quaternion

If nothing else, I finally know what a quaternion really is after going through this book.

You need quaternions to avoid gimbal lock. In my small experience with 3D graphics, you would only deal with linear algebra on building cameras.

Before vector notation became popular in the last 1800s, people used quaternions. The three independent square roots of -1 in quaternion notation are i, j, and k.

One example that springs to mind is the use, in 3D graphics programs, of an archaic math construct called the 'quaternion'. The quaternion was the loser in a long-ago war with 'vectors', but this did not stop hackers from using the idea with the most value.

It actually seems a little better than it used to be, but for example good luck learning what the hell a quaternion is by reading the Wikipedia article, even if you have sufficient background to understand and use them in practice. The German article is much better.

It also allows us to define division in the complex, quaternion and octonion numbers. However, it is possible to multiply two non-zero sedenions together and get zero.

You probably can score ok in a math SAT. Can you visualize the mathematics of a quaternion, or formulate the equation for an n-sided betzier patch? The work a linguistic/novelist/journalist or a mathematican is not on the highscool level.

Has anyone here ever seen a good explanation of why quaternion multiplication maps to rotation concatenation?

By implementing things like model->world->camera space transforms, quaternion rotation, shadow mapping, lightmaps, deferred rendering, etc yourself, you will end up intuitively understanding them.

When it comes to translations and rotations, most of the algorithms out there use 4x4 matrices, dual quaternions, axis-angle rotations, or Euler angles. There may be some emerging work using geometric algebras. So when you say "extremely complex" I think quaternions, because i^2 = j^2 = k^2 = ijk = -1 . I once had the opportunity to refactor a lot of disparate implementations of 3D manipulations into a single math library, and migrate all the various algorithms to quaternion-based calculations.

Solving a problem using category theory that gives you a combinatorial answer, using quaternion algebra to compute a rotation, etc. The other thing is that in Math you are often dealing with the same question but with very different objects or variable types.