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Re: CHAT: I'm back!

From:Apollo Hogan <apollo@...>
Date:Tuesday, September 21, 2004, 22:37
On Tue, 21 Sep 2004, Philippe Caquant wrote:

> --- "H. S. Teoh" <hsteoh@...> 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. > > Of course, if your original 3d-language already has > specific words for "approaching from below while > rotating anti-clockwise", then it will be a little > harder in 4d. Good luck.
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. 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? Do the people live on the "surface" of a hypersphere? What do they look like? Sorry I can't say more, but my brain is already falling apart trying to visualize this stuff. --Apollo 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. 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...)


H. S. Teoh <hsteoh@...>