Invisible wrote:
> 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

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

> 
> (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!)

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...

> 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

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

> 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, 

dimension adds a new(n+1) dimension, the irony is you already must know 
what direction to rotate to in order to build the n+1-dimension object 
in that dimension, I was also noticing the increasing number of P,L and 
S, great brain exercise, also it reminded me of a possible explanation 
for the Bermuda Triangle as a "door" to a Wormhole into hyperspace, man 
this is the most pleasing way of meditation I have ever found :-); yoga? 
way stay meditating in a 3D world when you can imagine the awesomeness 
of a 4D world? :-D

> 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. ;-)

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

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