Archive for the 'Occasional' Category

Metallic Means


The golden mean occurs repeatedly in the pentagram [image Wikimedia Commons]

Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by {\phi} and is the positive root of the quadratic equation

\displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)

with the value

{\phi = (1+\sqrt{5})/2 \approx 1.618}.

There is no doubt that {\phi} is significant in many biological contexts and has also been an inspiration for artists. Called the Divine Proportion, it  was described in a book of that name by Luca Pacioli, a contemporary and friend of Leonardo da Vinci.

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The Beginning of Modern Mathematics

The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg’s printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was discovering a new world.


Illustrations by Leonardo da Vinci in Pacioli’s De Divina Proportione.

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On Knots and Links

The picture below is of a sculpture piece called Intuition, which stands in front of the Isaac Newton Institute (INI) in Cambridge. It is in the form of the Borromean Rings, a set of three interlocked rings, no two of which encircle each other.


“Intuition”. A sculpture piece in front of the Isaac Newton Institute [Photograph courtesy of S J Wilkinson].

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Unsolved: the Square Peg Problem

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?

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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Raphael Bombelli’s Psychedelic Leap

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 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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Taylor Expansions from India


NPG 1920; Brook Taylor probably by Louis Goupy

FIg. 1: Brook Taylor (1685-1731). Image from NPG.

The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his Methodus Incrementorum Directa et Inversa, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).

It is noteworthy that the series for {\sin x}, {\cos x} and {\arctan x} were known to mathematicians in India about 400 years before Taylor’s time.
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