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 youThe problem was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra.multiplytriplets?" Whereto I was always obliged to reply, with a sad shake of the head: `No, I can onlyaddand subtract them".

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 thoughtAnd in a famous act of mathematical vandalism, he carved these equations into the stone of the Brougham Bridge:close; and the sparks which fell from it were thefundamental equations between i,j,k; exactly suchas I have used them ever since.

i^{2} = j^{2} = k^{2} = 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, so I'll tell you what we did. 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 knows 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!

It was a bit of an adventure. For one thing, nobody except the actual bus drivers knew we should take bus number 20. Even the Dublin Bus website claims - like the folks at the bus station, but contrary to what we discovered - that you should take the 120, not the 20, to Broombridge. To help guide yourself, print out a copy of this map, which shows Broombridge Road crossing the Royal Canal. But also do what we did, which is to ask the bus driver if their bus goes to Broombridge - and to tell you when you've arrived! The Irish tend to be very friendly, so before you know it, I bet lots of people will be helping you out.

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

i^{2} = j^{2} = k^{2} = ijk = -1

& cut it on a stone of this bridge

on the 16th of October 1843

Sir William Rowan Hamilton

in a flash of genius discovered

the fundamental formula for

quaternion multiplication

i

& 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!

© 2004 John Baez

Image copyrights reserved by Tevian Dray