Archive for the 'Occasional' Category

Geodesics on the Spheroidal Earth – I

Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not?

FlatEllipsoid

A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.

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Fourier’s Wonderful Idea – I

Breaking Complex Objects into Simple Pieces

“In a memorable session of the French Academy on the
21st of December 1807, the mathematician and engineer
Joseph Fourier announced a thesis which inaugurated a
new chapter in the history of mathematics. The claim of
Fourier appeared to the older members of the Academy,
including the great analyst Lagrange, entirely incredible.”

Introduction

Fourier

Joseph Fourier (1768-1830)

The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

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Sophus Lie

It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002).

Sophus-Lie

Sophus Lie (1842-1899)

Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no hardship for Lie to walk the 60 km from Oslo to Moss at the weekend to visit his parents. At school, Lie was a good all-rounder, though his mathematics teacher, Ludvig Sylow, a pioneer of group theory, did not suspect his great potential or anticipate his remarkable achievements in that field.

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Subtract 0 and divide by 1

We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that

\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1

This is an essential step in deriving an expression for the derivative of {\sin x}.

LHopital-Bernoulli

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The Evolute: Envelope of Normals

Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.

SinEvolute

Sin t (blue) and its evolute (red).

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Hardy’s Apology

Godfrey Harold Hardy’s memoir, A Mathematician’s Apology, was published when he was 63 years old. It is a slight volume at just 90 pages, but is replete with interesting observations and not a few controversial opinions. After 78 years, it is still in print and is available in virtually every mathematics library. Though many of Hardy’s opinions are difficult to support and some of his predictions have turned out to be utterly wrong, the book is still well worth reading.

Hardy-MathApology

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Kaprekar’s Number 6174

The Indian mathematician D. R. Kaprekar spent many happy hours during his youth solving mathematical puzzles. He graduated from Fergusson College in Pune in 1929 and became a mathematical teacher at a school in Devlali, north-east of Mumbai.

KaprekarFlowGraph495

Kaprekar process for three digit numbers converging to 495 [Wikimedia Commons].

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