I think there's a lot of fuss here to come up with something which has the same structure (isomorphic) and same representation / encoding as quaternions, while trying to avoid calling them quaternions. Part of math pedagogy is getting comfortable with terminology and when to say that two things are "the same thing". You understand a structures (like quaternions, like rotors) by using both axiomatic approaches (what are the properties of these things?) and from constructive approaches (how do we build these things?)<p>Take the real numbers, for example. They are intuitively simple, but if you take an analysis class, you'll be asked to rigorously prove that your different approaches to understanding real numbers are equal to each other, and therefore interchangeable. You can take your axiomatic approach (real numbers are an ordered field, where each nonempty set bounded above has a least upper bound). You can construct real numbers as decimal numbers with an infinite number of digits past the decimal point. You can construct real numbers as limits of series of fractions. All of these approaches "are" the real numbers. No one version is privileged over the others. By understanding these different approaches, you get a better understanding of the real numbers.<p>Same is true of quaternions, but here, it seems that we're being sold an idea that one version is superior and the other versions are inferior. This is, I believe, a bad pedagogical approach. There are certain advantages to thinking of quaternions in terms of a scalar component and 3D imaginary component. For one thing, the imaginary component points in the direction of the axis of rotation! That's pretty handy, for visualizing quaternions as rotations. This article takes the viewpoint that there is no reason to think of quaternions as anything other than rotors and plane reflections--again, this is bad pedagogy, because people should be encouraged to think of mathematical objects in different ways, and use whichever way that they find convenient for the problem, or convenient for their own mental model. No one approach is privileged. As a mathematician, reading this article, all I see at the end is "Oh, you want to use quaternions, but you don't want to <i>call</i> them quaternions."<p>Perhaps we need a more in-depth article on the notion of rotations explained for non-mathematicians, something that, at the very least, incorporates the relationship between axis-angle representations and quaternions/rotors, because these two different representations are <i>not</i> isomorphic, and they are related to each other through calculus, the exponential map, and Lie algebra. It's also worthwhile to think what the tangent space is for unit quaternions. If you incorporate the different approaches to representing rotations, and don't try to sell one as being "the best" representation, I'd love to read <i>that</i> article.