geometries I mention as derived from collections of axioms. By itself, it doesn’t favor any of them over the others. And why should we even prefer Euclidean geometry when we live on a sphere?
Realizing that the mathematical constant that I revere (so much that I smugly look down on those who celebrate it in a way that I think undercuts its real value) is actually contingent on a specific geometry which we designate as “the standard” one just because it’s most convenient to us is a sort of mathematical “checking my privilege.”
Checking your privilege just means to acknowledge the biases that arise from your personal experience and circumstances (which may differ hugely from those of other people) when you think about or discuss an issue. It’s become a sort of shibboleth of the left wing in American political discourse. A lot of people think it’s primarily a way to shut down conversations about the kinds of “hot button” topics where privilege is most relevant: race, class, gender, religious belief, disabilities. And indeed it can be used that way. But I think it’s a valuable thing to be aware of how your life experiences have been shaped by cultural forces and how those experiences shape your beliefs and assumptions in turn.
In the case of π, what I thought was a universal and fundamental value turned out only to be applicable to one specific set of axioms: axioms which were (not to put too fine a point on it) laid out by a white man in 300 BC. Of course I am not proposing that, had Euclid not been a white man, we would have discovered non-Euclidean geometry in less than the two millennia it actually took. But what if instead of following the paths laid down by a mathematician who did all his scratchwork on flat surfaces, we’d had other geometrical prodigies who used (cylindrical) tree trunks instead?
I follow a lot of disputes about pop culture and a common argument against adding “diverse” characters to media is that there’s no reason to “change” it or “try to make a statement.” People who make such arguments assume, based on privileges that they probably haven’t realized they enjoy, that the straight white male is the default and that using characters who don’t fit that mold is a conscious choice to deviate from that default. But maybe some people don’t fit that demographic and have different defaults. Maybe, like with Euclidean, spherical and hyperbolic geometry, there’s not really a good reason to prefer just one of them and disciplines flourish when they stop pretending there is, like geometry did at the beginning of the 19th century.
It seems like there are solid reasons to prefer Euclidean geometry, but it’s what we’re taught in school, and the mathematical canon looked at everything through that lens for almost two thousand years. Maybe math would be further along right now if we’d given those other geometries more credence between 300 B.C. and 1800 A.D. And maybe we’d make more progress faster even today if we aggressively gave more voice to people who don’t fit in with what people assume is the default.
Appendix: My Favorite Facts About π
Originally I omitted this section of this post, but I decided to add it as a palate cleanser and to demonstrate my solidarity with what I concluded earlier was the noble purpose of Pi Day: to spread the joy of mathematical knowledge. (Although I think that spreading awareness of privilege would be an even more noble purpose.)
two and a half one and a half facts about pi that leap to my mind due to their surprisingness and elegance whenever the subject of “how pi is interesting” comes up. (One and a half because one of them is only sort of “about” pi, down from two and a half because in writing this I learned that one isn’t actually true.) Here they are:
The Basel Problem, a.k.a. the sum of the reciprocals of the squares of the natural numbers
Soooo one of those names doesn’t tell you anything about this problem and the other is maybe a little dense to try to unpack, so this is a maybe slightly more friendly version of it:
(Clicking that image will take you to the Wikipedia page for the problem, which goes way more in depth than I intend to here.)
I love this fact because it’s easy to state precisely— twelve English words is far fewer than many interesting math problems take to state, and the mathematical representation is tidier still— but the result is extremely surprising and ties into some math so deep that I’d quickly be way over my head if I attempted to do more than gloss over it here.
If you’re not familiar with the notation, you only need to worry about the stuff after the second equals. The three things in between the equals signs all say the same thing. If you’re not familiar with infinite sums, they basically ask two questions: “if you keep adding those fractions up, will the sum eventually ‘even out’ at a particular number? If so, what number is that?”
Here, the answer to the first question is “yes,” and the answer to the second one is “π²/6”.
Yes! You read that right. If you keep adding up those fractions, with the squares of the natural numbers in the denominators, the sum will keep getting closer and closer to π²/6. Where does the π come in when we’re adding up these simple fractions? Well, as you’ll see if you check that Wikipedia article, the answer involves some pretty sophisticated mathematical machinery and (you may have to take my word for this part) a sizable dollop of cleverness as well. Just to add up some (ok, infinitely many) simple fractions, and get an answer that involves an irrational number! The problem took 90 years to solve after it was first posed and another 6 to prove rigorously. It’s beautiful and elegant, one of my favorite mathematical facts. And as a bonus, it also relates to the function ζ, a.k.a. the Riemann Zeta Function: one of the most important functions in mathematics, with still more surprising implications relating to the distribution of prime numbers, one of math’s most enduring mysteries.
Average river sinuosity
The sinuosity of a river is the ratio of its length with all its twists and turns included to its length “as the crow flies.” Up until this very day, I believed that the average sinuosity of the world’s rivers was approximately equal to pi.
But thanks to the Internet, it’s not difficult to check this and find that it’s almost certainly not true. A gentleman who writes for the Guardian used the data crowdsourced in that last link to write a column about the debunking. It turns out that the “fact” was based in part on a simulation, not actual measurements of real-world rivers. Perhaps in a more mathematically ideal world, where rivers are free to go where they will without being disrupted by weather or other large-scale geographic events, it would actually be true. But in this one, it appears it’s actually not.
This is my “half a fact” about pi, which includes pi but isn’t really about it, but for my money it’s the most beautiful and elegant fact in mathematics. Behold:
You don’t have to take my word for it. The Wikipedia article linked from the picture above has an entire section entitled “Mathematical beauty.” Euler’s identity is even more elegant to state than the Basel problem, and it ties together five of the most fundamental and important mathematical constants— e, i, π, 0 and 1— in a totally unexpected and concise way. If there’s one thing that you as a (presumed) layperson can do to understand why mathematicians so often refer to their discipline as “beautiful,” it’s to learn about the different parts of Euler’s identity and why it’s so remarkable that they fit together in such a simple way. That would be an excellent way to celebrate Pi Day.
Actually, I know the reason for this. It’s the same reason that we use Newtonian mechanics for when you throw a ball rather than relativity or quantum mechanics: it’s a hell of a lot simpler and more familiar, and for most of your purposes it’ll be more than good enough.
Also, Euclidean geometry does the job on a spherical Earth because, much to the detriment of many B.C. cultures, a sphere is an awful lot like a flat plane if you’re only looking at a tiny piece of it. ↩︎
- Well, the Western canon certainly represents him as white. But then, it does the same for the middle-Eastern Jesus… ↩︎