Brougham Bridge Next: Conway and Smith's On Quaternions and Octonions Up: The Octonions Previous: Octonions Online

## Brougham Bridge

Most mathematicians have heard the story of how Hamilton invented the quaternions. In 1835, at the age of 30, he had discovered how to treat complex numbers as pairs of real numbers. Fascinated by the relation between complex numbers and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry. In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October 1843. He later wrote to his son:

Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: "Well, Papa, can you multiply triplets?" Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them".
The problem was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra.

Finally, on the 16th of October, 1843, while walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy in Dublin, he made his momentous discovery:

That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i,j,k; exactly such as I have used them ever since.
And in a famous act of mathematical vandalism, he carved these equations into the stone of the Brougham Bridge:

i2 = j2 = k2 = ijk = -1

He spent the rest of his life working on quaternions. He even wrote a poem about them which presaged the unification of space and time in 4-dimensional spacetime:

THE TETRACTYS

Or high Mathesis, with her charm severe,
Of line and number, was our theme; and we
Sought to behold her unborn progeny,
And thrones reserved in Truth's celestial sphere:
While views, before attained, became more clear;
And how the One of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,
Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again,
In later age, and occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.

Given such a romantic story, Tevian Dray and I could not resist trying to find Brougham Bridge when we were attending the 17 International Conference on General Relativity and Gravitation in Dublin - especially because we're both fans of the quaternions and their less-beloved kin, invented by Hamilton's friend John Graves: the octonions! So, on July 21st, 2004, we set out with a group of friends to find it.

Tevian had figured out that the bridge is on Broombridge Road. This is less mysterious if you know that "Brougham" is pronounced "broom", and was also spelled "Broome" at one point.

We took a bus from the city center to Broombridge Road. It was a bit difficult to find the right bus. We took bus number 20 from O'Connell Street just south of Parnell Square - an area packed with tourists and full of bus stops, but nobody who knew how to get to Broombridge Road. The 20 picked us up at one of the northernmost of the bus stops on the west side of this street. It then wound north on Parnell Square Street, west on Mountjoy Street, then Berkeley, then left on the North Circular Road, then took a right fork on Capra road, went right on Dowth Avenue, left on Fassaugh Road, right on Carnlough Road... and we got off at the Broombridge stop!

So, it was quite an adventure. But I hear it's gotten much simpler since 2004: now the 'Luas' green line tram runs through the city center directly to its terminus at Broombridge.

When we arrived, Tevian took some pictures. Here's the the first thing we saw at the bus stop:

From the bus stop we walked north a couple of blocks to the bridge itself. The bridge is not very impressive as you approach it this way:

You also don't see much of it as you cross it:

But, once you cross over to the north side, you can take a slanted walkway eastwards down to the canal.

From here the bridge looks a bit more impressive, though covered with graffiti:

You can then cross under to the west side of the bridge:

And here, on the bridge itself, is a plaque in honor of Hamilton!

Unfortunately it's covered with graffiti, perhaps a form of poetic justice.

The text says:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i2 = j2 = k2 = ijk = -1
& cut it on a stone of this bridge

Having come all this way, you can't help taking a closer look:

In fact you can't resist having your picture taken in front of it!

Here, from left to right, are John Baez, Jan Åman, Tevian Dray and Al Agnew:

Before leaving we took one last even closer look...

... but then, as the rest of us were walking off, Tevian couldn't resist adding the definition of his own favorite algebraic structure: the octonions!

Next: Conway and Smith's On Quaternions and Octonions Up: The Octonions Previous: Octonions Online