[math-fun] Geometry question
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
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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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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My Latin skills aren't up to the task of decoding Dan's coinage. (I did a Google search, but got "Your search - *auropileous definition* - did not match any documents" for my pains.) Anyone care to parse this for me? Thanks, Jim On Mon, Jun 12, 2017 at 6:08 PM, 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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I think it means "goldenhaired", but I don't understand the application. On Mon, Jun 12, 2017 at 8:03 PM, James Propp <jamespropp@gmail.com> wrote:
My Latin skills aren't up to the task of decoding Dan's coinage. (I did a Google search, but got "Your search - *auropileous definition* - did not match any documents" for my pains.) Anyone care to parse this for me?
Thanks,
Jim
On Mon, Jun 12, 2017 at 6:08 PM, 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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I assume it's... auro- golden pileous hairy On 12-Jun-17 20:03, James Propp wrote:
My Latin skills aren't up to the task of decoding Dan's coinage. (I did a Google search, but got "Your search - *auropileous definition* - did not match any documents" for my pains.) Anyone care to parse this for me?
Thanks,
Jim
On Mon, Jun 12, 2017 at 6:08 PM, 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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participants (5)
-
Allan Wechsler -
Dan Asimov -
Fred Lunnon -
James Propp -
Mike Speciner