[ogogmad]

> Geometric Algebra provides a compelling alternative to quaternions

No it doesn't. Please stop repeating this. Quaternions are literally a part of geometric algebra. His page uses the algebra Cl_{3,0}, whose even subalgebra is Cl_{0,2}, which is the quaternions.

Just to clarify what he's actually saying, because he said it poorly: He's not criticising the quaternions at all. He's criticising a philosophy of the quaternions that goes back to Hamilton: That they are formally scalars plus vectors. He's instead promoting a view that they're formally scalars plus bivectors. You can go from the author's POV to Hamilton's POV by replacing the word "bivector" with "vector" and "exterior product" with "cross product". Mathematically, this replacement constitutes an isomorphism, which shows that the author's algebra is isomorphic to - which for mathematicians means the same as - Hamilton's quaternions. The author's philosophy has the advantage that it shows that the quaternions are a subalgebra of Geometric Algebra. But this is only a different way of talking about the quaternions from Hamilton's. And Hamilton's view isn't even wrong - it leads to the octonions, which the GA approach doesn't - and it doesn't need you to know what a bivector is.

[hgomersall]

The GA view is IMO much easier to understand if you think visually. The fact you can subsume complex numbers and quaternions into a common framework is incredibly neat.

[jacobolus]

> Hamilton's view isn't even wrong

Hamilton’s view (and later the view of Gibbs, Heaviside, et al.) is wrong insofar as in models of the 3-dimensional physical world we have two kinds of "vectors" which behave fundamentally differently under reflection (because one type are actually bivectors called by the wrong name). Physics textbooks usually consider these both “vectors” but call one type “pseudovectors” (or call them “axial” and “polar” vectors), and (ideally) keep careful track of which vector is of each type. If you take the cross product of two regular vectors (yielding a pseudovector), it is fundamentally meaningless to add that to another ordinary vector, even though that seems like an entirely reasonable thing to do, algebraically.

Hamilton got there by trying to generalize “complex numbers” – mathematicians had been eliding the difference between points in the plane, vectors in the plane, and quotients of vectors in the plane (“complex numbers”) for a century or so by his time, and didn’t yet know the territory well enough to understand which points they were confused about.

Even in the plane, separating vectors from complex numbers is very powerful and useful. Nearly any time you see something like z̄w between complex numbers you can profitably consider z and w to be vectors u and v and use the geometric product instead (by comparison, when you see zw that implies a scalar+bivector was the kind of object you wanted already). Being able to use the dot and wedge products of planar vectors u·v and u∧v is clearer than the version mathematicians use, Re(z̄w) = ½(z̄w + zw̄) and Im(z̄w)i = ½(z̄w – zw̄), and makes the other applicable vector identities a lot easier to learn, recognize, and use. Bonus: they keep working in higher dimensional spaces or spaces of other signature.

The “cross product” is sort of interesting as a historical curio or abstract puzzle, but it is an abomination of a tool to teach to students, because it (1) creates widespread conceptual confusion, and leads people to make wrong inferences about the world, and (2) completely fails to generalize to the obvious analogous situations like the Euclidean plane or 4-dimensional spacetime (so people end up needing to learn a new notation for each new context they find). Replacing the cross product with the wedge product (as the anti-commuting part of a geometric product) clears up the confusion and builds up a whole shed full of effective general-purpose tools.