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MathGroup Archive 2006

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Re: need mathematica's help for exploring a certain type of mapping

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68582] Re: need mathematica's help for exploring a certain type of mapping
  • From: "Nabeel Butt" <nabeel.butt at gmail.com>
  • Date: Wed, 9 Aug 2006 23:57:56 -0400 (EDT)
  • References: <200608090820.EAA21373@smc.vnet.net> <NDBBJGNHKLMPLILOIPPOMELCFBAA.djmp@earthlink.net>
  • Sender: owner-wri-mathgroup at wolfram.com

David,
        Thanks for the reply.You seem to have very good ideas.
      The mapping does not neccessarily map xy-plane to  a 3D-surface in
R^3.It is also not neccessarily continous.*ALL *what is specified of it is
that given two points which are at a unit distance in R^2 they are mapped on
to points in R^3 that are also at a unit distance.What we need to show is
that given there is a mapping that preserves unit distance does it preserve
all distances?.If we are able to demonstrate a map that preserves unit
distances but not all distances we have actually disproven a very important
hypothesis in mathematics.And your ability to create graphics will be
useful.
   Also there may be many possible unit distance preserving maps.
     Look forward for your views.
        regards,
              Nabeel



On 8/9/06, David Park <djmp at earthlink.net> wrote:
>
> Nabeel,
>
> I thought that the definition of an isometry was that it preserved
> distances.
>
> A mapping that did a 3D rotation of the xy-plane, plus translations in 3D
> space, plus reflections in a plane would cover the isometries if we use
> the
> Euclidean metric in both 2D and 3D.
>
> But suppose you wanted to map onto a specific curved surface. Then are you
> going to allow a different distance measuring function on the surface, a
> different metric? Are we allowed to design a metric for any given mapping?
> Is it possible to have an isometry then? I don't know.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
> From: Nabeel Butt [mailto:nabeel.butt at gmail.com]
To: mathgroup at smc.vnet.net
>
>
> Dear Users,
>               I need to use mathematica's graphics to explore a certain
> kind of problem.The following theorem is not yet proven nor disproven and
> mathematica might proof  useful in disproving it though.
> Hypothesis:If a mapping from R^2->R^3 is unit distance preserving then it
> must be an isometry.
>     The real issue at hand is for mathematica to generate a mapping that
> preserves unit distance but is not an isometry so in the process
> disproving
> the theorem.
>     The real problem is that R^2 consists of infinite points and it might
> not be possible to check all of them.What i suggest is that you apply the
> unit preserving maps to special type of figures in R^2 like the
> circumfrence
> of the circle,square,isoceles triangle etc.
>     Any ideas are welcome.Thanks in advance.
>         regards,
>           Nabeel
>
> --
> Nabeel Butt
> LUMS,Lahore
>
>
>


-- 
Nabeel Butt
LUMS,Lahore



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