## Unexpected geekery

15 July, 2009I used to be a Real Person with a Job and a Car and everything, and I worked with a bunch of geeks (software, hardware, electronics, physics, maths…). When I announced that I was leaving for Dublin, Rhys said “Aha! You should go to Broome Bridge!” This is where, walking and thinking in 1843, Hamilton scratched the formula for quaternions onto the stone so he wouldn’t forget it.

Well, I am always up for a bit of geekery. I found the bridge on the map. Hmm, wait, though, that’s in a random industrial estate on the Northside. Isn’t it dangerous up there? Maybe I won’t be making that particular pilgrimage.

Time passed, and I forgot all about it.

On Sunday, Charlie and I went for a leisurely bike ride along the Royal Canal. We stopped to laugh at the ducks walking on top of the copious amounts of algae and admire the bicycles that had fallen into the water.

And there, on the bridge:

Sadly, though I tried to understand, I am none the wiser as to what quaternions are for. Can anyone more mathematically minded explain it to this poor ex-code-monkey?

Apologies from a Physicist to the Mathematicians among you for my sloppy terminology but I find helpful a view of quaternions where they are generated by a complex rotation of a complex number, in the similar way that a complex number is generated by a real rotation of a real;

by White Pudding 16 July, 2009 at 7:45 amir=>i.

To explain:

Given the two complex numbers z=r+i and w=r+j (where r is the real direction, j is an imaginary direction orthogonal to the imaginary direction i)then

zw=(r+i)(r+j)= rr+ir+rj+ij= r+i+j+ij.

Define k=ij, ii=-1, jj=-1 (j acting like i) and

ij=-ji (reversing order of i,j rotations points in opposite direction) then the remaining quaternion identities follow:

kk= ijij= -jiij= -j(-1)j= jj= -1 and

ijk= (ij)k= kk= -1

To be pedantic, note associativity under multiplication is assumed, ie. ijk = (ij)k = i(jk)

Octonions are therefore a quaternion rotation of a quaternion…

Hope that helps Becky!

Thanks! But… what’s it for?

(Octonions? Aaargh…)

by Becky 16 July, 2009 at 10:52 am