Posts Tagged 'Hamilton'

Terence Tao to deliver the Hamilton Lecture

Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let’s try 6: it is even, so we halve it to get 3, which is odd so we triple and add 1 to get 10. Thereafter, we have 5, 16, 8, 4, 2 and 1. From then on, the value cycles from 1 to 4 to 2 and back to 1 again, forever. Numerical checks have shown that all numbers up to one hundred million million million reach the 1–4–2–1 cycle  [TM197 or search for “thatsmaths” at irishtimes.com].

Fields Medalist Professor Terence Tao.

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The Mathematics of Fair Play in Video Games

Video games generate worldwide annual sales of about $150 billion. With millions of people confined at home with time to spare, the current pandemic may benefit the industry. At the core of a video game is a computer program capable of simulating a range of phenomena in the real world or in a fantasy universe, of generating realistic imagery and of responding to the actions and reactions of the players. At every level, mathematics is crucial [TM184 or search for “thatsmaths” at irishtimes.com].

League-Of-Legends

League of Legends, from Riot Games.

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Maths and Poetry: Beauty is the Link

Mathematicians are not renowned for their ability to reach the deepest recesses of the human soul. This talent is usually associated with great artists and musicians, and a good poet can move us profoundly with a few well-chosen words [TM173 or search for “thatsmaths” at irishtimes.com].

Hamilton-Stamp

Irish postage stamp issued in 2005, on the 200th anniversary of the birth of William Rowan Hamilton.

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Symplectic Geometry

Albert-EinsteinFor many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum “physics is geometry” is a guiding principle of modern theoretical physics. Einstein’s General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:

\displaystyle \mbox{Matter tells space how to curve. Space tells matter how to move.}

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Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of additional circles, or epicycles. He stuck with the circular model [TM162 or search for “thatsmaths” at irishtimes.com].

Hodograph-AB

Left: Elliptic orbit with velocity vectors. Right: Hodograph, with all velocity vectors plotted from a single point.

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The Many Modern Uses of Quaternions

Hamiltons-Bridge-PlaqueThe story of William Rowan Hamilton’s discovery of new four-dimensional numbers called quaternions is familiar. The solution of a problem that had bothered him for years occurred to him in a flash of insight as he walked along the Royal Canal in Dublin. But this Eureka moment did not arise spontaneously: it was the result of years of intense effort. The great French mathematician Henri Poincaré also described how sudden inspiration occurs unexpectedly, but always following a period of concentrated research [TM148, or search for “thatsmaths” at irishtimes.com].

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The next Hamilton

William Rowan Hamilton was Ireland’s greatest mathematician. His name is heard thousands of times every day throughout the world when researchers use the Hamiltonian function that encapsulates the dynamics of a vast range of physical systems. He achieved fame early in life and remains one of the all-time great scientists. [TM099, or search for “thatsmaths” at irishtimes.com].

hamilton-stamp

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Old Octonions may rule the World

This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton.

Multiplication table for octonions, of the formz=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

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Robots & Biology

The article in this week’s That’s Maths column in the Irish Times ( TM037 ) is about connections between robotics and biological systems via mechanics.

The application of mathematics in biology is a flourishing research field. Most living organisms are far too complex to be modelled in their entirety, but great progress is under way in simulating individual organs and modelling specific functions such as blood-flow and locomotion.

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Admirably Appropriate

The topic of the That’s Maths column ( TM026 ) in the Irish Times this week is the surprising and delightful way in which mathematics developed for its own sake turns out to be eminently suited for solving practical problems.

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Santa’s TSP Algorithm

This week’s That’s Maths column ( TM011 ) discusses the challenge faced by Santa Claus: he has about a billion homes to visit in one night, so he needs to be smart in picking his route. The challenge he faces is called the Travelling Salesman Problem, or TSP. Continue reading ‘Santa’s TSP Algorithm’


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