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Re: Re: need mathematica's help for exploring a certain type of mapping
*To*: mathgroup at smc.vnet.net
*Subject*: [mg68593] Re: [mg68578] Re: need mathematica's help for exploring a certain type of mapping
*From*: Murray Eisenberg <murray at math.umass.edu>
*Date*: Fri, 11 Aug 2006 04:40:00 -0400 (EDT)
*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst
*References*: <200608090820.EAA21373@smc.vnet.net> <NDBBJGNHKLMPLILOIPPOMELCFBAA.djmp@earthlink.net> <200608100357.XAA21852@smc.vnet.net>
*Reply-to*: murray at math.umass.edu
*Sender*: owner-wri-mathgroup at wolfram.com
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
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