## Quaternions

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In honor of St. Patrick’s Day, it seems fitting to commemorate Sir William Rowan Hamilton and his discovery/invention of the quaternions, an extension of the complex numbers. This happened while he was walking with his wife near Dublin, his native city. He described the experience in an 1865 letter to his son, the Reverand Archibald H. Hamilton:

But on the 16th day of the same month – which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse – unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely,

$i^2 = j^2 = k^2 = ijk = -1$

which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.

While the inscription is gone (presumably he didn’t have the opportunity to cut very deeply into the stone — wouldn’t that ruin a knife anyway?), there is a plaque by the bridge (now called Broome Bridge) memorializing the event:

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

(For more pictures, see this site on the National Curve Bank.)

Gwen Fisher has made a wonderful quilt showing the Cayley table of the quaternions, which you can see here. If you want to design your own quilt based on the quaternions or other groups, visit the Cayley quilt maker: you select the size and properties, choose the View Tiles and “paint” the squares, then choose the View Table or Quilt to see what you’ve created!