2 × 2 × 2 × 2, part 4

I don’t know about you, but for me the results of yesterday’s experiment in “seeing a cube like a Flatlander” were discouraging.

After all, you and I have something in common: We are very familiar with three dimensional objects like cubes. All of us can understand what is going on when we see a cube rotating, and some people can even mentally rotate a cube entirely in their mind, without needing to look at a cube.

Yet when I look at the representations of a rotated cube that I made in yesterday’s post, I experience a disconnect. Intellectually I know that the rotations are correct, because I can trace the path of each of the cube’s eight corners as it travels to its new position.

But looking at this visual representation does not give me the automatic intuition that I get from viewing the usual perspective view of a rotating cube. My spatial intuition does not quite survive the change in visual representation — even though the new representation is indeed spatial and geometric.

The conclusion I reach from this (since I am an optimist by nature) is the following: In order to use a technique like this to gain an intuition about four dimensions, we would first need to train ourselves to develop a spatial intuition for nested “outer/inner” as a stand-in for “front/back”.

One way to do this would be to develop a game that portrays a 3D cube using “outer/inner” to represent depth, and then challenge the player to solve a progression of puzzles that require rotating that cube. Eventually the player’s existing spatial intuition about cubes and 3D rotation would transfer over to this new representation.

And then we would be ready to try to tackle the 4D hypercube. To do this, we would start with a true 3D cube, and add “outer/inner” as a graphical stand-in for a fourth dimension.

I’m not saying this multi-stage plan would work. I’m just saying that it seems like a reasonable thing to try. I’d love to get the opinions of others on this.

3 thoughts on “2 × 2 × 2 × 2, part 4”

  1. So part 3 really made me think. What got me wondering was that there seemed to only be one intuitive axis for a flatlander to rotate a square around. Yet there are 3 intuitive axes for us solidlanders to rotate a cube around. It struck me as odd because most other dimensional properties follow a simple mathematical progression. An example would be 2^n – the number of vertices defining the unit area/volume/length/hypervolume for dimension n. But the 4th dimension only has 4 axes to rotate around. It looks like from the 2nd dimension onward, we can rotate in-universe around 1, 3, 4, 5, … axes. That didn’t seem right at all. To make matters worse, that one axis that flatlanders can rotate around is the one axis that doesn’t exist in their world. The one they’re shuffling these squares around to try and understand!

    So then I wondered: Well, what does intuitive rotation look like to a flatlander? They think they rotate about a single 0-dimensional point. The axis they’re “really” rotating around just looks like the point at an origin. So what does that say about our view of rotation? Do we “think” we’re rotating around an axis, when “really” we’re rotating around a plane? And that’s when I saw the solution to the 1 → 3 problem.

    Rotation is always viewed in-dimension as about an n-2 feature. In 2D, it’s about a 0D point. In 3D, it’s about a 1D line. In 4D, it must be about a 2D plane. The number of possible rotation types, then, is the number of orthogonal (n-2)D structures that can exist in nD space. So since there are only n orthogonal axes to work with, then there are a limited number of ways to use them to define a (n-2)D structure. There’s only one way to use 0 out of 2 axes to define a point of rotation in 2D. There’s only 3 ways to use 1 out of 3 axes to define a line of rotation in 3D. And there’s then only 6 ways to use 2 out of 4 axes to define a plane of rotation in 4D.

    What gets me about this understanding of rotation is that although rotation about a point is sufficient for flatlanders to understand 2D roation, they have to learn the concept of rotation around a line, redefining what even a “normal” rotation to them really means. And to bring that to us solidlanders, although rotation about a line is sufficient for 3D, we need to understand rotation about a plane, even that our familiar 3 rotations must also be rotations about planes that only intersect our universe (w=0) at one orthogonal line each.

    Sorry if I described all that with too much granularity and repetition. It’s likely you might already have understood all of this, but it does serve as a context for my thoughts on a 4D visualization/puzzle. What I think, which might sound daunting at first, is that creating a visualization/game that somehow demonstrates that rotation about an axis is the same as rotation about a plane in 4D might be a good step. I think even if I got adept at manipulating an “outer/inner” environment, my brain might create some sort of non-euclideon ruleset to describe how the environment behaves. Hard to say, really, whether that’d put me closer to or farther from being able to “get” 4D rotation. That said, it at least seems like a direction to move in. I’m finding myself in the shoes of our flatland friends, trying to understand how rotation about a point is really rotation about an axis that intersects their world orthogonally at only one point. I guess the key to understanding 4D rotation might be to be able to observe how a 3D rotation about an axis is really rotation about a plane that intersects 3-space orthogonally at one line. Then we can break free from boring axis-based rotation and rotate about planes.

    Meanwhile, those in the 5th dimension are trying to understand how 6-dimensional objects rotate about hyper-volumes. But that may be a story for another lifetime…

  2. You know, reading back over the previous parts, and then this part, with the background of what I had just discovered, I see you were already talking about the same thing I was. So, since we already understand rotation about an axis, then we can use a 2D inner/outer environment to show the relationship between the flatland 2D point rotation and already-understood 3D axis rotation that goes with it, and how that axis rotation relates to the two non-flatland rotations. Then we’d maybe get a feel for what kind of thinking it takes to see the relationship between the 3 3D axis rotations and 4D plane rotations that go with them, and how those plane rotations can relate to the 3 non-solidland rotations. So yeah, I think your plan sounds like a well-directed start. Let’s do it.

Leave a Reply

Your email address will not be published. Required fields are marked *