What is the maximum number of territories which may all be bordering each other?
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I'm working on a map for a game and I'm having trouble laying things out and am wondering if I'm up against a topological limitation.
What is the maximum number of different territories (non-overlapping regions) that can exist on a 2D surface such that every territory is touching (shares a border of non-zero length) every other territory?
Does the answer change if we allow 0-length (corner) "touches"?
general-topology
asked Dec 12 '18 at 4:27
GuestGuest
1
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add a comment |
$begingroup$
I'm working on a map for a game and I'm having trouble laying things out and am wondering if I'm up against a topological limitation.
What is the maximum number of different territories (non-overlapping regions) that can exist on a 2D surface such that every territory is touching (shares a border of non-zero length) every other territory?
Does the answer change if we allow 0-length (corner) "touches"?
general-topology
asked Dec 12 '18 at 4:27
GuestGuest
1
$endgroup$
2
$begingroup$
If corners touch, then you can have infinitely many territories (imagine a circle cut by an infinite number of diameters; each wedge is a territory, and they all meet in the center). Otherwise, the four color theorem seems relevant. Of course, this assumes that we are talking about a plane. The answer is different on a torus, for example.
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– Xander Henderson
Dec 12 '18 at 4:36
3
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In other terms, you are asking what is the largest complete graph which is also planar. It is $K_{color{red}{4}}$.
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– Jack D'Aurizio
Dec 12 '18 at 4:37
1
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But if you play on the surface of a torus, it is $K_{color{red}{7}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
Thanks for the help! I'll have to consider whether I want to allow wrapping/toruses.
$endgroup$
– Guest
Dec 12 '18 at 17:41
add a comment |
$begingroup$
I'm working on a map for a game and I'm having trouble laying things out and am wondering if I'm up against a topological limitation.
What is the maximum number of different territories (non-overlapping regions) that can exist on a 2D surface such that every territory is touching (shares a border of non-zero length) every other territory?
Does the answer change if we allow 0-length (corner) "touches"?
general-topology
asked Dec 12 '18 at 4:27
GuestGuest
1
$endgroup$
I'm working on a map for a game and I'm having trouble laying things out and am wondering if I'm up against a topological limitation.
What is the maximum number of different territories (non-overlapping regions) that can exist on a 2D surface such that every territory is touching (shares a border of non-zero length) every other territory?
Does the answer change if we allow 0-length (corner) "touches"?
general-topology
general-topology
asked Dec 12 '18 at 4:27
GuestGuest
1
asked Dec 12 '18 at 4:27
GuestGuest
1
edited Dec 12 '18 at 4:28
Guest
asked Dec 12 '18 at 4:27
GuestGuest
1
asked Dec 12 '18 at 4:27
GuestGuest
1
asked Dec 12 '18 at 4:27
GuestGuest
1
1
2
$begingroup$
If corners touch, then you can have infinitely many territories (imagine a circle cut by an infinite number of diameters; each wedge is a territory, and they all meet in the center). Otherwise, the four color theorem seems relevant. Of course, this assumes that we are talking about a plane. The answer is different on a torus, for example.
$endgroup$
– Xander Henderson
Dec 12 '18 at 4:36
3
$begingroup$
In other terms, you are asking what is the largest complete graph which is also planar. It is $K_{color{red}{4}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
1
$begingroup$
But if you play on the surface of a torus, it is $K_{color{red}{7}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
Thanks for the help! I'll have to consider whether I want to allow wrapping/toruses.
$endgroup$
– Guest
Dec 12 '18 at 17:41
add a comment |
2
$begingroup$
If corners touch, then you can have infinitely many territories (imagine a circle cut by an infinite number of diameters; each wedge is a territory, and they all meet in the center). Otherwise, the four color theorem seems relevant. Of course, this assumes that we are talking about a plane. The answer is different on a torus, for example.
$endgroup$
– Xander Henderson
Dec 12 '18 at 4:36
3
$begingroup$
In other terms, you are asking what is the largest complete graph which is also planar. It is $K_{color{red}{4}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
1
$begingroup$
But if you play on the surface of a torus, it is $K_{color{red}{7}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
Thanks for the help! I'll have to consider whether I want to allow wrapping/toruses.
$endgroup$
– Guest
Dec 12 '18 at 17:41
2
2
$begingroup$
If corners touch, then you can have infinitely many territories (imagine a circle cut by an infinite number of diameters; each wedge is a territory, and they all meet in the center). Otherwise, the four color theorem seems relevant. Of course, this assumes that we are talking about a plane. The answer is different on a torus, for example.
$endgroup$
– Xander Henderson
Dec 12 '18 at 4:36
$begingroup$
If corners touch, then you can have infinitely many territories (imagine a circle cut by an infinite number of diameters; each wedge is a territory, and they all meet in the center). Otherwise, the four color theorem seems relevant. Of course, this assumes that we are talking about a plane. The answer is different on a torus, for example.
$endgroup$
– Xander Henderson
Dec 12 '18 at 4:36
3
3
$begingroup$
In other terms, you are asking what is the largest complete graph which is also planar. It is $K_{color{red}{4}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
In other terms, you are asking what is the largest complete graph which is also planar. It is $K_{color{red}{4}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
1
1
$begingroup$
But if you play on the surface of a torus, it is $K_{color{red}{7}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
But if you play on the surface of a torus, it is $K_{color{red}{7}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
Thanks for the help! I'll have to consider whether I want to allow wrapping/toruses.
$endgroup$
– Guest
Dec 12 '18 at 17:41
$begingroup$
Thanks for the help! I'll have to consider whether I want to allow wrapping/toruses.
$endgroup$
– Guest
Dec 12 '18 at 17:41
add a comment |
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$begingroup$
If corners touch, then you can have infinitely many territories (imagine a circle cut by an infinite number of diameters; each wedge is a territory, and they all meet in the center). Otherwise, the four color theorem seems relevant. Of course, this assumes that we are talking about a plane. The answer is different on a torus, for example.
$endgroup$
– Xander Henderson
Dec 12 '18 at 4:36
3
$begingroup$
In other terms, you are asking what is the largest complete graph which is also planar. It is $K_{color{red}{4}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
1
$begingroup$
But if you play on the surface of a torus, it is $K_{color{red}{7}}$.
$endgroup$
– Jack D'Aurizio
Dec 12 '18 at 4:37
$begingroup$
Thanks for the help! I'll have to consider whether I want to allow wrapping/toruses.
$endgroup$
– Guest
Dec 12 '18 at 17:41