On Tue, 21 Sep 2004, Philippe Caquant wrote:

> --- "H. S. Teoh" <[log in to unmask]> wrote:
> Hi.
> >
> > What think ye of this idea? What kind of terms might
> > a language spoken
> > in a 4D world have, which we do not have? At least
> > one curious mind
> > thirsts to know. :-)
> Why, all spatial terms, of course (prepositions,
> adverbs...) I guess one has to consider the relation
> between a cube and hypercube, for ex. A cube has, let
> us see, 6 faces, 8 vertices, 12 edges. A hypercube (or
> a supercube, as Gamow says) has 24 faces, 16 vertices
> and 32 edges (Gamow has drawn a nice 'supercube' on
> page 67 of "One, two, three... infinity", Dover Ed.).
> So the number of spatial words should be multiplied
> more or less in the same proportions. Instead of:
> before / behind / on the right side / on the left side
> / above / under, for ex, you should have 24 different
> words.
> Of course, if your original 3d-language already has
> specific words for "approaching from below while
> rotating anti-clockwise", then it will be a little
> harder in 4d. Good luck.

Don't forget also, that strange things can happen in the world of
4-dimensional geometry: there are no such things as knotted strings,
but you can tie spheres into knots... Klein bottles also live in this

Prepositions will be strange: in/out/etc. will have to have new meanings,
and new prepositions: if you tie a rope "around" something then do you
also tie a sphere "around" something?

Do the people live on the "surface" of a hypersphere?  What do they
look like?

Sorry I can't say more, but my brain is already falling apart trying to
visualize this stuff.


A useful trick that I use when I try to visualize 4-dimensional geometry
is to imagine objects in 3-space but with color (say running from red to
blue) where the color indicates location in 4-space.  This means that
if two things are colored differently, they don't actually intersect.
Thus it is clear that there are no knotted strings in 4-space, as you
can take a string (say red) and push a bit of it until it is blue and
then the blue part can cross any red parts, unknotting the string easily.
(Easier to see with a picture...)