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On Tue, 21 Sep 2004, H. S. Teoh wrote:

> On Tue, Sep 21, 2004 at 03:37:43PM -0700, Apollo Hogan wrote:
> [...]
>
> 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.

Well, I meant S^2, the _skin_ or surface of a ball, not the solid ball.
(I.e. the unique-up-to-homeomorphism compact two-manifold without boundary
and with genus -2, :-)  But I do think that you can tie it into a knot
in 4-space.  You can also link together spheres (S^2) inseparably, just as we
can link rings inseparably in 3-space.

One way (not the only way) to generate knotted spheres in 4-space is to
"suspend" ordinary knots in 3-space... The idea is something like the knot
becomes the "equator" of the sphere... just take two hemispheres, sew them
together along the knot (of course you can't do this in 3-space, because of
the twisting of the knot, but there's room in 4-space).  Voila, you've got
a knot.  (Warning: I've not thought very much about this, so I'm bound to
say something wrong, but the idea is something like this.)

> > 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 EbisÚdian-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.

Well, just pull out the geometry books and static/dynamic mechanics and
generalize everything to dimension N!  Then it'll be easy to plug in N=4
and you'll have the answers to all your questions :-)

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

See above for one method.  Another method is to "spin" a 3d knot around
a plane in 4d... I guess you want the center inside the knot.  Good luck
visualizing this :-)

> 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

Nice.  I haven't had time to read it all, but it looks very interesting.  Needs
more pictures, though :-)

> 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...)

Well, I wouldn't get too carried away, as we have one dimension of teeth,
but we don't have much variance with them.  (Basically non-dental vs dental
vs lateral but we don't have front-dental vs front-side-dental vs side-dental,
etc.)  And the vowel chart is really based on the fundamental formants of
the speech signal, so depending on the nature of 4-dimensional sound-waves...
and ears... maybe it's the same general shape?

> If I were to take all this into account, I could end up with a
> language easily more pathological than EbisÚdian, 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.)

Quite fun :-)

--Apollo

PS
Have you looked at the book "Knotted surfaces and their diagrams" (?).  It's
a hefty mathematical tome that I've only browsed in the bookstore, but it seems
to give ways to draw knotted surfaces in 4d and techniques for manipulating
them... It may be worth a gander just for some ideas.