Pi at PI: How do theoretical physicists use pi?

Humans have been investigating the number pi for thousands of years. Let’s find out what it is, how theoretical physicists use it today, and why it’s so beloved.

Pop quiz: What is pi?

Pi is the ratio between the circumference and the diameter of a circle.

To find pi, take a circular object and wrap a string around the outside of it to find the circle’s circumference. Then, stretch the string from one side of the circle to the other to find the diameter. Divide the circumference by the diameter, and voila: you’ll find pi.  

Pi is a mathematical constant. No matter how big or small your circle, from a button to a planet, the ratio between the circumference and diameter, and thus pi, remains the same.  

Pi is commonly shortened to 3.14. But pi is an irrational number, which means that the numbers after its decimal point never end. Currently, researchers have calculated trillions of digits of pi – but that level of precision isn’t necessary for most scientific calculations. NASA, for example, reports that 15 decimals of pi is sufficient for calculating interplanetary navigation.

How do theoretical physicists use pi today?

“Every modern theoretical physicist uses pi – everyone in this building uses it,” says Damian Pope, Outreach Scientist at Perimeter Institute.

Pi is woven into the fabric of the universe, from the smallest quantum scale to the entire cosmological universe. It shows up when researchers are dealing with circles, waves, and spheres. These shapes are ubiquitous in both modern physics and everyday life. 

Pi at the quantum level

It may surprise you to know that pi even comes up in quantum computing, including in the qubit, the quantum computer bit. In classical computers, information is stored as bits, which can exist as either 0 or 1. Quantum computers, however, use qubits. Qubits can exist in superposition –  a blend of probabilities between 0 and 1.

One way that researchers model superposition is by using a Bloch sphere. On a Bloch sphere, the states of 0 and 1 are represented as the ‘north and south poles’ of the sphere.

To help researchers visualize superposition, imagine you have a pen anchored at the middle of the Bloch sphere. As the pen pivots and points to different points on the outside of the sphere, this represents the different superposition states of the qubit.  

“The Bloch sphere helps quantum computing researchers as it gives them a model, or a framework, in which to think about qubits, which are the basic building blocks of quantum computers,” says Pope. “Pi comes in again because you have a sphere.”

“Pi is built into the description of your qubit,” he adds.

Pi at the universe level

Pi shows up constantly when modelling the universe. Stars, planets, orbital paths, and plenty of other objects all require pi. Pi is even useful for modelling some of the most interesting structures in the universe: black holes. “People often talk about the boundary of black holes like it’s the surface of a ball or a sphere,” says Pope. “Pi is built into the formula for calculating that surface area.”

Why are scientists such big fans of pi?  

Pi is one of the most important mathematical constants. It’s pervasive and useful, says Pope. From mathematical physics to modelling quantum particles and fields, pi is always there for us.  

And it’s not just physicists who encounter pi, says Pope. Architects, structural engineers, and even artists, engage with pi.  

“We use it a lot, and it’s very useful,” says Pope. “You learn to know and love it.”