Archive for December, 2016

Unsolved: the Square Peg Problem

Published December 29, 2016 Occasional Closed
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The idiom “square peg in a round hole” expresses a mismatch or misfit, often referring to somebody in the wrong profession. It may also indicate a difficult or impossible task but, of course, it is quite simple to fit a square peg in a round hole, hammering it in until the corners are tight against the circular boundary of the hole. Since the peg may be oriented at any angle, there are an infinite number of ways to fit a square within a circle. In contract, for a boomerang-shaped hole, there is just one way to draw a square with its vertices on the curve.

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Twenty Heads in Succession: How Long will we Wait?

Published December 22, 2016 Occasional Closed
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If three flips of a coin produce three heads, there is no surprise. But if 20 successive heads show up, you should be suspicious: the chances of this are less than one in a a million, so it is more likely than not that the coin is unbalanced.

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The Edward Worth Library: a Treasure Trove of Maths

Published December 15, 2016 Irish Times Closed
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Infinite Riches in a Little Room.  Christopher Marlowe.

The Edward Worth Library may be unknown to many readers. Housed in Dr Steevens’ Hospital, Dublin, now an administrative centre for the Health Service Executive, the library was collected by hospital Trustee Edward Worth, and bequeathed to the hospital after his death in 1733. The original book shelves and cases remain as they were in the 1730s. The collection is catalogued online. [TM105 or search for “thatsmaths” at irishtimes.com].

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Raphael Bombelli’s Psychedelic Leap

Published December 8, 2016 Occasional Closed
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The story of how Italian Renaissance mathematicians solved cubic equations has elements of skullduggery and intrigue. The method originally found by Scipione del Ferro and independently by Tartaglia, was published by Girolamo Cardano in 1545 in his book Ars Magna. The method, often called Cardano’s method, gives the solution of a depressed cubic equation t3 + p t + q = 0. The general cubic equation can be reduced to this form by a simple linear transformation of the dependent variable. The solution is given by

cardano-formula

Cardano assumed that the discriminant Δ = ( q / 2 )2 + ( p / 3 )3, the quantity appearing under the square-root sign, was positive.

Raphael Bombelli made the psychedelic leap that Cardano could not make. He realised that Cardano’s formula would still give a solution when the discriminant was negative, provided that the square roots of negative quantities were manipulated in the correct manner. He was thus the first to properly handle complex numbers and apply them with effect.

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The Shaky Foundations of Mathematics

Published December 1, 2016 Irish Times Closed
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The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].

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Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]

The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.

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