How can anyone not recognize Pi (π) which is the most well known transcendental number in mathematics? It is defined as a ratio of the circumference to the diameter of any circles and it appears frequently in geometry when we need to calculate area and volume of various geometrical structures. Every year on March 14 which is the closest calendrical approximation of Pi, people around the world celebrate Pi-day and they aren’t just mathematicians, engineers, scientists, but the general population overall in which pie eating contest is a very popular activity among them. Yes, Albert Einstein’s birthday was on March 14, 1879 as well. (π) cannot be expressed as an exact number or as a ratio of two integers.        π≈ 3.14     or       π≈ 22/7       π is associated with mathematics.

John Wallis, a seventeenth century British mathematician, wrote in his book Arithmetica Infinitorum defining Pi as the product of an infinite string of ratios made up of integers. ( Wallis derived this formula by a method of successive  interpolations in 1655. Proofs of this formula have been written using many different approaches such as combinatorics, probability, and calculus involving trigonometric integrals.

In August of 2015, two scientists, Tamar Friedmann from Department of Mathematics, Rochester University, and C.R. Hagen from Department of Physics and Astronomy, Rochester University, discovered Pi hiding in a quantum mechanics formula for the energy states of the hydrogen atom. As a matter of fact, they didn’t just find Pi, but also the classic seventeenth century Wallis formula for Pi, making them the first to derive it from physics, namely, quantum mechanics. This is a beautiful connection between Pi and Quantum Mechanics, the fact that they were not even looking for Pi during the calculation process for the energy states of the hydrogen atom underscores pi’s omnipresence in math and science.

The Danish physicist Niels Bohr who developed quantum calculations back in the early 20th century and provided accurate values for the energy states of hydrogen atom while Hagen applied a different method called the variational principle to approximate the value for the ground state of the hydrogen atom. Hagen and Friedmann were able to calculate the values for the different energy states and compare them with the values obtained by Bohr even when the energy levels were getting higher. The ratio of the lowest energy state to the excited state for any given orbital angular momentum quantum number yielded the Wallis formula for Pi. Their calculations resulted in a special mathematical function called gamma functions leading to the formula,    ( and it can be reduced to the Wallis formula listed in Figure 2. Mathematician Moshe Machover of King’s College London calls the finding a “cunning piece of magic.” He further commented ” this derivation of Pi is a surprise of the familiar, much like a magician’s trick. A child who sees a trick for the first time may be only surprised. But an adult , who has seen numerous tricks over the years, experiences both surprise and familiarity.” Doug Ravenel, a professor of mathematics at the University of Rochester says ” This is a beautiful connection between Pi and quantum mechanics that could have been found 80 years ago, but was not discovered until now.”