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Re: The Very Very First Sentence

From:Joe <joe@...>
Date:Saturday, February 14, 2004, 9:36
Philippe Caquant wrote:

>Thank you, it was quite interesting reading it. It >remembers me very much of George Gamow's book, 'One, >two, three... infinite', which I read when I was in my >teens and was stolen from me some day, or maybe >disappeared in 4th dimension. (But it looks like we >can make it appear again by the power of Amazon and a >few bucks). > >Gamow of course wrote no romance, but mathematical >vulgarization. He also uses very much Analogy to make >us understand for instance what could be an >'hypercube', which Abbott called an extra-cube I >think. > >At the moment, my concern is not so much about >hypothetical 2-dimensional beings (looks like the >Cards in Carroll's Alice), but about language >primitives we need, we being 'normal' 3-d people, to >describe a 2- (or less) -d world. So I'm not trying to >fancy about 2d living beings, because nobody ever saw >any one (let's except some actresses out of charity): >I think of a 2d-world as a purely geometrical one, and >that is quite well known to everybody who learned a >little of geometry at school. (NB: there is also >3d-geometry, of course). The question is: what >concepts, and what words, do we need to describe such >a word, or rather: which ones do we NOT need ? > >
Well, first, we need to ignore 'forwards' and 'backwards' We only have up, down, left(or west, perhaps), and right(or east). I really cant think of anything else. Incidentally, surely, just as we can project 3D things into 2D space(in drawings), we can project 4D things into 3D space - so, we can create a sculpture of a hypercube. If we were very clever, we could construct it so we can rotate it in 4D space(not through actual 4D rotation, of course, but through mechanical morphing). A hypercube, from whatever the 4-D equivalent of 'the top' is, looks a lot like a cube within another cube. Though very exactly proportioned. It ought also be possible to fold the net of a hypercube, in 4D space, again, using clever mechanics.