> I guess the question is, why does it then stop. Why not a 4D alternative.
It doesn't stop. That's what motivated geometric algebra, which works in any dimension. Quaternions are a sub-algebra of geometric algebra. They represent 3D rotations, which makes them interesting in their own right.
Asterisk: I believe there's a sign convention issue in mapping between quaternions and the even subalgebra of the 3D geometric algebra, so they aren't identical, just isomorphic.
> Also, is there as easy of a problem to understand introducing the 3D technique (be it quarternions or be it Gemoetric Algebra) that works as well as using sqrt(-1) for imaginary numbers?
That's an extraordinarily high bar. I don't believe anything reaches it. Part of the problem is that complex numbers are one of the most successful concepts in all of mathematics. The other part of the problem is that most of the useful facets of geometric algebra escaped the field of abstract mathematics under their own name before the unifying structure was discovered. The dot and cross product, quaternions, differential forms and the general Stokes' theorem are all examples. The remaining value proposition of geometric algebra lies mostly in getting rid of minor annoyances that come from this half-baked nature of traditional vector calculus tools:
* Cross products break in more than 3 dimensions and they break if you reflect them (see: pseudovectors). Bivectors have no such issues. They represent rotations in any dimension, reflected or not.
* Vector algebra with dot and cross products involves memorizing lots of new identities and applying creativity to work around the absence of division, while geometric algebra just has division and the same bunch of algebra tricks you already know. The geometric product isn't commutative, so it isn't perfect in this sense, but learning to deal with non-commutative algebra is a much more fundamentally useful thing than learning a bunch of 3D-specific identities.
* Dot and Cross with one argument fixed "destroy information" mapping from their input to their output. If you put them into an equation, the equation does not fully constrain the free vector, so you are often going to need more than one equation to represent any single geometric concept. Not so with geometric algebra. Many concepts map to a single equation. Including Maxwell's Equation (I use the singular intentionally)!
do you have a good reference for this? i've looked into GA bit but don't remember seeing anything like this. e.g. what would dividing a bivector by a vector mean?
It doesn't stop. That's what motivated geometric algebra, which works in any dimension. Quaternions are a sub-algebra of geometric algebra. They represent 3D rotations, which makes them interesting in their own right.
Asterisk: I believe there's a sign convention issue in mapping between quaternions and the even subalgebra of the 3D geometric algebra, so they aren't identical, just isomorphic.
> Also, is there as easy of a problem to understand introducing the 3D technique (be it quarternions or be it Gemoetric Algebra) that works as well as using sqrt(-1) for imaginary numbers?
That's an extraordinarily high bar. I don't believe anything reaches it. Part of the problem is that complex numbers are one of the most successful concepts in all of mathematics. The other part of the problem is that most of the useful facets of geometric algebra escaped the field of abstract mathematics under their own name before the unifying structure was discovered. The dot and cross product, quaternions, differential forms and the general Stokes' theorem are all examples. The remaining value proposition of geometric algebra lies mostly in getting rid of minor annoyances that come from this half-baked nature of traditional vector calculus tools:
* Cross products break in more than 3 dimensions and they break if you reflect them (see: pseudovectors). Bivectors have no such issues. They represent rotations in any dimension, reflected or not.
* Vector algebra with dot and cross products involves memorizing lots of new identities and applying creativity to work around the absence of division, while geometric algebra just has division and the same bunch of algebra tricks you already know. The geometric product isn't commutative, so it isn't perfect in this sense, but learning to deal with non-commutative algebra is a much more fundamentally useful thing than learning a bunch of 3D-specific identities.
* Dot and Cross with one argument fixed "destroy information" mapping from their input to their output. If you put them into an equation, the equation does not fully constrain the free vector, so you are often going to need more than one equation to represent any single geometric concept. Not so with geometric algebra. Many concepts map to a single equation. Including Maxwell's Equation (I use the singular intentionally)!