```On Tue, Sep 21, 2004 at 01:55:22PM -0700, Philippe Caquant 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.

Hmm. I've done a little bit of research myself, and I believe that
it's probably more accurate to consider the hypercube as being the 4D
volume (super-volume?) surrounded by 8 cubical cells, rather than
considering the 24 faces directly. Considering that a 4D being would
necessarily see in 3D (i.e., have a 3D retina), this seems to lead me
to just 4 pairs of directions: up/down, left/right, front/back,
"in"/"out" (or whatever you call it... this is different from being
actually "inside" or "outside" a 4D volume).

But of course, once you add behind/beside, above, below, etc., it does
get quite a bit hairier than that.

> specific words for "approaching from below while
> rotating anti-clockwise", then it will be a little
> harder in 4d. Good luck.
[...]

Heh... what I'm afraid of would be various necessarily distinct words
for 'twist', 'turn', etc., since there are 6 perpendicular planes of
rotation in 4D (as opposed to only 3 in 3D), and 4D does not have the
nice 3D property that the combination of two rotations equals a 3rd
rotation. I don't even know what two simultaneous rotations in 4D
looks like yet, let alone what they might correspond with.

Yikes, I think I need to understand 4D a lot more than I currently do,
before I can even start considering a conlang for it! :-(

But besides spatial words, I'm struggling with this concept of either
having no knots in my 4D con-world, or having knots that can only be
tied by twisting 2D sheets (because knotted 1D strings in 4D are
identical to the unknotted string, so you can't tie a knot with a 1D
string without it coming apart trivially in 4D). So the conlang would
either have NO words for tie, knot, etc., or it would need a new set
of words for whatever bizarre operations one would need to perform to
get 2D sheets to "knot" in 4D. (I'm not even sure this is possible to
begin with.)

T

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