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   7 points in a plane
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   Author  Topic: 7 points in a plane  (Read 1933 times)
thedkl
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7 points in a plane  
« on: Mar 13th, 2007, 4:46am »
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draw 7 points on a 2d plane such that for every 3 of them there is a pair of a 1 unit distance
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towr
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Re: 7 points in a plane  
« Reply #1 on: Mar 13th, 2007, 5:49am »
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I half suspect there isn't a solution.
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Grimbal
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Re: 7 points in a plane  
« Reply #2 on: Mar 13th, 2007, 7:14am »
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I also half suspect.  Does it make it certain?  Wink
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towr
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Re: 7 points in a plane  
« Reply #3 on: Mar 13th, 2007, 7:49am »
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I just found a solution.
 
Say a connects to b,c,d,e; b and c connect to f; d and e to g; and f and g are connected. [edit]Also b connect to c, and d connects to e.[/edit] (Where a connection means there's a distance of 1 between them)  
 
There may be more solutions, but this was enough work.
« Last Edit: Mar 13th, 2007, 9:40am by towr » IP Logged

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Re: 7 points in a plane  
« Reply #4 on: Mar 13th, 2007, 7:57am »
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Damn, you are right.  Congratulations!
 
Note that I haven't changed my mind ... I still think you are right.   Roll Eyes
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rmsgrey
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Re: 7 points in a plane  
« Reply #5 on: Mar 13th, 2007, 8:33am »
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on Mar 13th, 2007, 7:49am, towr wrote:
I just found a solution.
 
Say a connects to b,c,d,e; b and c connect to f; d and e to g; and f and g are connected. (Where a connection means there's a distance of 1 between them)
 
There may be more solutions, but this was enough work.

d,e,f?
b,c,g?
Any three of b,c,d,e?
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SMQ
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Re: 7 points in a plane  
« Reply #6 on: Mar 13th, 2007, 8:35am »
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I came up with the same solution (all colored links are unit length):
 

 
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rmsgrey
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Re: 7 points in a plane  
« Reply #7 on: Mar 13th, 2007, 8:50am »
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Ah, the solution was incompletely specified...
 
There's (at least one) configuration that has only the links towr mentioned unit length, and not the additional pair of links...
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Re: 7 points in a plane  
« Reply #8 on: Mar 13th, 2007, 8:51am »
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on Mar 13th, 2007, 8:33am, rmsgrey wrote:

d,e,f?
b,c,g?
Any three of b,c,d,e?

Are you interpreting tha problem statement the same way as the rest of us?  I read it as: plot 7 points in a plane such that for any three chosen points, at least one pair of points is exactly a unit distance apart.
 
Edit: Ah, I see, towr forgot to mention the b-c and d-e links...
 
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« Last Edit: Mar 13th, 2007, 8:54am by SMQ » IP Logged

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towr
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Re: 7 points in a plane  
« Reply #9 on: Mar 13th, 2007, 9:38am »
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on Mar 13th, 2007, 8:51am, SMQ wrote:
Edit: Ah, I see, towr forgot to mention the b-c and d-e links...
Indeed. I'll go edit them in..
 
Are there other solutions? I had some 1100 possible solutions left, when I tried a random one to see if it was an actual solution.
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Re: 7 points in a plane  
« Reply #10 on: Mar 13th, 2007, 2:27pm »
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This graph is called the Moser spindle; it cannot be 3-colored, and so it is impossible to color the points of R2 with 3 colors such that no two points a unit distance apart are the same color.  It can, however, be done with 7 colors, so the chromatic number of the plane is somewhere between 4 and 7, inclusive.  (Even though these bounds are easy to prove, they are still the best known after more than 50 years!)
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Altamira_64
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Re: 7 points in a plane  
« Reply #11 on: Apr 12th, 2017, 6:31am »
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How about 8 points? Is there a solution?
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