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

*To*: mathgroup at smc.vnet.net*Subject*: [mg68628] RE: [mg68559] need mathematica's help for exploring a certain type of mapping*From*: "David Park" <djmp at earthlink.net>*Date*: Sun, 13 Aug 2006 05:52:45 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Nabeel, I know how to easily make a counter-example going from 1D to 2D. We need a statement that any such function must be periodic and isotropic, with period 1, about all points in the 2D plane. And then a theorem that says there are no such functions. 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 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 8/10/06, David Park <djmp at earthlink.net> wrote: Nabeel, This is interesting, but probably beyond me. But I like to do graphics so if you have some ideas I could probably implement the graphics. The first question is: Does the mapping have to map the xy-plane to a planar surface in 3D space? Is the mapping continuous? Suppose you draw a straight line in the plane and mark out points on the line all separated by unit distance. What can this map into? Then suppose you shift all the points on the line by a small amount. Or say by half a unit. They will still have to be equidistant but separated by something other than 1/2 in 3D from the original points. How would one achieve that? I think I could envision a periodic stretching and squeesing along the line, with a period of 1, but could one make that isotropic about all points? 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 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, 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

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