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.
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
I’ve been hinting all month about rotation, and the time has finally come to dig into the topic. As mentioned, my interest began with wanting to understand what it means to rotate a tesseract — particularly what’s really going on in a common animation that I’ve seen. What’s the math there?
This interest in rotation is part of a larger interest: trying to wrap my head around the idea of a fourth physical dimension. (Time is sometimes called the fourth dimension, but not here.) To make it as easy as possible, for now I’m focusing only on tesseractae, because “squares” are an easy shape.
After chewing at this for a while (the tesseract post was late 2016), just recently new doors opened up, and I think this journey is almost over!
Last week we celebrated Albert Einstein’s birthday (he turned 140). Now we need another cake so we can celebrate the other March major mathematician’s birthday — Emmy Noether turns 137 today.
To my regret, despite that I frequently invoke her name (she co-starred with Albert in the Special Relativity series), her work in mathematics is pretty far above my head, and I’m simply not qualified to write about it. I can say that her work connects mathematical symmetry with physical conservation laws. She also made significant contributions to abstract algebra.
Just recently, I’ve begun to nibble at the edges of the latter in the form of group theory as a part of studying rotation.