>>12307777I think you are thinking of it wrong. Dimension loosely only means the number of basis vectors. They don't necessarily need to be lines, and indeed in Cl(R^3), they are not. There are {1,i,j,k,ij,jk,ki,ijk}, and they all reside in R^3. This is why there should be no problem with the visualization. You are still just working in the 3 dimensional space, but you are now working with different objects to only vectors. The way the different objects interact is intuitive in the way that I've described above, and provides a more natural structure than, for example, saying that the product of two vectors gives you a new perpendicular vector. Indeed, in 3 dimensional space, we can imagine more than just lines, but we can imagine points, lines, planes and volumes, so why wouldn't it make more geometric sense to work in a system that utilizes all of these rather than just lines?
If this is different to what you are referring to, then sorry, I misinterpreted.
