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







share|cite|improve this question











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  • 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


















0












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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.
    $endgroup$
    – 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
















0












0








0


1



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 4:28







Guest

















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
















  • 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












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