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Re: Re: need mathematica's help for exploring a certain type of mapping
*To*: mathgroup at smc.vnet.net
*Subject*: [mg68610] Re: [mg68578] Re: need mathematica's help for exploring a certain type of mapping
*From*: "Nabeel Butt" <nabeel.butt at gmail.com>
*Date*: Fri, 11 Aug 2006 04:41:04 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Dear Murray,
Suppose f:R^2->R^2 and f is unit distance preserving then it has
been proven that f is an isometry.
Infact, it has been proven for f:R^n->R^n.
Sorry i mistakenly type f:R^3->R^2.
I welcome any ideas for f:R^2->R^3.
regards,
Nabeel
On 8/10/06, Murray Eisenberg <murray at math.umass.edu> wrote:
>
> Unless I misunderstand what you say, it is not possible that a map R^3
> -> R^2 could be an isometry. After all, such a map would be a
> homeomorphism -- a topological embedding of R^3 into R^2. As such, the
> map would preserve topological dimension. But dim(R^3) = 3 whereas
> dim(S) <= 2 for every subspace of R^2.
>
> Nabeel Butt wrote:
> > Dear David,
> > Keep the metric Eucilidean(easy to visualise in
> mathematica).
> > Also,i want a map that preserves unit distances but not necessarily
> all
> > distances.
> > It may happen that a mapping from R^2->R^3 that preserves unit
> distances
> > preserves all distances and hence,is an isometry.
> > It is proven in mathematical literature that a unit preserving map
> from
> > R^3->R^2 is an isometry.
> > However, for f:R^2->R^3 this is still an open question.
>
> --
> Murray Eisenberg murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower phone 413 549-1020 (H)
> University of Massachusetts 413 545-2859 (W)
> 710 North Pleasant Street fax 413 545-1801
> Amherst, MA 01003-9305
>
--
Nabeel Butt
LUMS,Lahore
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