Not that anyone should care, but it’s Friday the 13th today! More to the point, it’s Friday, and I’ve been remiss about Friday Notes this year. The one last January is the only one I’ve posted so far. Big part of that is having, at long last, reduced my note pile enough that I don’t feel so (word) pressed.
Some of it is the ongoing problem of ennui. The eleven-year blog anniversary is approaching, and that tends to ignite old questions about why I bother to do this. By now I’ve left some sort of small scrawl on the internet wall and explored many of the topics that drove me to blogging.
On the other hand, the pile isn’t by any means gone, so let’s get to it…
Four years ago I started pondering the tesseract and four-dimensional space. I first learned about them back in grade school in a science fiction short story I’d read. (A large fraction of my very early science education came from SF books.)
Greg Egan touched on tesseracts in his novel Diaspora, which got me thinking about them and inspired the post Hunting Tesseracti. That led to a general exploration of multi-dimensional spaces and rotation within those spaces, but I continued to focus on trying to truly understand the tesseract.
Today we’re going to visit the 4D space inside a tesseract.
Here on the 4th day of the 4th month, I feel I really should be writing about the 4th dimension. I did say that I would during March Mathness, and I tried to set the math foundation here and here.
But two problems: Firstly, I’m kinda burned out. Those three posts were a bit of work, diagrams & models & math (oh, my!), and then trying to explain them clearly. Secondly, obviously no one finds this interesting except me, so not much motivation for the effort involved. Which was expected (kinda the story of my life). I also said these posts were as much recording my notes as attempts to share.
But it is 4/4 (and no Twins game today), so I thought I’d try winging it anyway.
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.
Time for math!
I have a special fondness for the month of March. For one thing, it contains the Vernal Equinox — one of my favorite days, because it heralds six months of light. (As a Minnesotan, Spring has much more impact than it did when I lived in Los Angeles.)
March is when the weather elves begin preparing for the April Showers that create May Flowers. It’s when baseball Spring Training is in full swing with the regular season looming (lately, even at the end of the month; this year on the 28th).
It also contains some important birthdays: Albert Einstein (3/14) and Emmy Noether (3/23), to name two, and in their honor I have myriad math posts planned!
If you’re anything like me, you’ve probably spent a fair amount of time wondering what is the deal with tesseracts? Just exactly what the heck is a “four-dimension cube” anyway? No doubt you’ve stared curiously at one of those 2D images (like the one here) that fakes a 3D image of an attempt to render a 4D tesseract.
Recently I spent a bunch of wetware CPU cycles, and made lots of diagrams, trying to wrap my mind around the idea of a tesseract. I think I made some progress. It was an interesting diversion, and at least I think I understand that image now!
FWIW, here’s a post about what I came up with…