>> 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
> 
> Hahaha, yeah I've been reading... :-) didn't know about the 
> bio-molecules tho.

Remember Thalidomide?

One Thelidomide molecule is actually "safe", but it's mirror image 
causes... well, go Google it. (Unfortunately, it turns out that normal 
human metabolism can convert one molecule to the other, so it's not so 
"safe" after all...)

But there are lots and lots of biological molecules who's mirror images 
are at best inactive and at worst toxic.

> Isn't this thing alone just cool? This reminds me the transformation 
> between 3D rectangular and spherical coordinates, a circumference in the 
> former is a line in the latter, fascinating...

Have you read about non-Euclid geometry? You may find it interesting...

>> Let us not also forget that your body would have 0 thickness in 4D-land.
> 
> Imagine that, like a 2D being in a 3D world, great for a movie plot :-D

Assuming our hypothetical 4D world has 4D atoms and 4D gravity, I 
imagine your 0-thickness body would slip between the microscopic gaps 
between atoms...

>> The logic is fairly simple;
> 
> agree
> 
>> the trouble starts when you try to "visualise" what it means.
> 
> exactly, like how does a 2D being thinks about a 3D object or being, how 
> do they "see" it, is just bizarre...

And that's just about the best way for us 3D simpletons to imagine the 
4D world; try to imagine how you'd describe a cube or a torus to a 2D 
simpleton.

>> 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.
> 
> yeah, I was thinking for about 3 hours how to get to a 4D hypercube 
> following the logic of building a 3D cube from a single line segment, 
> then a square and then a cube.

Start with 1 point.

Extrude. Now you have 2 points + 1 line.

Extrude. Now you have a square consisting of 4 points + 4 lines + 1 surface.

Extrude. Now you have a cube with 8 points + 12 lines + 6 surfaces + 1 
volume.

Extrude. Now you have a hypercube with... uh... 16 points, 32 lines, 24 
surfaces, 8 volumes and 1 hypervolume.

More fascinating, 4D space apparently has a regular hypergon with 400 
sides...

>> 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. ;-)
> 
> In deed :-)
> http://en.wikipedia.org/wiki/Fourth_dimension

Once you realise that a frustrum is a squished cube, and that you're 
looking *through* the 8th side, it's quite easy to find al 8 sides of a 
wireframe projection...

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