Re: CHAT: I'm back!

On Tue, Sep 21, 2004 at 03:37:43PM -0700, Apollo Hogan wrote:
[...]
> 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
> world.
Ah yes. I've been waiting to train my mind enough that I can "see" a
Klein bottle. I think they're the coolest thing ever! :-)


> 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?
Interesting, can you actually tie a sphere into a knot in 4D? I
thought you could only do it to a 2D surface in 4D. But maybe I'm
wrong.


> Do the people live on the "surface" of a hypersphere?  What do they
> look like?
I'd like to think so, yes. Or at least, on some kind of "flat" 3D
volume. Maybe they walk around on a flat 3D hyperplane. I think having
a 4th spatial dimension is enough to make the language incredibly
weird, so I'd like to keep it as analogous to Earth as possible. No
need for Ebisédian-style weird physics here to add to the confusion.
:-P

As for what they look like... I'm hoping that I can get away with
humanoid creatures... but I have my doubts whether having 2 feet is
enough for one to stand on steadily in 4D. If 2 eyes were enough for
4D stereovision I'd go for it, but I'm not sure if it captures enough
parallax to be useful.

> Sorry I can't say more, but my brain is already falling apart trying to
> visualize this stuff.
LOL, I know what you mean... although I scare myself by having
developed an ability to rotate 4D objects in my mind.

(Well, kindof...  I definitely haven't gotten to the point I can
rotate arbitrary 4D objects or rotate things in arbitrary planes yet.
But I can definitely visualize rotating spherical hypercones, cubical
cylinders, cubical pyramids, cylindrical prisms, etc. I can see how
the tip of a hypercone "sticks out" at me from the center of its
spherical projection, how it moves outside the sphere when the
hypercone turns, and how it disappears "behind" the sphere when it
turns on the other side to point away from me. I haven't dared try
rotating the 600-cell in my head, though.  *That* one still makes
smoke come out my ears. :-P)


> PS
> 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.
Yep.


> 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...)
I know precisely what you mean. The interesting thing is that even if
the string has 4D width (i.e., it's not just a 3D rope, which would be
trivial to un-knot in 4D), it *still* can be unknotted in the same
manner.

What I want to visualize, though, is how exactly one knots a sphere...
what does it look like???

But speaking of 4D visualization, I've actually written up a webpage
describing the process by which I do this. It's actually not *that*
hard, believe it or not; the key to the whole thing being that a 4D
being would have a 3D retina and would see 3D images in its eye, and
that although we only see in 2D, our mind pretty much has a very good
grasp of 3D. So it's just a matter of imagining a 3D image in our
mind, and then inferring 4D depth from it. Well, I'm starting to
repeat what I wrote on the webpage here, so let me just post the URL
instead:

	http://eusebeia.dyndns.org/~hsteoh/4d/vis.html

But back to conlanging, one problem that suddenly occurred to me is
that a 4D being's mouth has a 4D cavity, which means that the tongue
has 4 degrees of freedom... and even if you assume it normally only
moves up/down in speech, that's still 3 degrees of freedom. Does that
mean the vowel chart is now 3D ????? (My aural perception explodes at
this proposition...) Also, unless I seriously deprive my 4D people of
teeth, they should probably have 2D areas of teeth. Does that mean
there are N^2 possible dental sounds now?? And worse, 4D lips would be
able to make 3D shapes... suddenly I have an incredible amount of new
labial sounds as well! And can you imagine having 2 degrees of freedom
with which to produce laterals? You'd have two distinct sets of
uni-laterals, and a set of bi-laterals. (Owie, my head hurts...)

If I were to take all this into account, I could end up with a
language easily more pathological than Ebisédian, without even trying!
(And I'm not even going to try thinking about the writing system,
which would necessarily involve 3D characters... this is worse than it
seems at first glance---just imagine the difference between a letter
represented by a (hollow) sphere and a letter represented by a ball.
They are completely different things to 4D eyes, but when viewed on
the outside by us, they look identical.)


T

--
If you think you are too small to make a difference, try sleeping in a closed
room with a mosquito. -- Jan van Steenbergen

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