Posts Tagged 'Algorithms'

Emergence of Complex Behaviour from Simple Roots

It is exhilarating to watch a large flock of birds swarming in ever-changing patterns. Swarming is an emergent behaviour, resulting from a set of simple rules followed by each individual animal, bird or fish, without any centralized control or leadership.

Flocking-Starlings-CyrilByrne

A murmuration of starlings at dusk near Ballywilliam, Co Wexford. Photograph: Cyril Byrne.

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ToplDice is Markovian

Many problems in probability are solved by assuming independence of separate experiments. When we toss a coin, it is assumed that the outcome does not depend on the results of previous tosses. Similarly, each cast of a die is assumed to be independent of previous casts.

However, this assumption is frequently invalid. Draw a card from a shuffled deck and reveal it. Then place it on the bottom and draw another card. The odds have changed: if the first card was an ace, the chances that the second is also an ace have diminished.

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Algorithms: Recipes for Success

The impact of computing on society is ever-increasing. Web-based commerce continues to grow and artificial intelligence now pervades our lives. To make wise choices, we need to understand how computers operate and how we can deploy them most constructively. Listen to any computer scientist and soon you will hear the word “algorithm” [TM168 or search for “thatsmaths” at irishtimes.com].

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Cumbersome Calculations in Ancient Rome

Arithmetica-WoodcutTypus Arithmeticae” is a woodcut from the book Margarita Philosophica by Gregor Reisch of Freiburg, published in 1503. In the centre of the figure stands Arithmetica, the muse of mathematics. She is watching a competition between the Roman mathematician Boethius and the great Pythagoras. Boethius is crunching out a calculation using Hindu-Arabic numerals, while Pythagoras uses a counting board or abacus (tabula) and – presumably – a less convenient number system. Arithmetica is looking with favour towards Boethius. He smiles smugly while Pythagoras is looking decidedly glum.

The figure aims to show the superiority of the Hindu-Arabic number system over the older Greek and Roman number systems. Of course, it is completely anachronistic: Pythagoras flourished around 500 BC and Boethius around AD 500, while the Hindu-Arabic numbers did not arrive in Europe until after AD 1200.

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Simple Curves that Perplex Mathematicians and Inspire Artists

The preoccupations of mathematicians can seem curious and strange to normal people. They sometimes expend great energy proving results that appear glaringly obvious. One such result is called the Jordan Curve Theorem. We all know that a circle has an inside and an outside, and that this property also holds for a much larger collection of closed curves [TM165 or search for “thatsmaths” at irishtimes.com].

Michaelangelo-RobertBosch-Hands

Detail from Michaelangelo’s The Creation of Adam, and a Jordan Curve representation [image courtesy of Prof Robert Bosch, Oberlin College. Downloaded from here].

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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.

Archimedes-Polygons

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Multiple Discoveries of the Thue-Morse Sequence

It is common practice in science to name important advances after the first discoverer or inventor. However, this process often goes awry. A humorous principle called Stigler’s Law holds that no scientific result is named after its original discoverer. This law was formulated by Professor Stephen Stigler of the University of Chicago in his publication “Stigler’s law of eponymy”. He pointed out that his “law” had been proposed by others before him so it was, in a sense, self-verifying. [TM157 or search for “thatsmaths” at irishtimes.com].

Axel Thue (1863-1922) and Marston Morse (1892-1977)
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