Archive for June, 2016

Lecture sans paroles: the factors of M67

In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause.

FrankNelsonColePrise

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Bending the Rules to Square the Circle

Squaring the circle was one of the famous Ancient Greek mathematical problems. Although studied intensively for millennia by many brilliant scholars, no solution was ever found. The problem requires the construction of a square having area equal to that of a given circle. This must be done in a finite number of steps, using only ruler and compass.

SquareCircle-00

Taking unit radius for the circle, the area is π, so the square must have a side length of √π. If we could construct a line segment of length π, we could also draw one of length √π. However, the only constructable numbers are those arising from a unit length by addition, subtraction, multiplication and division, together with the extraction of square roots.

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Bloom’s attempt to Square the Circle

The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com].

Joyces-Tower

Joyce’s Tower, Sandycove, Co Dublin.

The challenge is to construct a square with area equal to that of a given circle using only the methods of classical geometry. Thus, only a ruler and compass may be used in the construction and the process must terminate in a finite number of steps.

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Prime Generating Formulae

The prime numbers have challenged and perplexed the greatest mathematicians for millennia. Shortly before he died, the brilliant Hungarian number theorist Paul Erdös said “it will be another million years, at least, before we understand the primes”.

Jones-Prime-Generator

A remarkable polynomial: Theorem 1 from Jones et al., 1976.

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Big Data: the Information Explosion

The world is awash with data. Large data sets have been available for many decades but in recent years their volumes have grown explosively. With mobile devices and internet connections data capture is simple and with powerful computers the analysis of “big data” is feasible [see TM092, or search for “thatsmaths” at irishtimes.com].

BigDataImagesGoogle

Google image search for “Big Data”

But there are challenges: many data sets are too large and too complex to be analysed or understood using traditional data processing methods. Our current armoury of analysis techniques is inadequate and new mathematical methods are needed.

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