What is a four dimensional cube?

2 three dimensional cubes, A and B, and each corner in A is connected by a new edge to it's equivalent corner in B. This is also called a tesseract.

What is a two dimensional cube?

A square.

In general, if you have an n-dimensional cube, you can get an n+1 dimensional cube by doubling it, and connecting each corner with it's equivalent corner.

A 0-dimensional cube is just a single dot.

A 1-dimensional cube is a two dots, connected by a line.

A 2-dimensional cube is 2 lines, connected. Also called a square.

A 3-dimensional cube is 2 squares, connected. Also called, well, a cube.

A 4-dimensional cube is 2 cubes, connected. Also called a tesseract.

A 5-dimensional cube is 2 tesseracts, connected.

In general, this is called the n-dimensional hypercube.

And continuing this, each edge on an n-dimensional cube will, together with its copy and the two edges connecting it to its copy, form another face on the n+1 dimensional cube. This new face will border exactly one face on the original cube, and one face on the copy. It's also bordered by two other connecting faces.

So, basically, if I start out on a face on the original cube, I can walk onto a connecting face, and then walk around all four connecting faces, and walk to the copy face. From there, I walk to another copy face, over the four connecting faces, and then back to the original cube. This way I should be able to continue going back and fourth between the cube and the copy, always walking across all the connecting faces.

I got a rubics cube here, with sides white, red, blue, orange, green, and yellow. For each of these faces, there exist 5 additional ones on the tesseract. For white: white, white_copy, and white_c1 to white_c4.

The solution is white, white_c1, white_c2, white_c3, white_c4, white_copy, red_copy, red_c1, red_c2, red_c3, red_c4, red, green, green_c1, green_c2, green_c3, green_c4, green_copy, orange_copy, orange_c1, orange_c2, orange_c3, orange_c4, orange, blue, blue_c1, blue_c2, blue_c3, blue_c4, blue_copy, yellow_copy, yellow_c1, yellow_c2, yellow_c3, yellow_c5, yellow.

I guess I'll never get that $14/hr job...

4th dimension is time?

5th dimension is your soul, and the 11th dimension, well we don't talk about the 11th dimension.

But can you prove that this solution wouldn't work on a 4 dimensions cube?

You can easily add additional dimensions to a three dimensional cube: color, sound, wall thickness, surface material etc.