What is the largest number of intersecting circles such that every pair of circles has an overlapping area?
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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 Nov 19 at 21:37
zooby
961616
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 Nov 19 at 21:37
zooby
961616
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
Nov 19 at 21:57
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
Nov 19 at 21:59
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 Nov 19 at 21:37
zooby
961616
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 Nov 19 at 21:37
zooby
961616
asked Nov 19 at 21:37
zooby
961616
asked Nov 19 at 21:37
zooby
961616
asked Nov 19 at 21:37
zooby
961616
asked Nov 19 at 21:37
zooby
961616
961616
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
Nov 19 at 21:57
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
Nov 19 at 21:59
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
Nov 19 at 21:57
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
Nov 19 at 21:59
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
Nov 19 at 21:57
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
Nov 19 at 21:57
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
Nov 19 at 21:59
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
Nov 19 at 21:59
add a comment |
1 Answer
1
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$mathbf{Lemma}$
-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 Nov 19 at 21:59
Dr. Mathva
742114
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
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 Nov 19 at 21:59
Dr. Mathva
742114
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
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 Nov 19 at 21:59
Dr. Mathva
742114
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
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 Nov 19 at 21:59
Dr. Mathva
742114
$mathbf{Lemma}$
-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 Nov 19 at 21:59
Dr. Mathva
742114
answered Nov 19 at 21:59
Dr. Mathva
742114
answered Nov 19 at 21:59
Dr. Mathva
742114
answered Nov 19 at 21:59
Dr. Mathva
742114
742114
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
add a comment |
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Yes but I can create a Venn diagram with 4 circles such that every pair is represented. (Not other combinations).
– zooby
Nov 19 at 22:01
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
Oh, all right... Sorry, I didn't understand the answer then...
– Dr. Mathva
Nov 19 at 22:03
add a comment |
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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
Nov 19 at 21:57
Only 2 circles should overlap at once. Like a Venn diagram these areas would correspond to 2 properties.
– zooby
Nov 19 at 21:59