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

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.
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

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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