Robyn Arianrhod is a science writer and a mathematician affiliated with Monash University’s School of Mathematics, where she researches general relativity and history of science. She is the author of the critically acclaimed books 'Einstein’s Heroes', 'Seduced by Logic', and 'Thomas Harriot'.
On October 16 1843, the Irish mathematician William Rowan Hamilton had an epiphany during a walk alongside Dublin's Royal Canal. He was so excited he took out his penknife and carved his discovery right then and there on Broome Bridge.i ²=j ²=k ²=ijk=–1
He was inspired by a remarkable way of representing two-dimensional rotations. The trick was to use what are called"complex numbers", which have a"real" part and an"imaginary" part. The imaginary part is a multiple of the number i,"the square root of minus one", which is defined by the equation i ²=–1.
Hamilton was mightily impressed by this connection between complex numbers and geometry, and set about trying to do it in three dimensions. He imagined a 3D complex plane, with a second imaginary axis in the direction of a second imaginary number j, perpendicular to the other two axes. Quaternions and vectorsHamilton called his 4D numbers"quaternions", and he used them to calculate geometrical rotations in 3D space. This is the kind of rotation used today to move a robot, say, or orient a satellite.
Half a century later, the eccentric English telegrapher Oliver Heaviside helped inaugurate modern vector analysis by replacing Hamilton's imaginary framework i, j, k with real unit vectors, i, j, k. But either way, the vector's components stay the same — and therefore the arrow, and the basic rules for multiplying vectors, remain the same, too.
Hamilton also developed a compact notation to make his equations concise and elegant. He used a Greek letter to denote a quaternion or vector, but today, following Heaviside, it is common to use a boldface Latin letter.Take, for example, one of Maxwell's equations relating the electric and magnetic fields:With just a handful of symbols , this shows how an electric field vector spreads through space in response to changes in a magnetic field vector .
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