In 1843, William Rowan Hamilton had a four-dimensional flash of insight that still shapes our three-dimensional world.
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.
It is the most famous graffiti in mathematical history, but it looks rather unassuming:
i ² = j ² = k ² = ijk = –1
Yet Hamilton’s revelation changed the way mathematicians represent information. And this, in turn, made myriad technical applications simpler — from calculating forces when designing a bridge, an MRI machine or a wind turbine, to programming search engines and orienting a rover on Mars. So, what does this famous graffiti mean?Rotating objects
The mathematical problem Hamilton was trying to solve was how to represent the relationship between different directions in three-dimensional space. Direction is important in describing forces and velocities, but Hamilton was also interested in 3D rotations.
Mathematicians already knew how to represent the position of an object with coordinates such as x, y and z, but figuring out what happened to these coordinates when you rotated the object required complicated spherical geometry. Hamilton wanted a simpler method.
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.
By the early 1800s several mathematicians, including Jean Argand and John Warren, had discovered that a complex number can be represented by a point on a plane. Warren had also shown it was mathematically quite simple to rotate a line through 90° in this new complex plane, like turning a clock hand back from 12.15pm to 12 noon. For this is what happens when you multiply a number by i.
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.
It took him many arduous months to realize that if he wanted to extend the 2D rotational wizardry of multiplication by i he needed four-dimensional complex numbers, with a third imaginary number, k.
