Hello:

This discussion has long since died, but raising the dead is sometimes fun, plus a few corrections may be in order.

A major disclaimer: I own the domain name quaternions.com, a research project where I try to get major results in physics expressed using only real quaternions. Why? The most useful tool in physics is the real numbers. The second most useful tool, in fact absolutely required to do quantum mechanics, are the complex numbers. What makes these things "numbers"? They can be added, subtracted, multiplied, divided, and have a norm (critical for defining topology stuff). Quaternions also have these properties. There are no other finite associative structures in math (up to an isomorphism to be technical, proof by Frobenius). So two types of numbers are ubiquitous, the third is obscure. I see no reason for this logically, so am trying to flush things out.

This is a project just for fun. I don''t seek converts, I''m here to play. One thing I generally avoid is the history of quaternions. It was not at all pretty. Gauss invented them, but his work in this area went unnoticed. Hamilton was trying to find triplets that could be divided in a sensible way for a decade before he thought to include another dimension. The words scalar, vector, dot product, cross product, divergence, gradient, curl were I believe all coined by Hamilton in his initial investigations. Rodrigues developed the rotation work independently. The pathfinder mission used quaternions in their software to do the rotations (I think this is standard practice these days).

As we know, Gibbs notation won the day. There are less than twenty five books in the Harvard library with the word quaternions in the title (half from last century), and over a thousand with the word vector, most of which do not mention quaternions. Why did it go this way? The clearest reason is this assertion.

> Simply this. You can draw a picture of a vector.

There are two types of vectors out there: polar and axial. I draw the same kind of stick arrow picture for both even though they transform differently. There should be something troubling about using precisely the same visual representation for two very different things. People are flexible, so only sensitive types get upset. The bigger problem is that quaternions don''t draw a picture.

I have to agree with this. Quaternions can be used however to draw pictures, meaning used directly to generate animations. Since most of my work is in physics, not gaming, I have always thought of a quaternions as representing an event in spacetime, a point of light at this time in this location. Put together a bunch of such events, and an animation is the result. I have wanted to see quaternion equations drive an animation for quite some time. Recently I have made progess in this area.

There are three animations to look at that illustrate Newton''s first law:
http://theworld.com/~sweetser/quaternions/qemation/inertia/inertia.html

[requires an SVG viewer, one is available at adobe.com]. Compared the the action that goes on in today''s games, this is lamo, twenty seconds of dots that move closer or farther away. The reason I think it may be important is that it starts a concrete discussion about what inertia means for an ordinary object, a photon, and quantum mechanics. My next step is to develop a generalization of the inertia law. Hopefully that will make more interesting animations.


doug
http://quaternions.com

Such an old topic, but still I found it and others may as well. Hopefully no one will take this article seriously. As can be seen from the discussion, it successfully confused most everyone. Apparently, John Blackburne is the only person who knows what a singularity is (p. 3.) Despite his state of correctness, no one seemed to pay his post any attention and continued to argue in the most hilarious and nonsensical manner. Then they proceed to make a mockery of mathematics, quoting names of mathematicians whose work is vastly above their comprehension. A 3-vector is not a 3 by 3 matrix. A quaternion is not a 3-vector. Do the words unavoidable singularity mean anything to anyone? renormalization group? augh good grief.