Can you put six points on the plane so that the distance between any two of them is an integer and no three...
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Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?
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Can you put six points on the plane so that the distance between any two of them is an integer and no three lie on the same line?
geometry discrete-mathematics graph-theory
edited Dec 19 '18 at 18:57
David G. Stork
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asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
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Dec 20 '18 at 22:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?
3 answers
Can you put six points on the plane so that the distance between any two of them is an integer and no three lie on the same line?
geometry discrete-mathematics graph-theory
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
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Dec 20 '18 at 22:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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if no three lie on the same line then any three must form a triangle. So... have you looked at hexagons?
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– postmortes
Dec 19 '18 at 18:37
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This question already has an answer here:
Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?
3 answers
Can you put six points on the plane so that the distance between any two of them is an integer and no three lie on the same line?
geometry discrete-mathematics graph-theory
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
$endgroup$
This question already has an answer here:
Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?
3 answers
Can you put six points on the plane so that the distance between any two of them is an integer and no three lie on the same line?
This question already has an answer here:
Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?
3 answers
geometry discrete-mathematics graph-theory
geometry discrete-mathematics graph-theory
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
edited Dec 19 '18 at 18:57
David G. Stork
11k41432
11k41432
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
asked Dec 19 '18 at 18:26
benjamin1234benjamin1234
111
111
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Dec 20 '18 at 22:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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if no three lie on the same line then any three must form a triangle. So... have you looked at hexagons?
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– postmortes
Dec 19 '18 at 18:37
add a comment |
$begingroup$
if no three lie on the same line then any three must form a triangle. So... have you looked at hexagons?
$endgroup$
– postmortes
Dec 19 '18 at 18:37
$begingroup$
if no three lie on the same line then any three must form a triangle. So... have you looked at hexagons?
$endgroup$
– postmortes
Dec 19 '18 at 18:37
$begingroup$
if no three lie on the same line then any three must form a triangle. So... have you looked at hexagons?
$endgroup$
– postmortes
Dec 19 '18 at 18:37
add a comment |
1 Answer
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You can do it with any finite number of points. See, e.g., this question which gives you infinitely many points on the unit circle with rational distance between any two. Then just pick $6$ (or however many so long as it is finite) and scale up the picture to get integer distances.
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
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but isn't that proof is for rational distances?
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– benjamin1234
Dec 19 '18 at 19:33
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As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
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– lulu
Dec 19 '18 at 19:35
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can do it with any finite number of points. See, e.g., this question which gives you infinitely many points on the unit circle with rational distance between any two. Then just pick $6$ (or however many so long as it is finite) and scale up the picture to get integer distances.
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
$endgroup$
$begingroup$
but isn't that proof is for rational distances?
$endgroup$
– benjamin1234
Dec 19 '18 at 19:33
$begingroup$
As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
$endgroup$
– lulu
Dec 19 '18 at 19:35
add a comment |
$begingroup$
You can do it with any finite number of points. See, e.g., this question which gives you infinitely many points on the unit circle with rational distance between any two. Then just pick $6$ (or however many so long as it is finite) and scale up the picture to get integer distances.
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
$endgroup$
$begingroup$
but isn't that proof is for rational distances?
$endgroup$
– benjamin1234
Dec 19 '18 at 19:33
$begingroup$
As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
$endgroup$
– lulu
Dec 19 '18 at 19:35
add a comment |
$begingroup$
You can do it with any finite number of points. See, e.g., this question which gives you infinitely many points on the unit circle with rational distance between any two. Then just pick $6$ (or however many so long as it is finite) and scale up the picture to get integer distances.
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
$endgroup$
You can do it with any finite number of points. See, e.g., this question which gives you infinitely many points on the unit circle with rational distance between any two. Then just pick $6$ (or however many so long as it is finite) and scale up the picture to get integer distances.
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
edited Dec 19 '18 at 18:38
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
answered Dec 19 '18 at 18:33
lulululu
41.6k24979
41.6k24979
$begingroup$
but isn't that proof is for rational distances?
$endgroup$
– benjamin1234
Dec 19 '18 at 19:33
$begingroup$
As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
$endgroup$
– lulu
Dec 19 '18 at 19:35
add a comment |
$begingroup$
but isn't that proof is for rational distances?
$endgroup$
– benjamin1234
Dec 19 '18 at 19:33
$begingroup$
As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
$endgroup$
– lulu
Dec 19 '18 at 19:35
$begingroup$
but isn't that proof is for rational distances?
$endgroup$
– benjamin1234
Dec 19 '18 at 19:33
$begingroup$
but isn't that proof is for rational distances?
$endgroup$
– benjamin1234
Dec 19 '18 at 19:33
$begingroup$
As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
$endgroup$
– lulu
Dec 19 '18 at 19:35
$begingroup$
As I say, for any finite collection you can scale the picture up. If $N$ is the least common denominator of all the distances between your $n$ chosen points, just scale the entire picture up by $N$ (multiply all coordinates by $N$). Clearly this doesn't work for infinite collections but it works for any finite subset.
$endgroup$
– lulu
Dec 19 '18 at 19:35
add a comment |
$begingroup$
if no three lie on the same line then any three must form a triangle. So... have you looked at hexagons?
$endgroup$
– postmortes
Dec 19 '18 at 18:37