Broom Bridge

Broom BridgeIn August I visited my son Alexey Radul, who currently works at the Hamilton Institute in Maynooth, Ireland. One of the greatest Irish attractions, Broom Bridge, is located there. It’s a bridge over the railroad that connects Maynooth and Dublin. One day in 1843, while walking over the bridge, Sir William Rowan Hamilton had a revelation. He understood how the formulae for quaternions should be written. He scratched them into a stone of the bridge. Now the bridge has a plaque commemorating this event. The plaque contains his formulae. I don’t remember ever seeing a plaque with math, so naturally I rushed off to make my pilgrimage to Broom Bridge.

Quaternions have very pronounced sentimental value for me, since my first research was related to them. Let’s consider a simple graph. We can construct an algebra associated with this graph in the following way. For each vertex we have a generator of the algebra. In addition we have some relations. Each generator squared is equal to −1. If two vertices are connected the corresponding generators anti-commute, and they commute otherwise. The simplest non-commutative algebra associated with a graph corresponds to a graph with two vertices and one edge. If we call the generators i and j, then the we get the relations: i2 = j2 = −1, and ij = −ji. I we denote ij as k, the algebra as a vector space has dimension 4 and a basis: 1, i, j, k. These are exactly the quaternions. In my undergraduate research I studied such algebras related to Dynkin diagrams. Thirty years later I came back to them in my paper Clifford Algebras and Graphs. But I digress.

I was walking on the bridge hoping that like Hamilton I would come up with a new formula. Instead, I was looking around wondering why the Broombridge Station didn’t have a ticket office. I already had my ticket, but I was curious how other people would get theirs. I asked a girl standing on the platform where to buy tickets. She said that there is no way to buy tickets there, so she sometimes rides without a ticket. The fine for not having tickets is very high in Ireland, so I expressed my surprised. She told me that she just says that she is from the town of Broombridge if she is asked to present her ticket.

Being a Russian I started scheming: obviously people can save money by buying tickets to Broombridge and continuing without a ticket wherever they need to go. If the tickets are checked, they can claim that they are traveling from Broombridge. Clearly Ireland hasn’t been blessed with very many Russians visitors.



  1. Oscar Cunningham:

    In England (and so I suspect in Ireland also) the ticket inspectors can sell tickets on the train. So whilst you can’t be fined if you claim you got on at a station with no ticket office or dispenser, you still have to pay the fare.

    My mum tells the story that she was once trying to ride without a ticket. Unfortunately there was a ticket inspector on the train. She had to claim that she got on at a station with no ticket office. But she was caught out! The station she claimed to have got on at had burned down the previous day.

  2. Leo:

    plaGue != plaQue

  3. Tanya Khovanova:


    Thanks, fixed.

Leave a comment