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Hamilton's Quaternions

William Rowan Hamilton’s 'Eureka moment' resulted in a new system of four-dimensional numbers which he called ‘quaternions’. But what use are they?

‘ONE SMALL SCRATCH FOR A MAN, ONE GIANT LEAP FOR MATHEMATICS’

On a bright Monday morning, on 16 October 1843, the world of mathematics was changed forever. William Rowan Hamilton was walking along the banks of the Royal Canal in Dublin with his wife, Helen. As he passed Broombridge in Cabra, he had a flash of inspiration. Hamilton described the ‘eureka’ moment in a letter to his son some years later:

‘Although your mother talked with me now and then, yet an undercurrent of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once an importance. An electric current seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work . . . Nor could I resist the impulse—unphilosophical as it may have been—to cut with a knife on a stone of Brougham Bridge as we passed it, the fundamental formula . . .’

Thus Hamilton’s mind gave birth to a strange new system of four-dimensional numbers called ‘quaternions’. In a nineteenth-century act of minor vandalism, Hamilton scratched his mathematical creation on the bridge.

Liberator of algebra
The quaternions are Hamilton’s most celebrated contribution to mathematics. Two-dimensional numbers had played a significant role in two-dimensional geometry and in solving practical problems in two dimensions. Hamilton had been trying to extend his theory of two-dimensional numbers to a theory of three-dimensional numbers (also called triplets). He hoped that these triplets would provide a natural mathematical structure for describing our three-dimensional world. He had difficulty finding a suitable theory of triplets (we now know why—it’s impossible!). Then, on 16 October 1843, his mind gave birth to quaternions as he walked along the banks of the Royal Canal. In this moment of revelation he realised that if he worked with number quadruples and an unusual multiplication operation he would get everything he wanted. He named his new system of numbers ‘quaternions’ because each number quad-ruple had four components.

Hamilton had created a completely new structure in mathematics. The mathematical world was stunned by his audacity in creating a system of numbers that did not satisfy the usual commutative rule for multiplication. (This is the rule in arithmetic which says that it does not matter in which order you multiply two ordinary numbers, e.g. two times three equals three times two. The quaternions did not satisfy this rule.) This did not bother Hamilton because this is what usually happens in nature. For example, consider an empty swimming pool and the two operations of diving into the pool head first and turning the water on. The order in which the operations take place is important!

Hamilton has been called the ‘liberator of algebra’ because his quaternions smashed the previously accepted notion that any useful algebraic number system should satisfy the rules of ordinary numbers in arithmetic. His quaternions opened up a new landscape where mathematicians could feel free to conceive new number systems that were not shackled by the rules of ordinary numbers in arithmetic. One could say that modern algebra was born on the banks of the Royal Canal in Dublin. I suppose one could also say that it was ‘one small scratch for a man, one giant leap for mathematics’! Hamilton’s creation of quaternions is commemorated by a plaque which was unveiled in 1958 by the taoiseach, Éamon de Valera, who was a mathematician himself and a fan of Hamilton. De Valera paid homage to Hamilton by scratching the quaternion formulas on his cell wall when he was in Kilmainham Jail in 1924.

Use of the word ‘liberator’ illustrates the fact that freedom plays a significant role in mathematics. The notion of freedom in mathematics can surprise some people, but the mathematician Cantor once said that freedom is the essence of mathematics. In fact, one is free to conceive of any new ideas one wants in mathematics (just as Hamilton was free to conceive of the quaternions, even though they broke with convention at the time). These new ideas may or may not lead to something interesting or useful. Historically (and probably also in the future) the major breakthroughs in mathematics have typically happened because the great mathematicians were free to conceive of any new ideas they wanted, even if their wild thoughts broke with conventions and seemed bizarre to other mathematicians and the general public.

Applications of Hamilton’s mathematics
Hamilton’s mathematics has been, and still is, crucial for many important applications to science, engineering, computer animation, computer games, special effects in movies, space navigation and many other areas.

For example:
(a) Lara Croft in Tomb Raider was created using quaternions.
(b) Quaternions now play an important role in special effects in movies. For example, an Irish company called Havok used quaternions in the creation of the acclaimed new special effects in The Matrix Reloaded, and also in Poseidon, which was nominated for an Oscar for its visual effects in 2007. Havok won an Emmy award in the US in 2008 for pioneering new levels of realism and interactivity in movies and games. The dramatic visual effects in the James Bond movie Quantum of Solace were also created by Havok.
(c) Space navigation uses quaternions. For example, quaternions were fundamental in the successful operation of Apollo 11, which landed the first man on the moon in 1969. Quaternions were also crucial in the landing of Curiosity on Mars in 2012.
(d) Quaternions played an important role in Maxwell’s mathematical theory and prediction of electromagnetic waves in 1864. Maxwell’s theory ultimately led to the detection of radio waves by Hertz. Thus the inventions of radio, television, radar, X-rays and many other significant products of our time are directly related to Hamilton’s mathematics. Maxwell’s work illustrates the magical power of mathematics because his mathematics made the invisible visible, since radio waves are invisible to our five senses.
(e) Hamilton’s fundamental theory of dynamics in 1834 was indispensable for the creation, in the early 1900s, of Quantum Mechanics, which is how we now understand the physical world at the microscopic level. Moreover, his famous Hamiltonian function is fundamental to many aspects of physics.
(f) Vector analysis, which is indispensable in physics, is an offspring of quaternions.

Reproduced with thanks to Fiacre Ó Cairbre, Senior Lecturer in Mathematics at Maynooth University.

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