Re: [math-fun] Moebius madness on Youtube
----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, January 11, 2008 12:54:07 PM Subject: Re: [math-fun] Moebius madness on Youtube Nicely done. And it grieves me to have to admit that, although I've been concerned with these groups for a good few years, I had no idea that the isomorphism of the Moebius group in n-space with the Euclidean group in (n+1)-space had such a straightforward demonstration! WFL On 1/11/08, Henry Baker <hbaker1@pipeline.com> wrote:
Very cool animation of Moebius transformations:
Those two groups are not isomorphic. The Euclidean group has a normal subgroup (the translations), while the Mobius group is semisimple. Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ
On 1/12/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:
----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, January 11, 2008 12:54:07 PM Subject: Re: [math-fun] Moebius madness on Youtube
Nicely done. And it grieves me to have to admit that, although I've been concerned with these groups for a good few years, I had no idea that the isomorphism of the Moebius group in n-space with the Euclidean group in (n+1)-space had such a straightforward demonstration! WFL
On 1/11/08, Henry Baker <hbaker1@pipeline.com> wrote:
Very cool animation of Moebius transformations:
Those two groups are not isomorphic. The Euclidean group has a normal subgroup (the translations), while the Mobius group is semisimple.
Gene
About ten years ago, I read somewhere (Coolidge?) that they were isomorphic (qua Lie groups, presumably); I remember being very surprised at the time, but never got around to investigating the matter until now. If this claim turns out to be wrong, that extricates me very conveniently from my elephant trap. I'd also have the consolation of company: Coolidge relates with sadistic relish how an unfortunate Ph. D. student by the name of Lohrl expended considerable time and effort in a futile attempt to prove that Moebius n-space and Lie-sphere (n-1)-space groups were isomorphic. [In that case at least, it seems perfectly obvious that they cannot possibly be; since only the latter group has a 1-parameter normal subgroup, to wit the offset transformations.] In the present case --- Moebius n-space and Euclidean (n+1)-space, continuous components say --- we have a transformation (stereographic projection) inducing a smooth map from isometries in (n+1)-space acting on the projecting sphere (though its north pole remains northwards), to conformalities in n-space. This map is a homomorphism, composition mapping to composition (I _think_ this is obvious!). It must surely be invertible, since its codomain contains generators for the whole Moebius group: translations, rotations, (proper) inversions. Therefore the groups are isomorphic ... no? Unless of course, there's a mistake somewhere in the reasoning above. And if there isn't, I'm right back there, still sitting in deep do-dos ... Fred Lunnon
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Eugene Salamin -
Fred lunnon