On 16 October 1843, walking the Royal Canal towpath to a Royal Irish Academy meeting with his wife, the Astronomer Royal of Ireland William Rowan Hamilton solved a problem that had stumped him for years: how to multiply in three dimensions. The answer needed a fourth term, and rather than lose it he cut the formula into the stone of the bridge with a penknife – i² = j² = k² = ijk = −1, the founding equation of the quaternions. It is, by common account, the most famous act of mathematical vandalism in history.
Be clear about what’s actually here. Broom Bridge is a plain two-arch limestone bridge carrying Broombridge Road over the Royal Canal in Cabra, in the Dublin suburbs. Hamilton’s original scratching wore away long ago. What you come to see is a plaque – set on the north-west corner, later moved up under the railing after vandals got at it – and the spot itself. If the maths means nothing to you, this is fifteen minutes and a photograph, not an afternoon. If it does, it’s a genuine pilgrimage, and people make it from all over the world.
The carving and the plaque
The quaternions Hamilton invented on this towpath turned out to be the clean way to describe rotation in three dimensions, which is why they sit, unseen, inside computer graphics, games engines, robotics and aircraft navigation today. The formula on the plaque is the same one. Taoiseach Éamon de Valera – a mathematician himself before politics – unveiled the commemorative stone on 13 November 1958, the year the bridge was officially renamed in Hamilton’s honour, though almost everyone still calls it Broom Bridge. The name itself is older and has nothing to do with the maths: it remembers William Broome, a director of the Royal Canal Company who lived nearby. The canal arch dates from around 1790, the railway arch from about 1845.
The Hamilton Walk
Every 16 October, mathematicians retrace Hamilton’s route from Dunsink Observatory, where he worked, to the bridge – an event now known as the Hamilton Walk, or Broomsday, in a nod to Joyce’s Bloomsday four months earlier. It draws serious names: past walkers include the Nobel physicists Murray Gell-Mann, Steven Weinberg and Frank Wilczek, and the mathematicians Andrew Wiles, Roger Penrose and Ingrid Daubechies. The walk is free and open to anyone; if you want to join, this is the one date to plan around.
Getting there and what’s nearby
The Luas Green Line ends at Broombridge, right beside the bridge, and the adjacent Broombridge railway station puts you on the commuter line into Connolly; a pedestrian bridge with lifts, finished in 2018, links the two and makes the whole interchange step-free. Dublin Bus route 120 (Parnell Square to Ashtown) stops at the Luas, roughly every half-hour at peak. Cyclists can ride in along the traffic-free Royal Canal Greenway, which runs past the bridge in both directions.
There are no toilets, café or visitor centre at the bridge – it’s a road crossing, nothing more. The real reason to linger in this corner of Dublin is next door: Glasnevin Cemetery and the National Botanic Gardens are about a kilometre and a half east and make the obvious pairing, turning a five-minute stop at the bridge into a proper half-day. Read the plaque, trace the formula once, and walk on to the gardens.