My best π day was in 1975 when I invented a new way to estimate π using elementary geometry and algebra. This wasn’t a particularly important result for the mathematical world because there were already so many ways to estimate π but it was one of those moments I can see now as watershed that strongly shaped the rest of my life.
Before that moment school work seemed to involve memorization of facts and accurate execution of mechanical procedures of little immediate or future relevance to my life. The discipline required for accuracy and memorization has never been a source of joy for me. At that time improvisation, exploration, speculation and experimentation were my sources of joy: playing pickup soccer games, riding my bike, noodling with organs and synthesizers in the local music store and roaming the recently-ploughed fields looking for ancient coins and other treasures. Finding that new way to estimate π involved joyful creative activities so it inspired me to make may way through high school academics with lightweight memorization and skill learning and heavy duty on-the-fly invention. Fortunately assessment in those days was by exams which I actually enjoyed takin. They were an opportunity for a concentrated time to invent the material I needed to answer the questions from foundational material I gleaned from text books.
The video below recreates the steps I took discovering this new method. I emphasize tools and techniques you can use to go on your own mathematical adventures. All you need is pencil/paper, straight edge and ruler and to recall that the square of the hypotenuse of a right angle triangle is the sum of the square of the other two sides.
I am indebted to my old mentor and high school mathematics teacher, Dr. Puritz, who confirmed that my method was new (to him) and worth something from his expert numerical analysis perspective.
Because my “new” method builds on elementary geometry and algebra it is very unlikely that I was the first person to discover it. The necessity that mothered the invention was to do the calculations on the programmable calculator I had at hand in 1975, an HP9810. As with non-programmable calculators of the day it had basic arithmetic functions and the requisite square root. Square roots appear in mechanical calculators in 1845 (Izrael Abraham Staffel) and in the 1960’s in electronic calculators (ELKA 6521).
What about people calculating π using these geometric considerations with hand calculations? Archimedes is credited with using a recurrence to calculate the sides of inscribed polygons starting from a single triangle inscribed in a half circle.
Liu Hui in the third century used areas of inscribed polygons starting from the hexagon.
Zhao Youqin in the 14th century started from the inscribed square as I did but used the square root recurrence to compute the perimeter approximation.
The key difference between these methods and mine is they all develop a sequence to approximate π where I develop a sum. As the sum converges rapidly (later terms get small quickly) compounded errors in the calculations have a smaller impact on the final approximation.
P.S. Numbers that arise using straight edge and campus are called Constructible Numbers and represent a natural bridge between geometry and algebra.