Cover the plane with closed disks
up vote
1
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Help with this Putname problem: Is it possible to find an infinite sequence of closed disks $D_1,D_2,...$ in the plane with centers $c_1,c_2,...$ such that
a) the $c_1$ have no limit point in the finite plane.
b) the sum of the areas of the $D_i$ is finite and
c) every line in the plane intersects at least one of the $D_i$?
contest-math recreational-mathematics
asked Nov 20 at 21:14
mathnoob
1,495319
add a comment |
up vote
1
down vote
favorite
Help with this Putname problem: Is it possible to find an infinite sequence of closed disks $D_1,D_2,...$ in the plane with centers $c_1,c_2,...$ such that
a) the $c_1$ have no limit point in the finite plane.
b) the sum of the areas of the $D_i$ is finite and
c) every line in the plane intersects at least one of the $D_i$?
contest-math recreational-mathematics
asked Nov 20 at 21:14
mathnoob
1,495319
What do you mean by “cover”?
– Michael Hoppe
Nov 20 at 21:53
Ok I edited it. It was written cover on my problem sheet. I was not sure at first as well. because how can you have condition b) satisfied and covers the plane.
– mathnoob
Nov 20 at 22:13
"putname" should be "Putnam" (a person on which a name has been put...)
– Jean Marie
Nov 20 at 22:19
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Help with this Putname problem: Is it possible to find an infinite sequence of closed disks $D_1,D_2,...$ in the plane with centers $c_1,c_2,...$ such that
a) the $c_1$ have no limit point in the finite plane.
b) the sum of the areas of the $D_i$ is finite and
c) every line in the plane intersects at least one of the $D_i$?
contest-math recreational-mathematics
asked Nov 20 at 21:14
mathnoob
1,495319
Help with this Putname problem: Is it possible to find an infinite sequence of closed disks $D_1,D_2,...$ in the plane with centers $c_1,c_2,...$ such that
a) the $c_1$ have no limit point in the finite plane.
b) the sum of the areas of the $D_i$ is finite and
c) every line in the plane intersects at least one of the $D_i$?
contest-math recreational-mathematics
contest-math recreational-mathematics
asked Nov 20 at 21:14
mathnoob
1,495319
asked Nov 20 at 21:14
mathnoob
1,495319
edited Nov 20 at 22:22
asked Nov 20 at 21:14
mathnoob
1,495319
asked Nov 20 at 21:14
mathnoob
1,495319
asked Nov 20 at 21:14
mathnoob
1,495319
1,495319
What do you mean by “cover”?
– Michael Hoppe
Nov 20 at 21:53
Ok I edited it. It was written cover on my problem sheet. I was not sure at first as well. because how can you have condition b) satisfied and covers the plane.
– mathnoob
Nov 20 at 22:13
"putname" should be "Putnam" (a person on which a name has been put...)
– Jean Marie
Nov 20 at 22:19
add a comment |
What do you mean by “cover”?
– Michael Hoppe
Nov 20 at 21:53
Ok I edited it. It was written cover on my problem sheet. I was not sure at first as well. because how can you have condition b) satisfied and covers the plane.
– mathnoob
Nov 20 at 22:13
"putname" should be "Putnam" (a person on which a name has been put...)
– Jean Marie
Nov 20 at 22:19
What do you mean by “cover”?
– Michael Hoppe
Nov 20 at 21:53
What do you mean by “cover”?
– Michael Hoppe
Nov 20 at 21:53
Ok I edited it. It was written cover on my problem sheet. I was not sure at first as well. because how can you have condition b) satisfied and covers the plane.
– mathnoob
Nov 20 at 22:13
Ok I edited it. It was written cover on my problem sheet. I was not sure at first as well. because how can you have condition b) satisfied and covers the plane.
– mathnoob
Nov 20 at 22:13
"putname" should be "Putnam" (a person on which a name has been put...)
– Jean Marie
Nov 20 at 22:19
"putname" should be "Putnam" (a person on which a name has been put...)
– Jean Marie
Nov 20 at 22:19
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
Put a unit disk at the origin. Now put disks with centers along each of the four coordinate axes. Put a disk of radius $frac 12$ tangent to the unit disk, so there are four with centers $(pm frac 32,0)$ and $(0,pm frac 32)$. Put disks of radius $frac 13$ tangent to the disks of radius $frac 12$ and in general put disks of radius $frac 1{n+1}$ tangent to the disks of radius $frac 1n$. The radii form the harmonic series, which is unbounded, so we cover all of the coordinate axes. Every line in the plane therefore hits at least one disk. The sum of the areas is $$pileft(4sum_{i=1}^infty frac 1{i^2}-3right)=pileft(frac {4pi^2}6-3right)$$
which is nicely finite. The first three stages of the construction are shown below.
answered Nov 21 at 0:04
Ross Millikan
289k23195367
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
1
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Put a unit disk at the origin. Now put disks with centers along each of the four coordinate axes. Put a disk of radius $frac 12$ tangent to the unit disk, so there are four with centers $(pm frac 32,0)$ and $(0,pm frac 32)$. Put disks of radius $frac 13$ tangent to the disks of radius $frac 12$ and in general put disks of radius $frac 1{n+1}$ tangent to the disks of radius $frac 1n$. The radii form the harmonic series, which is unbounded, so we cover all of the coordinate axes. Every line in the plane therefore hits at least one disk. The sum of the areas is $$pileft(4sum_{i=1}^infty frac 1{i^2}-3right)=pileft(frac {4pi^2}6-3right)$$
which is nicely finite. The first three stages of the construction are shown below.
answered Nov 21 at 0:04
Ross Millikan
289k23195367
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
1
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
add a comment |
up vote
3
down vote
accepted
Put a unit disk at the origin. Now put disks with centers along each of the four coordinate axes. Put a disk of radius $frac 12$ tangent to the unit disk, so there are four with centers $(pm frac 32,0)$ and $(0,pm frac 32)$. Put disks of radius $frac 13$ tangent to the disks of radius $frac 12$ and in general put disks of radius $frac 1{n+1}$ tangent to the disks of radius $frac 1n$. The radii form the harmonic series, which is unbounded, so we cover all of the coordinate axes. Every line in the plane therefore hits at least one disk. The sum of the areas is $$pileft(4sum_{i=1}^infty frac 1{i^2}-3right)=pileft(frac {4pi^2}6-3right)$$
which is nicely finite. The first three stages of the construction are shown below.
answered Nov 21 at 0:04
Ross Millikan
289k23195367
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
1
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Put a unit disk at the origin. Now put disks with centers along each of the four coordinate axes. Put a disk of radius $frac 12$ tangent to the unit disk, so there are four with centers $(pm frac 32,0)$ and $(0,pm frac 32)$. Put disks of radius $frac 13$ tangent to the disks of radius $frac 12$ and in general put disks of radius $frac 1{n+1}$ tangent to the disks of radius $frac 1n$. The radii form the harmonic series, which is unbounded, so we cover all of the coordinate axes. Every line in the plane therefore hits at least one disk. The sum of the areas is $$pileft(4sum_{i=1}^infty frac 1{i^2}-3right)=pileft(frac {4pi^2}6-3right)$$
which is nicely finite. The first three stages of the construction are shown below.
answered Nov 21 at 0:04
Ross Millikan
289k23195367
Put a unit disk at the origin. Now put disks with centers along each of the four coordinate axes. Put a disk of radius $frac 12$ tangent to the unit disk, so there are four with centers $(pm frac 32,0)$ and $(0,pm frac 32)$. Put disks of radius $frac 13$ tangent to the disks of radius $frac 12$ and in general put disks of radius $frac 1{n+1}$ tangent to the disks of radius $frac 1n$. The radii form the harmonic series, which is unbounded, so we cover all of the coordinate axes. Every line in the plane therefore hits at least one disk. The sum of the areas is $$pileft(4sum_{i=1}^infty frac 1{i^2}-3right)=pileft(frac {4pi^2}6-3right)$$
which is nicely finite. The first three stages of the construction are shown below.
answered Nov 21 at 0:04
Ross Millikan
289k23195367
edited Nov 21 at 0:18
answered Nov 21 at 0:04
Ross Millikan
289k23195367
answered Nov 21 at 0:04
Ross Millikan
289k23195367
answered Nov 21 at 0:04
Ross Millikan
289k23195367
289k23195367
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
1
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
add a comment |
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
1
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
Thanks so much that's soo cool!
– mathnoob
Nov 21 at 0:17
1
1
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
@mathnoob: needing to meet every line means we need it infinite in size, but having finite area means we need the circles to decrease in area faster than $frac 1n$. That leads to the thought of radii like $frac 1n$ and areas like $frac 1{n^2}$
– Ross Millikan
Nov 21 at 1:12
add a comment |
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What do you mean by “cover”?
– Michael Hoppe
Nov 20 at 21:53
Ok I edited it. It was written cover on my problem sheet. I was not sure at first as well. because how can you have condition b) satisfied and covers the plane.
– mathnoob
Nov 20 at 22:13
"putname" should be "Putnam" (a person on which a name has been put...)
– Jean Marie
Nov 20 at 22:19