Sir William Rowan Hamilton Irish Mathematician The Power and Value of Mathematics

Sir William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. His discovery of quaternions is perhaps his best known investigation. Hamilton's work was also significant in the later development of quantum mechanics. The Power and Value of Mathematics

The Power and Value of Mathematics Career William Rowan Hamilton's mathematical career included the study of geometrical optics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions, solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions, linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley-Hamilton Theorem). The Power and Value of Mathematics

The Power and Value of Mathematics Quaternions One of the great contributions Hamilton made to mathematical science was his discovery of quaternions in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He could not do so for 3 dimensions, and in fact it was later shown that it is impossible. Eventually Hamilton tried 4 dimensions and created quaternions. The Power and Value of Mathematics

The Power and Value of Mathematics Genius According to Hamilton, on October 16 he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation i2 = j2 = k2 = ijk = − 1 suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge), for fear he would forget it. Quaternion Plaque on Broom Bridge The Power and Value of Mathematics

Vector algebra and matrices had yet to be developed. Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton described a quaternion as an ordered quadruple (4-tuple) of one real number and three mutually orthogonal imaginary units with real coefficients, and described the first coordinate as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the dot product of the vector parts, while the vector part of the product is the cross product. But the significance of these was still to be discovered. The Power and Value of Mathematics

The Power and Value of Mathematics A man ahead of his time He spoke to his son of anticipated applications of Quaternions to Electricity, and to all questions in which the idea of Polarity is involved - applications which he never in his own lifetime expected to be able fully to develop. The Power and Value of Mathematics

The Power and Value of Mathematics Today Quaternions are often used in computer graphics to represent rotations and orientations of objects in three-dimensional space. Quaternions also see use in control theory, signal processing, attitude control, physics, bioinformatics, and orbital mechanics, mainly for representing rotations / orientations in three dimensions. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The Power and Value of Mathematics

The Power and Value of Mathematics Quotations "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Rowan Hamilton (Quoted in ' "Life of Sir William Rowan Hamilton" (3 vols., 1882, 1885, 1889)) The Power and Value of Mathematics

The Power and Value of Mathematics Broom Bridge Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including physicists Murray Gell-Mann in 2002 and Steven Weinberg in 2005 and mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains. The Power and Value of Mathematics

The Power and Value of Mathematics On 16th of October 1843, a bright October day, Sir William Rowan Hamilton discovered the numbers later called Quaternions. This incident might be as outstanding as Archimedes' discovery while bathing, or Poincare's while stepping onto a bus. The Royal Canal and the Bridge are still existing and tangible, this makes the scene so well imaginable, and let mathematicians of all times pilgrim to Broom Bridge. The Power and Value of Mathematics

The Power and Value of Mathematics Archimedes According to legend, when Archimedes got into his bath and saw it overflow, he suddenly realised he could use water displacement to work out the volume and density of the king's crown. Archimedes not only shouted "Eureka" - I have found it - he supposedly ran home naked in his excitement. The Power and Value of Mathematics

The Power and Value of Mathematics Henri Poincaré Jules Henri Poincare was born in 1854 in Nancy, into a prominent family -- his father was on the medical faculty at the University in Nancy and his cousin would become prime minister and President of the French Republic. He had been working on extending the result of his dissertation which dealt with differential equations, when he decided to go on a geological field trip for a diversion. The moment his foot hit the step to get on the bus, a thought shot through his mind, completely out of nowhere -- the equations he was playing with in order to solve a problem in algebra were actually identical in form to those which characterize the non-Euclidean geometry of Lobachevski. It was not a problem he had been working on. It was not a hunch that might be worth checking out. It was a sudden realization that arrived fully formed...and it was a biggie. The Power and Value of Mathematics

The Power and Value of Mathematics References: http://en.wikipedia.org/wiki/William_Rowan_Hamilton http://en.wikipedia.org/wiki/Quaternion http://philosophersplayground.blogspot.com/2006/08/smartest-people-youve-never-heard-of.html The Power and Value of Mathematics