William Rowan Hamilton carved quaternions into a Dublin bridge in 1843
Walking beside Dublin's Royal Canal on 16 October 1843, William Rowan Hamilton suddenly saw how to multiply points in space, and scratched the rule into Brougham Bridge with a pocket knife. The carving has faded, but mathematicians have walked to the bridge every year since 1989 to mark the moment.
Hamilton knew that complex numbers, with two parts, describe points on a flat plane, and he spent years trying to build a similar system for three dimensions. Adding and subtracting triples was easy; dividing them defeated him. His insight was that it could not be done with three components. He needed four: a real part plus three imaginary units, i, j and k, each of which squares to minus one, and whose product i times j times k is also minus one. He told a friend the next day that it felt as if an electric circuit had closed and a spark flashed out.
The price was that order matters: multiplying two quaternions one way round generally gives a different answer from the other way. Ferdinand Frobenius proved in 1877 that no three-dimensional version could ever have worked, and that the real numbers, the complex numbers and the quaternions are the only associative division algebras of their kind. Gauss had actually found quaternions in 1819, but his notes stayed unpublished until 1900.
Hamilton devoted the rest of his life to them, founding a school of quaternionists, and after his death Peter Tait carried the torch. For a time Dublin examined students on them compulsorily and physics was written in their language, including Maxwell's equations. From the mid-1880s, the simpler vector analysis of Gibbs, Heaviside and Helmholtz pushed them aside, leaving Hamilton's wordy books hard for later readers.
Their comeback came through rotation. Quaternions describe turning objects in three dimensions more compactly than matrices, compute faster, and avoid gimbal lock, the jam that can freeze angle-based systems. Computer graphics, robotics and spacecraft attitude control now routinely rely on them.
Source: Quaternion
