2 × 2 × 2 × 2, part 2

One of the problems in describing things in four dimensions is that they are hard to see. If I used the classic view, where Z and W are both in perspective, it would probably just be confusing.

So let’s ask ourselves: If we lived in a 2D Flatland, and we wanted to do puzzles on a 2 × 2 × 2 cube, how would we visualize the cube?

To be clear, we are talking about very smart Flatland people here, who know perfectly well what a cube is. The understand it has six faces, twelve edges and eight corners. They just have trouble visualizing it, since the world they grew up with — and therefore all their intuitions — is two dimensional.

If I were such a 2D individual, and I ran out of places to put that pesky third dimension — the one that doesn’t fit in my world — I might try nesting things one inside the other.

If you take that approach, then an intrepid Flatlander might visual a cube by doing something like this:

 
 
 
 

Our Flatland friend has found a way to position all eight subcubes of the 2 × 2 × 2 cube, by nesting one thing inside another.

In the above picture, a Back cube is represented by a pink inner square and a Front cube is represented by a blue outer square.

Maybe we three dimensional folks can use a similar trick to allow us to think about 2 × 2 × 2 × 2 hypercube puzzles. More tomorrow.

2 thoughts on “2 × 2 × 2 × 2, part 2”

  1. Whenever I think 4-dimensionally, I tend to replace that extra spatial dimension with something that I can understand. In your pink-nested cube example, I’d imagine you can be in pink mode or blue mode, and the rules of the world is, you can travel orthogonally to the rooms of your same color, or remain in a single room, but switch colors. I think this approach makes it possible to work with the 4D, but it also kind of robs 4D of its extra dimension. I find myself stuck either uncomprehending 4 spacial dimensions, or just strolling along familiar paths where really anything – color, pitch, transparency – can be a dimension, just not a spatial one. Maybe we’ll be able to find some bridges to help connect the entirely familiar to the instinctively incomprehensible.

  2. Good insight. As you say, the colors aren’t important — those were just labels. The important part was the position of each square. This will become more clear in the next post.

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