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

*To*: mathgroup at smc.vnet.net*Subject*: [mg68578] 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 23:57:40 -0400 (EDT)*References*: <200608090820.EAA21373@smc.vnet.net> <NDBBJGNHKLMPLILOIPPOMELCFBAA.djmp@earthlink.net>*Sender*: owner-wri-mathgroup at wolfram.com

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. 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

**Follow-Ups**:**Re: Re: need mathematica's help for exploring a certain type of mapping***From:*Murray Eisenberg <murray@math.umass.edu>

**References**:**need mathematica's help for exploring a certain type of mapping***From:*"Nabeel Butt" <nabeel.butt@gmail.com>