Re: 4d-world [Was: Re: I'm back!]

On Tue, Sep 21, 2004 at 08:46:41PM -0700, Apollo Hogan wrote:
> On Tue, 21 Sep 2004, H. S. Teoh 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.
Any suggestions as to how one might visualize such a thing? I've tried
all night and failed. :-P

> 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.)
Weird! I think I've a rough idea of how this might work now... but
still, it's pretty... *twisted*. :-P

[...]
> > 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 :-)
Whoa... I have trouble visualizing this one. :-P  I need to write a 4D
raytracer to render this thing...

[...]
> > 	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 :-)
Yeah it does. But I don't know how to illustrate some of the things
properly, 'cos it requires actual 3D, not just a 2D image of 3D. Such
as how a sphere is embedded between an inner cube and a frustum. Or
should I settle for wireframe diagrams?

[...]
> 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?
Perhaps I can just assume it's similar enough to not require obscene
amounts of exotic phonemes. :-) After all, I do want this language to
be comprehensible to us poor flat 3D beings. Plus, you do have a point
that it's probably not too far-fetched that they have essentially the
same hearing apparatus as we do, since theoretically our eardrum
should be able to resonate to a 2D wave, but we only hear everything
as combinations of 1D wave forms anyway.

Nevertheless, I think there should at least be 2 sets of laterals,
'cos there'd be two perpendicular planes where the tongue can touch
the sides of the mouth while still maintaining 1 free axis. But I
guess it's a question of whether this will make any difference in
sound.

[...]
> 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.
Hmm. I'll look it up and maybe purchase a copy. It's high time I
grounded my 4D intuitions on solid mathematical grounds anyway. :-)


T

--
Some days you win; most days you lose.