<< Huh? Distance between the two points is sqrt(2) in both cases. >> On third thoughts --- must've been a bit too late at night for me --- apologies, everybody! WFL

On 12-Jun-17 21:25, Fred Lunnon wrote:

...

Similar, not congruent. WFL

On 6/13/17, Mike Speciner <ms@alum.mit.edu> wrote:

...

But that's isomorphic to {<0,0>,<1,1>}

On 12-Jun-17 19:55, Fred Lunnon wrote:

...

I doubt that this is blond; I'm pretty certain it lacks hairy ears.

Regarding the next question, it ain't so: take n = 1 , X = { 0, sqrt(2) } .

WFL

On 6/12/17, Dan Asimov <dasimov@earthlink.net> wrote:

...

This may be trivial, hard, or auropileous; I'm not sure:

Suppose we are given a finite subset X of R^n such that

* The dot product of any two vectors between points of X is an integer.

(I.e., for all x,y,z,w in X, the real number

<x-y, z-w>

lies in Z.)

Question: --------- Does it follow that for any sufficiently high dimension d, there is a subset X' of the integer lattice Z^d such that X' is congruent to X ???

—Dan

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