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

From:H. S. Teoh <hsteoh@...>
Date:Wednesday, September 22, 2004, 17:48
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... [...]
> > > > 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.