I find it interesting that the idea seems to involve making a certain sort of abstraction-move twice; first you consider formal products of vectors up to alternatingness (i.e. take the nth exterior power of a vector space) and then take a formal sum over all of those...
So in three-dimensional space I've got not just vectors, and not just scalars, vectors, bivectors and trivectors (i.e. the inhabitants of the 0th, 1th, 2nd, 3rd exterior products of 3 dimensional euclidean space) but objects that have four components, one of which is a vector, one of which is a bivector, one of which is a trivector.
And then you get all the crazy connections to complex numbers and quaternions and all four of Maxwell's equations are reduced to "nabla F = J". Crazy!
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
krja![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
(no subject)
Date: 2005-11-14 06:22 pm (UTC)(no subject)
Date: 2005-11-14 06:35 pm (UTC)http://www.mrao.cam.ac.uk/~clifford/publications/ps/imag_numbs.pdf
I find it interesting that the idea seems to involve making a certain sort of abstraction-move twice; first you consider formal products of vectors up to alternatingness (i.e. take the nth exterior power of a vector space) and then take a formal sum over all of those...
So in three-dimensional space I've got not just vectors, and not just scalars, vectors, bivectors and trivectors (i.e. the inhabitants of the 0th, 1th, 2nd, 3rd exterior products of 3 dimensional euclidean space) but objects that have four components, one of which is a vector, one of which is a bivector, one of which is a trivector.
And then you get all the crazy connections to complex numbers and quaternions and all four of Maxwell's equations are reduced to "nabla F = J". Crazy!