## Posts Tagged 'Pi'

### The Power of the 2-gon: Extrapolation to Evaluate Pi

Lewis Fry Richardson

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 ${\pi}$, 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).]

### Bouncing Billiard Balls Produce Pi

There are many ways of evaluating ${\pi}$, 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.

### 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 might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).

### 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 C to diameter D has the same value for all?

Slicing a disk to estimate pi (Image Wikimedia).

### 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.

Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)

### 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 need to compute earlier digits. This is known as the BBP formula.
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 recognizing March 14 as National Pi Day.

Today is also the birthday anniversary of Albert Einstein, giving us another reason to celebrate. He was born on 14 March 1879, just 134 years ago today.
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 the lever, for designing the helical pump that bears his name, for devising engines of war, and for much more. Continue reading ‘Archimedes uncovered’