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MathGroup Archive 2006

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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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