Saul Luizaga wrote:

> I don't know if some day would be possible but would be great to visit a 
> 4D world and meet 4D people :-D

I should point out that a rotation in 4D can leave your body in mirror 
image. If you go to 4D-land, turn around the "wrong" way, and then come 
back, your body will be inverted.

Why would you care? Well... certain biological molecules are chiral, 
so... good luck assimilating your food. :-P

(If 4D rotation seems weird, consider a 2D figure rotated through 3D 
space. For example, take the letter "d", draw it on a transparency, and 
then flip the transparency over. You now have the letter "b" - it's 
mirrored in 2D space, but in 3D space it's just a rotation!)

Let us not also forget that your body would have 0 thickness in 4D-land.

> 4D makes my head go thinking pretty bizarre stuff, don't you?

Perhaps.

The logic is fairly simple; the trouble starts when you try to 
"visualise" what it means.

For example, when you extrude an object, the following happens:

- Each point becomes two matching points.
- Each line becomes two matching lines.
- Each pair of matching points as a new line added between them.
- Each pair of matching lines has a new surface added between them.

Thus, clearly, when extruding an object into a new dimension:

- P = 2P
- L = 2L + P
- S = 2 + L

By that reconing, extruding a square (4 points, 4 lines) into a cube 
should yield

- 2*4 = 8 points
- 2*4 + 4 = 12 lines
- 2 + 4 = 6 surfaces

...which turns out to be correct.

Applying the exact same formulas, extruding a cube into a 4D hypercube 
should yield

- 8*2 = 16 points
- 2*12 + 8 = 32 lines
- 2 + 6 = 8 hypersurfaces
- I have no idea how many "normal" surfaces

For extra fun, look at a 2D projection of a rotating 3D cube. 
Unfortunately, this won't look like some 2D lines moving around, it will 
*look like* a true 3D cube - because our brains are designed this way. 
But now consider a 3D perspective projection of a rotating 4D 
hypercube... You can see exactly the same effects. It's just more 
mind-blowing. ;-)

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