To describe how space could be flat, finite, and yet unbounded, science writers sometimes use an analogy involving the surface of a torus (the mathematical abstraction of the doughnut shape). Such a surface has no boundary — no edge. And despite being embedded in three-dimensional space, the torus surface, if seen in terms of compensating surface metric, is indeed flat.
Yet a natural issue people have is that the three-dimensional embedding is clearly curved, not flat. It’s easy to see how wrapping a flat 2D sheet into a cylinder doesn’t distort it, but hard to see why wrapping a cylinder around a torus doesn’t stretch the outside and compress the inside.
In fact it does, but there are ways to eat our cake (doughnut).
I’ll end these posts about the configuration space metaphor where I began: in a big cube. I started the series in the Neapolitan room, a three-dimensional space where we could indicate our feelings about vanilla, chocolate, and strawberry ice cream with a single marker. From there we visited the Baskin-Robbins 31 Flavors space (which is tasty but beyond our ability to visualize).
Then I focused on spaces with only two-dimensions (which are easy to visualize). These are probably the best use of the metaphor; they turn a tug-of-war into a sensible place to stand. They also strongly differentiate “don’t care” from “care about both.”
Now let’s see what we can do with three dimensions…
Last time we considered a cube-shaped room where we could indicate our opinion about Neapolitan ice cream with a single marker. That worked well because we were dealing with three flavors and the room has three dimensions: east-west, north-south, up-down.
Later I’ll explore other examples of a 3D “room” but while we’re talking ice cream, I want to give you an idea where this goes, I want to jump ahead for a moment and consider good old Baskin-Robbins, who famously featured “31 flavors!”
So now the question is, can we set a marker for all 31 flavors?
Have you ever had (or at least seen) Neapolitan ice cream? It’s the kind with chocolate, vanilla, and strawberry, usually as separate layers in one package. As a kid, I didn’t care for the strawberry. I loved the chocolate, and was fine with the vanilla (wouldn’t usually choose it, but don’t disdain it).
That’s just my take on it: one flavor liked, one not liked, and one that’s just okay. Someone else might have the same pattern with different flavors. Or love them all equally, or want just the strawberry. Some might not like ice cream at all — any combination is possible.
What if we wanted to describe our feeling about Neapolitan as a whole?
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!
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…