Newsgroups: sci.math
From: tao@saffron.Princeton.EDU (Terry Tao)
Subject: Re: Q: Distance-preserving mappings in R^n
Date: Wed, 27 Jul 1994 01:15:06 GMT

Here is a proof that there can be no unit-distance-preserving
non-isometries in the plane.  It shouldn't be too hard to extend this
proof to higher dimensions.

Suppose f is a map such that if |x-y|=1, then |f(x)-f(y)|=1.

Claim 1.  If |x-y|=sqrt(3), then |f(x)-f(y)|=sqrt(3).

Proof.  By considering what can happen to a diamond <|> made up of five
unit lengths, we can see that if |x-y|=sqrt(3), then |f(x)-f(y)| is
either sqrt(3) or 0.  However, 0 is impossible, because we can find
another point z such that |x-z|=sqrt(3) and |y-z|=1, and various
permutations with the triangle inequality then reach a contradiction.

Claim 2.  If d is a number of the form sqrt(a^2 + ab + b^2) where a, b,
are integers, and |x-y|=d, then |f(x)-f(y)| = d.

Proof.  From Claim 1, f must map a equilateral triangular lattice to
another equilateral triangular lattice isometrically.  Since d is a
distance on such a lattice, the result follows.

Claim 3.  If d is as above and n<d is an integer, then if |x-y| >= d-n,
then |f(x)-f(y)| >= d-n.

Proof.  One can find a z such that |x-z|=d and there are n unit lengths
connecting y and z.  The result then follows from Claim 2 and the
triangle inequality.

Claim 4.  The fractional parts of the numbers sqrt(a^2 + ab + b^2) are
dense in [0,1].

Proof.  This is messy and number theoretic; probably an exercise in Hady
& Wright anyway, so I'll omit it. :-)

Corollary: For all x and y, |f(x)-f(y)| >= |x-y|.

Claim 5.  f is an isometry.

For any x and y, extend the ray from x  through y until a point z is
reached such that |x-y|=d for some d of the form in Claim 2.  Then
|f(x)-f(z)| = d, but |f(x)-f(y)| >= |x-y| and |f(y)-f(z)| >= d - |x-y|
by the abov e claims.  Thus by the triangle inequality, |f(x)-f(y)| must
equal |x-y|, and f is an isometry.


-- 
Terry Tao	Math Dept., Princeton University (tao@math.princeton.edu)
i mi pacna da ni'o .i mi caki pacna di'e .itu'e ro ma'arbi'i ba galtu .i ro 
cmana ba dizlo .i ro rufsu tumla ba xutla .i ro korcu pluta ba sirji .i le 
la jegvon. kamymisno ba tolcanci .i ro se rectu ba simkansa viska ra

==============================================================================

Newsgroups: sci.math
From: tao@saffron.Princeton.EDU (Terry Tao)
Subject: Re: Q: Distance-preserving mappings in R^n
Date: Wed, 27 Jul 1994 01:54:49 GMT

Some addenda to my previous post:

>Here is a proof that there can be no unit-distance-preserving
>non-isometries in the plane.  It shouldn't be too hard to extend this
>proof to higher dimensions.

Actually, the proof is identical, replacing diamonds with two tetrahedra
glued together.

>Claim 4.  The fractional parts of the numbers sqrt(a^2 + ab + b^2) are
>dense in [0,1].
>

This turned out to be fairly easy, since all integer multiples of the
irrational sqrt(3) are of this form. :-)


This whole question reminds me of another problem I saw some time ago:
how many colors are needed to color the plane so that two points a unit
distance apart always have distinct colors?  I know the answer is
between  4 and 7 - has there been any more progress on the problem
since?


-- 
Terry Tao	Math Dept., Princeton University (tao@math.princeton.edu)
i mi pacna da ni'o .i mi caki pacna di'e .itu'e ro ma'arbi'i ba galtu .i ro 
cmana ba dizlo .i ro rufsu tumla ba xutla .i ro korcu pluta ba sirji .i le 
la jegvon. kamymisno ba tolcanci .i ro se rectu ba simkansa viska ra