Richardson's extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of $latex {\pi}&fg=000000$, and show how the estimates improve dramatically with higher order extrapolation. [This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).] … Continue reading The Power of the 2-gon: Extrapolation to Evaluate Pi
Tag: Pi
Bouncing Billiard Balls Produce Pi
There are many ways of evaluating $latex {\pi}&fg=000000$, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method. Historical Methods Archimedes used inscribed and circumscribed polygons to deduce that $latex \displaystyle \textstyle{3\frac{10}{71} < \pi < 3\frac{10}{70}} &fg=000000$ giving roughly … Continue reading Bouncing Billiard Balls Produce Pi
It’s as Easy as Pi
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see TM120 or search for “thatsmaths” at irishtimes.com]. The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You … Continue reading It’s as Easy as Pi
Who First Proved that C / D is Constant?
Every circle has the property that the distance around it is just over three times the distance across. This has been “common knowledge” since the earliest times. But mathematicians do not trust common knowledge; they demand proof. Who was first to prove that all circles are similar, in the sense that the ratio of circumference … Continue reading Who First Proved that C / D is Constant?
Fermat’s Christmas Theorem
Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions. Girard also showed how the area of a spherical triangle depends on its interior angles. If the angles of a triangle on the unit sphere … Continue reading Fermat’s Christmas Theorem
The remarkable BBP Formula
Information that is declared to be forever inaccessible is sometimes revealed within a short period. Until recently, it seemed impossible that we would ever know the value of the quintillionth decimal digit of pi. But a remarkable formula has been found that allows the computation of binary digits starting from an arbitrary position without the … Continue reading The remarkable BBP Formula
Happy Pi Day 2013
Today, 14th March, is Pi Day. In the month/day format it is 3/14, corresponding to 3.14, the first three digits of π. So, have a Happy Pi Day. Larry Shaw of San Francisco's Exploratorium came up with the Pi Day idea in 1988. About ten years later, the U.S. House of Representatives passed a resolution … Continue reading Happy Pi Day 2013
Archimedes uncovered
The That’s Maths column in this week's Irish Times ( TM012 ) describes the analysis of the ancient codex known as the Archimedes Palimpsest. Archimedes of Syracuse Archimedes (Ἀρχιμήδης, 287-212 BC) was a brilliant physicist, engineer and astronomer, and the greatest mathematician of antiquity. He is famed for founding hydrostatics, for formulating the law of … Continue reading Archimedes uncovered
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