I am not yet in 10th dimension. I am in 4th. Still some way to go. Desperately working on finishing my paper on quantum fractals in n-dimensions, with illustrations mainly in 4d. Anyway, right now, this very minute, I am busy with one of the simplest 4d "cristals" - the
hypercube, or
tesseract". You can find many pictures on the net, but here comes one from my own "picture factory" (I am programming in Borland's Delphi):
It is not very difficult to understand the meaning of this picture. If you take two parallel flat squares, move them away in the direction perpendicular
to their planes, and connect their vertices - you get a cube. Each square has 4 vertices, so the cube has 4+4=8 vertices.
Now, repeat the same with cubes. Take two cubes, each with 8 vertices, move them apart in 4th dimension, and connect together the corresponding vertices (easier said than done :) ). You get
hypercube. It has 8+8=16 vertices. Now look at the picture above: find two cubes there, and see that their vertices are connected. Good exercise in 3d visualization! So, this is a 2d representation of the four dimensional "not-so-sacred" polyhedron - the hypercube.
In my program I can rotate it in 4d, and I rotated it so that the projection looks "nice'.
The hypercube is also called the 8-cell. When you count carefully - you will be able to find 8 cubes altogether, though they will be somewhat distorted by two projections, first from 4d to 3d, and then from 3d to to 2d of the computer screen!