Pi / $\pi$

Prerequisites: Trigonometric Functions.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Take any circle, for example a coin or a wheel, and roll it once along a straight line. The distance it covers is a bit more than three times its width. The exact factor is the same for every circle, however large or small, and this number is called pi, about 3.14159.

The reason it is the same for all circles is that all circles have the same shape. Making a circle twice as wide makes everything about it, including its circumference, twice as large, so the ratio does not change.

Once pi is fixed by circles, it shows up wherever something is round or something repeats. The area of a disk, the surface of a sphere and the volume of a ball all contain pi. A full turn is two pi when angles are measured in radians, so anything that rotates or oscillates, such as a pendulum, a wave or an alternating current, carries a pi in its formulas. Even the bell curve of probability theory contains pi in its normalization, because computing its area can be turned into a question about circles.

The digits of pi never end and never repeat. People have computed trillions of them, but for physics a handful of digits is more than enough.

Concrete

The number $\pi$ is defined as the ratio of the circumference $C$ of a circle to its diameter $d = 2r$:

$$C = \pi d = 2\pi r .$$

The area follows by cutting the disk into thin rings. A ring at radius $r'$ with width $dr'$ has area $2\pi r'\, dr'$, so

$$A = \int_0^r 2\pi r' \, dr' = \pi r^2 .$$

The same idea in three dimensions, with spherical shells of area $4\pi r'^2$, gives the volume of a ball, $V = \int_0^r 4\pi r'^2\, dr' = \tfrac{4}{3}\pi r^3$.

Radians

An angle in radians is the arc length it cuts out of a circle of radius 1. A full turn is therefore $2\pi$, a right angle is $\pi/2$, and the arc length on a circle of radius $r$ is $s = r\theta$. Formulas like $\frac{d}{d\theta}\sin\theta = \cos\theta$ hold only with angles in radians, which is why physics uses them throughout.

How to compute it

Archimedes trapped the circle between inscribed and circumscribed regular polygons with 96 sides and found $\frac{223}{71} < \pi < \frac{22}{7}$. Modern methods use series, for example

$$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots ,$$

which follows from integrating $\frac{1}{1+x^2} = 1 - x^2 + x^4 - \dots$ from $0$ to $1$ and using $\arctan 1 = \pi/4$. This series converges very slowly; practical computations use faster series.

Where $\pi$ appears in physics

• Oscillations: a mass on a spring with angular frequency $\omega$ has period $T = 2\pi/\omega$. For a pendulum of length $L$ at small amplitude, $T = 2\pi\sqrt{L/g}$.
• Spheres: Coulomb's law is written $F = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}$ so that the $4\pi$ from the area of a sphere cancels in Gauss's law $\oint \vec E \cdot d\vec A = Q/\varepsilon_0$.
• Quantum mechanics: $\hbar = h/2\pi$ converts between frequency $\nu$ and angular frequency $\omega = 2\pi\nu$.

• Gaussian integrals: $\int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi}$. The quick proof squares the integral and switches to polar coordinates, where the factor $2\pi$ of a full turn appears.

• Prehistoric Calculus: Discovering Pi by Kalid Azad

Abstract

The definition through circles uses Euclidean geometry. Analysis gives a definition without geometry. The map

$$\mathbb{R} \to U(1) , \qquad t \mapsto e^{it} = \sum_{n=0}^\infty \frac{(it)^n}{n!}$$

is a surjective group homomorphism from the additive group of real numbers onto the unit circle in $\mathbb{C}$. Its kernel is a discrete subgroup of $\mathbb{R}$, and one can show it is nonzero, so it has the form $c\,\mathbb{Z}$ for a unique $c > 0$. One defines $\pi = c/2$. Equivalently, $\pi/2$ is the smallest positive zero of $\cos t = \sum_n (-1)^n t^{2n}/(2n)!$. Then $e^{i\pi} = -1$, and the length of the curve $t \mapsto e^{it}$ for $0 \le t \le 2\pi$ is $2\pi$, which reproduces the geometric definition. In this language $2\pi$ is the period of the characters $e^{int}$ of $U(1)$, which is why it appears in Fourier analysis and in quantization conditions such as the Bohr-Sommerfeld rule $\oint p\, dq = 2\pi\hbar\, n$.

Arithmetic nature

$\pi$ is irrational (Lambert, 18th century) and transcendental (Lindemann, 1882): it is not a root of any nonzero polynomial with rational coefficients. Transcendence implies that the circle cannot be squared with ruler and compass, since every constructible length is algebraic.

Geometry in higher dimensions

The volume of the unit ball in $\mathbb{R}^n$ and the area of the unit sphere $S^{n-1}$ are

$$\operatorname{vol}(B^n) = \frac{\pi^{n/2}}{\Gamma\!\left(\frac{n}{2}+1\right)} , \qquad \operatorname{area}(S^{n-1}) = \frac{2\pi^{n/2}}{\Gamma\!\left(\frac{n}{2}\right)} ,$$

with $\Gamma(\tfrac12) = \sqrt{\pi}$. Both follow from evaluating $\int_{\mathbb{R}^n} e^{-|x|^2} d^n x = \pi^{n/2}$ once in Cartesian and once in spherical coordinates. These factors appear in loop integrals in quantum field theory and in dimensional regularization.

For a compact oriented two-dimensional Riemannian manifold $M$ without boundary, the Gauss-Bonnet theorem states

$$\int_M K \, dA = 2\pi \chi(M) ,$$

where $K$ is the Gaussian curvature and $\chi$ the Euler characteristic. Here $2\pi$ is again the total angle of a full turn, now relating local geometry to topology.

Why is it interesting?

Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

An Intuitive Guide To Exponential Functions & e by Kalid Azad

• Why Pi Matters By Steven Strogatz