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i² = j² = k² = ijk = −1

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Somerset Times Edition

Week 2, Term Four, 2017

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There have been a number of ‘Eureka’ moments throughout history including Archimedes falling into his bath, Newton watching an apple fall to the ground and Einstein’s thought experiment of a man free-falling. One lesser known flash of inspiration came to Irish Mathematician, William Rowan Hamilton, while he was out taking a trip with his wife along the Royal Canal in Dublin.

Hamilton’s quaternions describe spatial rotations and these days have application in computer graphics and altitude control systems on spacecraft. This all happened on October 16 1843, so next week will be the 174th anniversary of the revelation. Furthermore, his epiphany challenged what is known as multiplicative commutativity, where the order in multiplication does not matter, ie, 3 x 4 = 4 x 3.

2017 - T4 - W2 - maths

Complex numbers of the form a + bi (doubles) extend the number line to a number plane where a represents the real part of a number and bi the imaginary part, with i = √(-1). Hamilton was looking to find triples of the form, a + bi + cj to represent three dimensional space but this proved impossible. He had to use quadruples of the form a + bi + cj + dk and include a fourth dimension which he assumed was time. He also had to accept the non-commutativity of i, j and k and so using a penknife, he scratched the famous equation into Dublin’s Broom Bridge as follows: i² = j² = k² = ijk = -1. This also meant that ij = k, but ji = -k.

This 19th century mathematical graffiti has long since disappeared but there is now a commemorative plaque on the bridge over the canal to remember the moment.

Graphic

On 16 October, mathematicians from all over the world take part in the annual commemorative walk from Dunsink Observatory (where Hamilton was Director) to Broom Bridge. The date is sometimes referred to as Broomsday and matches Bloomsday on 16 June which celebrates the life of another famous Dubliner, novelist James Joyce.

While we are on a literary track, it should be noted that quaternions also unexpectedly feature in Alice in Wonderland’s Chapter Seven; A Mad Tea Party. Lewis Carroll (real name, Charles Dodgson) was a top mathematician and often incorporated mathematical ideas into his story telling. The tea party features the three characters (spatial dimensions) the Mad Hatter, the March Hare and the Dormouse. The fourth dimension, Time, has fallen out with the Hatter and won’t let the clocks go past six o’clock. The three characters therefore shuffle round the table looking for clean cups and saucers and are restricted to moving only in the plane. This relates to the fact that with just three spatial dimensions, Hamilton was unable to make progress, and the fourth dimension of Time was needed for three dimensional movement.

The Hatter’s nonsensical question “Why is a raven like a writing desk?” correlates with Hamilton’s idea that in pure time, cause and effect are no longer linked. When Alice attempts to solve the riddle, the Hatter tells her to “...say what you mean”, to which Alice responds “...at least I mean what I say – that’s the same thing”.

Not the same thing a bit!” replies the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!” This is a reference to the non-commutativity of quaternion algebra.

Later in 1843, inspired by Hamilton’s work, John Graves discovered octonions which have eight dimensions and have applications in the fields of string theory, special relativity, and quantum logic. Octonions are also non-commutative and also non-associative which means that i(jk) ≠ (ij)k.

Eureka moments are generally dramatic insights into a problem which then require further investigation and application. Archimedes was able to measure the volumes of irregular shaped objects (including the King’s crown) and design much larger ships using this Law of Buoyancy. Newton went on to formulate the principle of universal gravitation while Einstein unified gravity and his ideas of space and time to develop General Relativity. Hamilton’s quaternions helped enable space exploration, virtual reality and much, much more.

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