What Hamilton had intuited was the quotient property of his quaternions; i2 + j2 +k2 = ijk = -1, or in geometric terms the rotation of a body through the 3D plane. What was remarkable about his achievement was his realization that aspects of his invented algebra were “non-commutative” i.e. the product of two quaternions differed depending on the order; a*b ≠ b*a. To make his breakthrough, Hamilton had to imagine what a world would look like without the 2,000 year old accepted principles of maths. In the history of civilisation, it was the first ever invention of a non-commutative algebra. It could not have been done without a dream of what never was, followed by the question ‘why not?’ In a spectacular fashion, Hamilton’s creativity and mental gymnastics disproves Shaw’s argument that the scientific method is uncreative (only the ‘why?’) and that the Irish cannot bring their talents to this sphere.
In a further delightful symmetry, Hamilton was also a poet himself. And better yet he further undermined Shaw’s opinion by being generally considered the worst poet Ireland ever produced. In fact, he was friends with William Wordsworth who, having read a volume of his poetry, charitably advised him not to write any more poems.
The maths of quaternions was briefly fashionable, before falling out of favour due to its clunky notation. It was eventually incorporated into linear analysis as it was extended into a model to incorporate any numbered dimensions. However, since the latter half of the 20th century, there has been a renaissance of interest in Hamilton’s work. While linear analysis has simpler notation and is more flexible in the abstract world, it was realized that quaternions were much more efficient at describing rotations in the 3D plane. This is the dimension, as it happens, of the human physical world, and hence a key component of multiple everyday processes.
Next Tuesday, it is the 169th anniversary of Hamilton’s epiphany under Broom Bridge. We should celebrate Hamilton, our finest scientist, and also Ireland’s rich legacy of scientific endeavour.
Vincent Kelly is a regular contributor to Science Calling. He works as an actuary in Dublin, and is interested in science and mathematics, particularly applied mathematics. He has a BSc in Financial and Actuarial Mathematics from DCU.
Maths Week Ireland runs from today until 21st October. One of the many events taking place is the annual Hamilton Walk. This starts next Tuedsay at the Dunsink Observatory and proceeds to Broom Bridge. For more informtion see about this event and others around the country see mathsweek.ie.