Do infinitely many points in a plane with integer distances lie on a line?
up vote
13
down vote
favorite
Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid question and point me in the right direction?
'Given an infinite set of points in a plane, if the distance between any two points is an integer, prove that all these points lie on a straight line.'
geometry
edited Apr 19 '11 at 0:35
joriki
170k10183341
asked Apr 19 '11 at 0:32
xbonez
1686
|
show 4 more comments
up vote
13
down vote
favorite
Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid question and point me in the right direction?
'Given an infinite set of points in a plane, if the distance between any two points is an integer, prove that all these points lie on a straight line.'
geometry
edited Apr 19 '11 at 0:35
joriki
170k10183341
asked Apr 19 '11 at 0:32
xbonez
1686
Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from.
– Alex B.
Apr 19 '11 at 0:34
How what is possible?
– Qiaochu Yuan
Apr 19 '11 at 0:35
Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles.
– joriki
Apr 19 '11 at 0:37
4
You can't have a circle of infinite radius in euclidean space. You can though have an infinite line.
– JSchlather
Apr 19 '11 at 0:46
2
@xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"?
– Qiaochu Yuan
Apr 19 '11 at 1:01
|
show 4 more comments
up vote
13
down vote
favorite
up vote
13
down vote
favorite
Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid question and point me in the right direction?
'Given an infinite set of points in a plane, if the distance between any two points is an integer, prove that all these points lie on a straight line.'
geometry
edited Apr 19 '11 at 0:35
joriki
170k10183341
asked Apr 19 '11 at 0:32
xbonez
1686
Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid question and point me in the right direction?
'Given an infinite set of points in a plane, if the distance between any two points is an integer, prove that all these points lie on a straight line.'
geometry
geometry
edited Apr 19 '11 at 0:35
joriki
170k10183341
asked Apr 19 '11 at 0:32
xbonez
1686
edited Apr 19 '11 at 0:35
joriki
170k10183341
asked Apr 19 '11 at 0:32
xbonez
1686
edited Apr 19 '11 at 0:35
joriki
170k10183341
edited Apr 19 '11 at 0:35
joriki
170k10183341
edited Apr 19 '11 at 0:35
joriki
170k10183341
170k10183341
asked Apr 19 '11 at 0:32
xbonez
1686
asked Apr 19 '11 at 0:32
xbonez
1686
asked Apr 19 '11 at 0:32
xbonez
1686
1686
Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from.
– Alex B.
Apr 19 '11 at 0:34
How what is possible?
– Qiaochu Yuan
Apr 19 '11 at 0:35
Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles.
– joriki
Apr 19 '11 at 0:37
4
You can't have a circle of infinite radius in euclidean space. You can though have an infinite line.
– JSchlather
Apr 19 '11 at 0:46
2
@xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"?
– Qiaochu Yuan
Apr 19 '11 at 1:01
|
show 4 more comments
Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from.
– Alex B.
Apr 19 '11 at 0:34
How what is possible?
– Qiaochu Yuan
Apr 19 '11 at 0:35
Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles.
– joriki
Apr 19 '11 at 0:37
4
You can't have a circle of infinite radius in euclidean space. You can though have an infinite line.
– JSchlather
Apr 19 '11 at 0:46
2
@xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"?
– Qiaochu Yuan
Apr 19 '11 at 1:01
Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from.
– Alex B.
Apr 19 '11 at 0:34
Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from.
– Alex B.
Apr 19 '11 at 0:34
How what is possible?
– Qiaochu Yuan
Apr 19 '11 at 0:35
How what is possible?
– Qiaochu Yuan
Apr 19 '11 at 0:35
Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles.
– joriki
Apr 19 '11 at 0:37
Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles.
– joriki
Apr 19 '11 at 0:37
4
4
You can't have a circle of infinite radius in euclidean space. You can though have an infinite line.
– JSchlather
Apr 19 '11 at 0:46
You can't have a circle of infinite radius in euclidean space. You can though have an infinite line.
– JSchlather
Apr 19 '11 at 0:46
2
2
@xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"?
– Qiaochu Yuan
Apr 19 '11 at 1:01
@xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"?
– Qiaochu Yuan
Apr 19 '11 at 1:01
|
show 4 more comments
2 Answers
2
active
oldest
votes
up vote
10
down vote
accepted
MR0013511 (7,164a)
Anning, Norman H.; Erdős, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 598–600.
The authors show that for any n there exist noncollinear points $P_1,dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property.
Reviewed by I. Kaplansky
I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.
edited Mar 9 '13 at 13:41
user642796
44.4k558115
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
add a comment |
up vote
9
down vote
Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| leq AB$ and $|AP-PC| leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m leq AC$). There are finitely many such points $P$ (at most $4;AB;AC$).
(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)
edited Mar 9 '13 at 13:44
user642796
44.4k558115
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
10
down vote
accepted
MR0013511 (7,164a)
Anning, Norman H.; Erdős, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 598–600.
The authors show that for any n there exist noncollinear points $P_1,dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property.
Reviewed by I. Kaplansky
I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.
edited Mar 9 '13 at 13:41
user642796
44.4k558115
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
add a comment |
up vote
10
down vote
accepted
MR0013511 (7,164a)
Anning, Norman H.; Erdős, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 598–600.
The authors show that for any n there exist noncollinear points $P_1,dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property.
Reviewed by I. Kaplansky
I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.
edited Mar 9 '13 at 13:41
user642796
44.4k558115
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
add a comment |
up vote
10
down vote
accepted
up vote
10
down vote
accepted
MR0013511 (7,164a)
Anning, Norman H.; Erdős, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 598–600.
The authors show that for any n there exist noncollinear points $P_1,dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property.
Reviewed by I. Kaplansky
I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.
edited Mar 9 '13 at 13:41
user642796
44.4k558115
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
MR0013511 (7,164a)
Anning, Norman H.; Erdős, Paul
Integral distances.
Bull. Amer. Math. Soc. 51, (1945). 598–600.
The authors show that for any n there exist noncollinear points $P_1,dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property.
Reviewed by I. Kaplansky
I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.
edited Mar 9 '13 at 13:41
user642796
44.4k558115
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
edited Mar 9 '13 at 13:41
user642796
44.4k558115
edited Mar 9 '13 at 13:41
user642796
44.4k558115
edited Mar 9 '13 at 13:41
user642796
44.4k558115
44.4k558115
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
answered Apr 19 '11 at 1:16
Gerry Myerson
145k8147298
145k8147298
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
add a comment |
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
The article and a follow-up.
– J. M. is not a mathematician
Apr 19 '11 at 1:26
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
@J.M. It appears both links point the same place, probably the follow-up
– Ross Millikan
Apr 19 '11 at 1:44
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
– JavaMan
Apr 19 '11 at 1:49
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021).
– Gerry Myerson
Apr 19 '11 at 1:51
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
@Ross: yes, I copied wrongly; DJC gives the correct link.
– J. M. is not a mathematician
Apr 19 '11 at 1:55
add a comment |
up vote
9
down vote
Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| leq AB$ and $|AP-PC| leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m leq AC$). There are finitely many such points $P$ (at most $4;AB;AC$).
(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)
edited Mar 9 '13 at 13:44
user642796
44.4k558115
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
add a comment |
up vote
9
down vote
Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| leq AB$ and $|AP-PC| leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m leq AC$). There are finitely many such points $P$ (at most $4;AB;AC$).
(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)
edited Mar 9 '13 at 13:44
user642796
44.4k558115
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
add a comment |
up vote
9
down vote
up vote
9
down vote
Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| leq AB$ and $|AP-PC| leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m leq AC$). There are finitely many such points $P$ (at most $4;AB;AC$).
(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)
edited Mar 9 '13 at 13:44
user642796
44.4k558115
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| leq AB$ and $|AP-PC| leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m leq AC$). There are finitely many such points $P$ (at most $4;AB;AC$).
(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)
edited Mar 9 '13 at 13:44
user642796
44.4k558115
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
edited Mar 9 '13 at 13:44
user642796
44.4k558115
edited Mar 9 '13 at 13:44
user642796
44.4k558115
edited Mar 9 '13 at 13:44
user642796
44.4k558115
44.4k558115
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
answered Apr 19 '11 at 2:46
Omar Antolín-Camarena
4,3921929
4,3921929
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
add a comment |
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..."
– user641
Apr 19 '11 at 6:18
add a comment |
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Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from.
– Alex B.
Apr 19 '11 at 0:34
How what is possible?
– Qiaochu Yuan
Apr 19 '11 at 0:35
Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles.
– joriki
Apr 19 '11 at 0:37
4
You can't have a circle of infinite radius in euclidean space. You can though have an infinite line.
– JSchlather
Apr 19 '11 at 0:46
2
@xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"?
– Qiaochu Yuan
Apr 19 '11 at 1:01