come back! i have more facts about hypercubes.
A 3-cube has 6 faces, 12 edges, and 8 vertices. A 3-cube has six square faces. Squares are also known as 2-cubes! a 3-cube's surface is made up of six 2-cubes.
A 4-cube has 8 cells, 24 faces, 32 edges, and 16 vertices. By cells, I mean 3-cubes. a 4-cube's surface is made up of eight 3-cubes. Look at this diagram of a 4-cube. Can you see the eight 3-cubes? You might only see four at first.
... On the 3-cube, each edge borders 2 faces. if you were a 2D creature, you could live on the surface of a 3-cube. when you got to the edge, you would have to turn a 90° corner, and then you could see more of your world.
This is nice and predictable.
However, on a 4-cube, each edge borders 3 faces. Scroll back up to the diagram of the 4-cube and convince yourself that this is true. If Mario lived on the 4-cube's faces, navigating edges would be extra confusing. Mario is 2-dimensional, so the 4-cube has two extra dimensions beyond the ones he lives in. When he reaches the edge, Mario can turn 90° in either of two directions, reaching two different destinations.
Are you ready? Now you are going to be Mario. You are a 3-dimensional being, so you could live on the 3-dimensional surface of a 4-cube. Here's a picture of you, Mario, standing inside one of the eight 3-cells that make up the surface of the 4-cube.
From your position you can travel in any of 6 directions: up, down, left, right, forward, or back. Whichever direction you choose, you will come to a square face. The hypercube's surface folds at these faces. You will have to turn your body 90° into the fourth dimension, then you can continue through the face and into a new 3-cell.
To summarize, the 3-cell you currently occupy has six neighbors. That makes seven 3-cells. Where is the eighth? if you move into any of your six neighbors, then continue straight on through, you will enter the eighth 3-cell. It is antipodal to your current position. If you go straight 4 times, you will end up back where you started.
Is that straightforward? good. Now things are going to get weird.
The 5-cube has 10 4-cells. These, taken together, are the 5-cube's surface. The 5-cube also has 40 3-cells, 80 faces (2-cells), 80 edges (1-cells), and 32 vertices.
The 40 3-cells are not the surface of the 5-cube. They are more analogous to the 1D wireframe edges of a 3-cube. Nevertheless, they form a contiguous 3-dimensional space, and as a 3-dimensional creature, you could live inside them.
Except. Remember 2D Mario's predicament on the 4-cube earlier? You are now in the same predicament. Your cell is 3-dimensional, but your universe is 5-dimensional. That means there are two extra dimensions beyond the three you occupy.
Whenever you reach the boundary of your cell, you meet a face. Passing through the face, you can turn your body 90° into the fourth dimension and enter a new cell. Or you can turn your body 90° into the fifth dimension and enter a different cell!
Since your cell has six faces, your cell has twelve neighbors! On the 5-cube, each face borders 3 cells. How can that be? I can't visualize it. I'll leave it to you to map out the 40-cell compound you are now trapped in. I'm sorry.
hint: each edge borders 4 faces and 6 cells. that's because any 2 faces define a cell, and 4 choose 2 is 6!