This is a Sideband to the previous post, The 4th Dimension. It’s for those who want to know more about the rotation discussed in that post, specifically with regard to axes involved with rotation versus axes about which rotation occurs.
The latter, rotation about (or around) an axis, is what we usually mean when we refer to a rotation axis. A key characteristic of such an axis is that coordinate values on that axis don’t change during rotation. Rotating about (or on or around) the Y axis means that the Y coordinate values never change.
In contrast, an axis involved with rotation changes its associated coordinate values according to the angle of rotation. The difference is starkly apparent when we look at rotation matrices.
I was gonna give us all the day off today, honestly, I was! My Minnesota Twins start their second game in about an hour, and I really planned to just kick back, watch the game, have a couple of beers, and enjoy the day. And since tomorrow’s March wrap-up post is done and queued, more of the same tomorrow.
But this is too relevant to the posts just posted, and it’s about Special Relativity, which is a March thing to me (because Einstein), so it kinda has to go here. Now or never, so to speak. And it’ll be brief, I think. Just one more reason I’m so taken with matrix math recently; it’s providing all kinds of answers for me.
Last night I realized how to use matrix transforms on spacetime diagrams!
In the last installment I introduced the idea of a transformation matrix — a square matrix that we view as a set of (vertically written) vectors describing a new basis for a transformed space. Points in the original space have the same relationship to the original basis as points in the transformed space have to the transformed basis.
When we left off, I had just introduced the idea of a rotation matrix. Two immediate questions were: How do we create a rotation matrix, and how do we use it. (By extension, how do we create and use any matrix?)
This is where our story resumes…
For me, the star attraction of March Mathness is matrix rotation. It’s a new toy (um, tool) for me that’s exciting on two levels: Firstly, it answers key questions I’ve had about rotation, especially with regard to 4D (let alone 3D or easy peasy 2D). Secondly, I’ve never had a handle on matrix math, and thanks to an extraordinary YouTube channel, now I see it in a whole new light.
Literally (and I do mean “literally” literally), I will never look at a matrix the same way again. Knowing how to look at them changes everything. That they turned out to be exactly what I needed to understand rotation makes the whole thing kinda wondrous.
I’m going to try to provide an overview of what I learned and then point to a great set of YouTube videos if you want to learn, too. Continue reading