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https://mathshistory.st-andrews.ac.uk/Biographies/Van_Ceulen/ | https://mathshistory.st-andrews.ac.uk/Biographies/Van_Ceulen/ | ||
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[[Category:MA279Fall2022Swanson]] | [[Category:MA279Fall2022Swanson]] | ||
Revision as of 12:28, 8 November 2022
Introduction
Pi is possibly the most famous constant in mathematics. Interestingly, it has also been one of the most difficult to calculate throughout history and this has lead to some very creative solutions as problems such as this usually do.
- This needs revision
Basic Approximations
- Is this necessary as a separate heading? Should this come at a different spot in this page? - 22/7 - Rolling a wheel out on a road to determine pi - Bounds between 3 and 4 and how they are determined
Early Calculation
In some of the earliest historical records known of pi, it was recorded to be somewhere roughly between 3 and 4. Specifically, the Babylonians nearly 4,000 years ago seem to have believed pi was 25/8 or 3.125. The Egyptians (at around the same time) noted the pi was (16/9)^2 or roughly 3.16. The most well-known early algorithm for computing pi is popularly known from the mathematician Archimedes (and a similar, iterative method was used in China known as Liu Hiu's Pi Algorithm). Archimedes knew that pi was described as the ratio of a circle's circumference to its diameter and to him the most significant limiting factor in calculating pi was knowing the exact circumference of some theoretical circle defined by a known radius. He could however find a close approximation by creating polygons both inside and outside the circle with known side lengths to calculate pi. This grew ever closer to pi the more sides were added to these polygons but took a significant amount of work. There is a famous example of a man who dedicated decades to the calculation of pi using this method, his name was Ludolph van Ceulen. He calculated pi up to 35 places using this method and when he died the digits for his upper and lower bounds of pi (from the inner and outer polygons around the circle in the approximation) were inscribed on his tombstone.
Newtonian Calculation
Might want to find more references on this or use this video as the reference: https://www.youtube.com/watch?v=gMlf1ELvRzc Also might consider renaming this section "Infinite Series Calculation"
- Pascal's Triangle realization - Integration of unit circle - Infinite series and convergence rate optimization to find pi - Fun quote by Newton: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."
Modern Calculation
- Algorithms by a computer - Is there a closed form known to find a particular digit of pi? - Reference the book Contact by Carl Sagan? - Largest number of digits of pi currently known
Odd Calculation
- Throwing pens on the ground or similar? - Probability related?
Applications
- Drink containers? - Super computing benchmarks? - Car tires? - Others?
References
- Likely clean these up to look more APA
https://en.wikipedia.org/wiki/Pi https://en.wikipedia.org/wiki/Approximations_of_π - might need to use this more heavily https://en.wikipedia.org/wiki/Liu_Hui%27s_π_algorithm https://en.wikipedia.org/wiki/Chronology_of_computation_of_π - unused currently, might be a useful graphic https://mathshistory.st-andrews.ac.uk/Biographies/Van_Ceulen/ https://www.universetoday.com/110331/happy-pi-day-5-ways-nasa-uses-pi/#:~:text=In%20basic%20mathematics%2C%20Pi%20is,flight%2C%20to%20name%20a%20few. - unused currently, might be useful
