Mixing
What is the largest number of intersecting circles such that every pair of circles has an overlapping area?
up vote
0
down vote
favorite
Think of a Venn diagram made of circles. You need to draw one such that every pair of sets is represented (and areas with more than 2 circles overlapping don't count!). What is the largest number of circles possible? I can do at least 4. (Is there a way to prove it?)
For the higher dimensional problem...
What is the smallest number of N-balls such that every M-tuple of sets is represented as shared volume?
e.g. for the 1-ball which is a line segment you can only have 2 sets to represent all pairs with the lines overlapping once.
geometry elementary-set-theory
asked 2 days ago
zooby
946616
add a comment |
up vote
0
down vote
favorite
Think of a Venn diagram made of circles. You need to draw one such that every pair of sets is represented (and areas with more than 2 circles overlapping don't count!). What is the largest number of circles possible? I can do at least 4. (Is there a way to prove it?)
For the higher dimensional problem...
What is the smallest number of N-balls such that every M-tuple of sets is represented as shared volume?
e.g. for the 1-ball which is a line segment you can only have 2 sets to represent all pairs with the lines overlapping once.
geometry elementary-set-theory
asked 2 days ago
zooby
946616
What exactly are the constraints you’re looking for? Can you give an example and a non-example in $mathbb{R}^2$? I see a construction where I place a circle of radius $2$ centered at each point on the unit circle. Each pair of circle overlaps since they all contain $(0,0)$, but this doesn’t seem as though it would fit your criteria.
– Santana Afton
2 days ago
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
2 days ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Think of a Venn diagram made of circles. You need to draw one such that every pair of sets is represented (and areas with more than 2 circles overlapping don't count!). What is the largest number of circles possible? I can do at least 4. (Is there a way to prove it?)
For the higher dimensional problem...
What is the smallest number of N-balls such that every M-tuple of sets is represented as shared volume?
e.g. for the 1-ball which is a line segment you can only have 2 sets to represent all pairs with the lines overlapping once.
geometry elementary-set-theory
asked 2 days ago
zooby
946616
Think of a Venn diagram made of circles. You need to draw one such that every pair of sets is represented (and areas with more than 2 circles overlapping don't count!). What is the largest number of circles possible? I can do at least 4. (Is there a way to prove it?)
For the higher dimensional problem...
What is the smallest number of N-balls such that every M-tuple of sets is represented as shared volume?
e.g. for the 1-ball which is a line segment you can only have 2 sets to represent all pairs with the lines overlapping once.
geometry elementary-set-theory
geometry elementary-set-theory
asked 2 days ago
zooby
946616
asked 2 days ago
zooby
946616
asked 2 days ago
zooby
946616
asked 2 days ago
zooby
946616
asked 2 days ago
zooby
946616
946616
What exactly are the constraints you’re looking for? Can you give an example and a non-example in $mathbb{R}^2$? I see a construction where I place a circle of radius $2$ centered at each point on the unit circle. Each pair of circle overlaps since they all contain $(0,0)$, but this doesn’t seem as though it would fit your criteria.
– Santana Afton
2 days ago
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
2 days ago
add a comment |
What exactly are the constraints you’re looking for? Can you give an example and a non-example in $mathbb{R}^2$? I see a construction where I place a circle of radius $2$ centered at each point on the unit circle. Each pair of circle overlaps since they all contain $(0,0)$, but this doesn’t seem as though it would fit your criteria.
– Santana Afton
2 days ago
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
2 days ago
What exactly are the constraints you’re looking for? Can you give an example and a non-example in $mathbb{R}^2$? I see a construction where I place a circle of radius $2$ centered at each point on the unit circle. Each pair of circle overlaps since they all contain $(0,0)$, but this doesn’t seem as though it would fit your criteria.
– Santana Afton
2 days ago
What exactly are the constraints you’re looking for? Can you give an example and a non-example in $mathbb{R}^2$? I see a construction where I place a circle of radius $2$ centered at each point on the unit circle. Each pair of circle overlaps since they all contain $(0,0)$, but this doesn’t seem as though it would fit your criteria.
– Santana Afton
2 days ago
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
2 days ago
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
$mathbf{Lemma}$
-Two circles intersect each other in at most two points-
Consider a Venn-Diagram with three circles. Here you would have in total 7 regions $(A, B, C, Acap B, Acap C, Bcap C, Acap Bcap C)$.
Adding a new circle you will get at most 6 intersections (two with every initial circle), which will create, at most, 6 new regions. Nevertheless, those are in total 14 regions at most, and you're looking for ${2^4}=16>14$ regions.
$mathbf{Remark}$
It is possible to create a Venn-Diagram with four ellipses.
answered 2 days ago
Dr. Mathva
606
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$mathbf{Lemma}$
-Two circles intersect each other in at most two points-
Consider a Venn-Diagram with three circles. Here you would have in total 7 regions $(A, B, C, Acap B, Acap C, Bcap C, Acap Bcap C)$.
Adding a new circle you will get at most 6 intersections (two with every initial circle), which will create, at most, 6 new regions. Nevertheless, those are in total 14 regions at most, and you're looking for ${2^4}=16>14$ regions.
$mathbf{Remark}$
It is possible to create a Venn-Diagram with four ellipses.
answered 2 days ago
Dr. Mathva
606
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
add a comment |
up vote
0
down vote
$mathbf{Lemma}$
-Two circles intersect each other in at most two points-
Consider a Venn-Diagram with three circles. Here you would have in total 7 regions $(A, B, C, Acap B, Acap C, Bcap C, Acap Bcap C)$.
Adding a new circle you will get at most 6 intersections (two with every initial circle), which will create, at most, 6 new regions. Nevertheless, those are in total 14 regions at most, and you're looking for ${2^4}=16>14$ regions.
$mathbf{Remark}$
It is possible to create a Venn-Diagram with four ellipses.
answered 2 days ago
Dr. Mathva
606
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
add a comment |
up vote
0
down vote
up vote
0
down vote
$mathbf{Lemma}$
-Two circles intersect each other in at most two points-
Consider a Venn-Diagram with three circles. Here you would have in total 7 regions $(A, B, C, Acap B, Acap C, Bcap C, Acap Bcap C)$.
Adding a new circle you will get at most 6 intersections (two with every initial circle), which will create, at most, 6 new regions. Nevertheless, those are in total 14 regions at most, and you're looking for ${2^4}=16>14$ regions.
$mathbf{Remark}$
It is possible to create a Venn-Diagram with four ellipses.
answered 2 days ago
Dr. Mathva
606
$mathbf{Lemma}$
-Two circles intersect each other in at most two points-
Consider a Venn-Diagram with three circles. Here you would have in total 7 regions $(A, B, C, Acap B, Acap C, Bcap C, Acap Bcap C)$.
Adding a new circle you will get at most 6 intersections (two with every initial circle), which will create, at most, 6 new regions. Nevertheless, those are in total 14 regions at most, and you're looking for ${2^4}=16>14$ regions.
$mathbf{Remark}$
It is possible to create a Venn-Diagram with four ellipses.
answered 2 days ago
Dr. Mathva
606
answered 2 days ago
Dr. Mathva
606
answered 2 days ago
Dr. Mathva
606
answered 2 days ago
Dr. Mathva
606
606
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
add a comment |
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
2 days ago
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005558%2fwhat-is-the-largest-number-of-intersecting-circles-such-that-every-pair-of-circl%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
What exactly are the constraints you’re looking for? Can you give an example and a non-example in $mathbb{R}^2$? I see a construction where I place a circle of radius $2$ centered at each point on the unit circle. Each pair of circle overlaps since they all contain $(0,0)$, but this doesn’t seem as though it would fit your criteria.
– Santana Afton
2 days ago
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
2 days ago