>Is there an easy explanation of what problems quaternions solve? Sure. Unit quaternions form a double-cover of SO(3). In other words, you can encode a rotati

>Is there an easy explanation of what problems quaternions solve? Sure. Unit quaternions form a double-cover of SO(3). In other words, you can encode a rotation of a 3-dimensional object with a single unit quaternion. But wait, there's more! You could do the same with a matrix, or a triple of angles. Why not do that? Answer: interpolation. The "natural" way you want to go from one rotation to another corresponds to exponentiation of quaternions. If you linearly interpolate matrices, the intermediate steps will do something nasty: they won't even be rotations! The natural way to implement the Arcball interface for rotations is using quaternions. Here[1], I have implemented it in ProcessingJS and wrote up the math behind it. Quaternions (like complex numbers) can do other things too, but this alone is a good start. Also gives you intuition why they aren't commutative: because rotations in 3-space aren't. TL;DR: Unit complex numbers = rotations of plane. Unit quaternions = rotations of 3-space. PS: you shouldn't think of complex numbers as the solution to the problem of "taking the square root of -1". Think of them as "how can I multiply/divide a 2D vector by another 2D vector?" - there's only one way to do it sanely (multiply/divide lengths, add/subtract angles). This is what the complex numbers are. Hamilton was trying to solve the same problem in 3D, and couldn't (turns out, it is not possible[2]), but solved it in 4 dimensions, and later found many applications for them. [1] http://www.math.tamu.edu/~romwell/arcball_js/index.html http://www.math.tamu.edu/~romwell/arcball_js/index.html [2] https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras) https://en.wikipedia.org/wiki/Frobenius_theorem_(real_divisi...