Of the Curved Universe and Pi
A Cockroach’s Approach
Is our Universe curved? Who knows. I’ll tell you who: someone who knows how to measure the value of π.
Let me start by reminding the reader that, by definition, π is the ratio of a circle’s circumference C and its diameter D: π = C/D = C/2R. Also, a circle is a set of points located at the same distance from another point, Z. Point Z is called the circle’s center. The said distance is the circle’s radius, half of the circle’s diameter. I hope we are on the same page.
To measure the π, one needs to draw a perfect circle, perfectly measure its circumference and diameter or radius, and divide one by the other. Problem solved. (Except that drawing and measuring the circle are outside this article’s scope.)
<Enters a Sceptisict.> But why would we want to measure the value of π if it is a constant and can be calculated to 62,831,853,071,796 decimal digits?
<Enters the Author.> Because it is a constant only if the Universe is flat. If the Universe is curved, not only is π not a constant anymore, but we can use its variation to estimate the curvature.
For presentability, let us assume that our Universe is two-dimensional, like a piece of paper. We move on it, not unlike cockroaches, and do not know or do not care that the piece of paper is embedded into a higher-dimensional space. If we draw a circle on a piece of paper, its circumference would be π0 times its diameter. (Here, π₀ is the value of π as we know it, 3.14159 and 62,831,853,071,791 more digits.)
Now, let us pretend that the Universe is curved (say, uniformly in two directions) and looks like a ball of radius 1/k. The Universe still looks flat for cockroaches because they “bend” together with it. But not so for the π.
Imagine an inquisitive cockroach that was charged with measuring the value of π. The cockroach selects a random point, Z, and carefully marks all the points that are precisely R units away from Z. The cockroach then follows the circle, carefully measures its circumference C, and divides C by 2R to obtain π. To the cockroach’s shock, the value of π depends on R.
How is that possible? When the Universe is curved (not a piece of paper but a ball), the radius of a circle is not R (the surface distance from Z) but R0, as shown in the picture: C = 2π₀R₀.
However, from the cockroach’s point of view, it is still C=2πR, by the definition of π. Since we are talking about the same circle, naturally, 2πR = 2π₀R₀ and, hence, π = π₀R₀/R.
R₀ can be calculated using high-school trigonometry: R₀ = 1/k sin(a) = 1/k sin(R / (1/k)) = sin(Rk)/k. Combining this with the previous equation, we get π = π₀sin(Rk)/(Rk). By the way, k is the curvature of the Universe, and it is the reciprocal of the curvature radius Rc: k = 1/Rc.
As one can see, the measured value of π depends on the radius of the circle used for the measurement. The cockroach has no reason to be surprised.
For the flat Universe, k=0, and the equation becomes π = π₀sin(R×0)/(R×0) = π₀sin(0)/0 = π₀×0/0, a nightmare in algebra but merely a minor annoyance in calculus. Taking the limit as k tends to 0, we get that π = π₀sin(x)/x = π₀∎
As the Universe curves, k increases, and π and π₀ become different from each other (though probably not significantly different). Knowing the “true” value of π₀ and measuring R and π, the inquisitive cockroach can calculate the curvature as the solution to the non-linear equation π/π₀ = sin(Rk)/(Rk). Solving this equation is outside this article’s scope, too, and trust me, it is the least of the cockroach’s problems.