Happy Pi Day: Pi Is an Imposter

​The constant value of π is held aloft with often almost mystical reverence—a transcendental, irrational number that nonetheless describes some enormous part of our reality. Within π, we find not just "circles" but all of the things that are of circles, like waves, the Gaussian curve (the "bell curve"), the whole of trigonometry.

π gives quantitative life to so many things that might seem to be unquantifiable, yet it itself is in a fairly real sense, unquantifiable. π is as deeply precise as anything, but never truly resolved. It may be some comfort then to know that π is an imposter.

π is certainly a very, very useful imposter, but to see my point we might start by imagining π in another, very different universe. Could π be different? In a sense, no. π is the ratio between a circle's circumference (C) and its diameter (d), given by C/d, and if we were to export that concept to really anywhere imaginable, it would be the same.

(A good discussion: ​"Could PI have a different value in a different universe?")

That's a bold claim, but it winds up holding because to export the concept of π, as C/d, is to export the idealized framework of π. This idealized framework is known as Euclidean space, which is not a real space, but an abstraction of space. There is a "real" map of the New York City subway system, which shows where the various subway lines actually run in reality to the city above, and then there is the colorful map diagram, showing an idealized, highly functional view of that system. The first view is out there in the wild, carved inch by inch underneath the city's concrete, while the other tells riders how to get around by offering the subway system as a set of relationships.

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π exists in the latter realm, which is Euclidean space. This space is governed not so much by measurements but by axioms, which are just a set of rules that everyone agrees on. In Euclidean space, everyone agrees that to get from one set of coordinates in two dimensions to another set, we can follow a straight path. Any given pair of coordinates will be found on exactly one line. In three dimensions swap line for plane, which is just a flat surface where any pairing of a single line and a single point can be found on exactly one flat plane. All of this stuff seems pretty obvious.

But it's also not real. It's just what we agree as real. We know that out there in the wild, Euclidean space doesn't exist. Instead, there is Riemannian geometry, which is the geometry of curved space. Here, it is not the case that there exactly one line between two points; instead, there is a curve (or curves). This is the space that we live in because gravity bends space-time, according to the theory of relativity. You can see this via gravitational lensing, where some massive cosmic body (a black hole or binary star system) warps space-time such that the path of light itself is bent inwards.

In the warped world, which is the real one, π doesn't exactly do the job. Or, rather, it needs to change all of the time depending on the local curvature of space-time. π as we know it is the ratio of a circle's circumference to its radius in an idealized universe. We have no such luxury in the real world, though pi will be quite good enough most of the time. The conceptual subway map gets us where we need to go.

As for pi in other universes, figure that it would probably still work out alright. We can imagine that gravity in another universe might be a much, much stronger force and the local space-time curvature would be way more warped, but a universe with such an extreme gravitational force would be a lot less likely to host intelligent life of the sort likely to daydream about alternative πs and hypergeometries.