The above formula is known as the “Wallis product,” named after John Wallis, a English mathematician who first recorded it in 1655. Wallis was apparently quite prolific and, among other contributions, is known as the mathematician that introduced the symbol for infinity:
The interest of this blog post, however is the Wallis product. The Wallis product provides a definition for pi, besides the standard definition of the ratio of a circle’s circumference to its diameter:
Pi is typically one of the first “irrational” number learned in school, others being e (2.17182…) and the square root of 2 (1.41421…). “Irrational” simply means the number cannot be written as a simple fraction (like 1/2 or 5/11). This additionally means the number has an infinite number of digits, so you can never write down the exact value.
The Wallis product is an infinite product that shows another way pi can be written and calculated. So why am I bringing up a formula that been around for 361 years? Because recently scientists found the Wallis product in nature. Specifically in the energy levels of a hydrogen atom.
University of Rochester particle physicist Carl Hagen discovered the connection while looking at the error in calculating the energy levels of a hydrogen atom. He said he immediately noticed a pattern and, upon investigation, discovered the pattern matched the Wallis product. What this says as that as energy levels increase, the hydrogen atom model approaches the Bohr model, that of a perfect sphere. Therefore it makes sense that pi makes an appearance.