Quarternions aren't weird. With a 2D angle we use 2 numbers (sin and cos) to calculate vector rotations. It turns out that in 3D we use 4 numbers.
Geometric algebra explains that in a succinct way that also appeals to our intuition about geometry. Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra. This will have dimension 2^(N-1)... so 2 for 2D, 4 for 3D, and 8 for 4D.
This, to me, takes the mystery out of why quaternions can represent rotations, and it places quaternions in a coherent theory of geometry that works in any number of dimensions, not just 3D. Alternatively, we could accept that the math just happens to work out that way, or we could even show that quaternions are a double cover of SO(3), but all that does is analyze why something works, whereas the geometric algebra version is a bit less of a leap and builds quaternions from the ground up.
"Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra."
It reminds me of the running joke we had in graduate school. Any book whose title starts off with "An Elementary Introduction to..." was going to be very difficult.
"The area of the plane will depend on the length of the pencils". Surely the area of the plane is infinite? The area of the _parallelogram_ will depend on the length of the pencils.
And I can't see how "you can assign an orientation to the plane" other than by changing the directions of the pencils. Again this description sounds like it refers to the parallelogram, not the plane.
And I don't know what a rotor is.
But other than that, I'm doing great.
Think of a scalar. It has a magnitude but it doesn't have a direction. It's 0-dimensional.
Think of a vector. It has a magnitude (the size of the vector) and it also has a direction, which points in a straight line through the origin. It's 1-dimensional. For example, the vector (2,0,0) has magnitude 2 and points along the X-axis. You could write that as 2 * x, if x is the vector (1,0,0).
A bivector also has a magnitude, but instead of being 0-dimensional (like a scalar) or 1-dimensional (like a vector, it's 2-dimensional. So you could have a bivector that "points" along the entire XY-plane (remember: two-dimensional) and has some magnitude, say, 5. You could write that as 5 * x * y, if x is (1,0,0) and y is (0,1,0).
If you attach physical units to these things, then you might have units of meters for vectors, and square meters for bivectors.
Having an understanding of subspaces in linear algebra is helpful.
A vector is an oriented (+,-) magnitude(length) _in_ a line. A Bivector is an oriented (+,-) magnitude(area) _in_ a plane.
That area does not have any particular shape.
Imagine putting a 2d rectangle into 3d space with some orientation. Starting from some corner of the rectangle, you have two sides coming out from it. In 3d space, those sides make 3d vectors. You can generalize it to 3 vectors making a cube with magnitude (volume). And these don't have to be perfect rectangles and cubes, they can be parallelograms and parallelopipeds (3d parallelograms) and higher dimensional analogues.
Mathematics education is hard. In my experience, you start out with no understanding of a subject and can't understand it when people explain it to you, and at some point it clicks and you can't understand why it was ever difficult. I could be intentionally obtuse and, for example, describe a vector space as an "abelian group, field, and homomorphism from the field to group endomorphisms", but I feel that's the only people who would use that definition already have a good understanding of vector spaces.
The reason that I consider the non-GA approach to quaternions as rotations "hand wavy" is because it's not constructive, or perhaps just because I personally don't understand it. Using GA, I can construct a representation for rotations in any Euclidean space, not just 3D space, but 2D, 4D, 5D, whatever. However, without GA at my disposal, the fact that unit quaternions are a double cover for SO(3) seems like some kind of black magic that came from the void.
I have a few drafts of an introductory article I was writing on geometric algebra sitting on my hard drive, but I've never been able to get the article into a state I'd consider publishable. So instead, I'm trying to inject what I know into HN discussions.
This is analogous to the problem of finding a coordinate system for the globe: specifying latitude and longitude tells you were you are, but there's a degeneracy at the poles. And no possible coordinate system can solve this problem entirely. Contrast this to the situation of giving a coordinate system for the circle, which we do with it's angle. This isn't quite a coordinate system, due to the problem we already encountered that X and X + 2pi are the same, but that's OK because the these two points are separated from each other. On the sphere, the latitude/longitude pair (pi/2, x) gives the north pole for any value of x, even ones that are arbitrarily close together. That maps not even locally invertible!
You suggest we think of points on the circle as point in 2D space that happen to lie on the circle (i.e. cos and sin of the angle corresponding to that point). Analogously, we can think of points on the sphere as points in 3D space that happen to lie on the sphere (like some point (x,y,z) with x^2 + y^2 + z^2 = 1). And analogously, we can think of rotations of 3D space as a point in 4D space (that happens to satisfy some conditions), and the quaternions give that 4D point. This is fantastic and convenient in both 2D and 3D! But in 2D we didn't need to do this, but could if we wanted to. For 3D rotations, we do need to, or else we have this terrible degeneracy that never rears its head in 2D. In that sense, 2D and 3D are very different!
There are plenty, it's just that you can't have a 3-dimensional one without singularities.