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Topic: 7 points in a plane (Read 3292 times) |
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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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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? 
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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.
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| « Last Edit: Mar 13th, 2007, 9:40am by towr » |
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Grimbal
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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. 
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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. |
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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):
--SMQ
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--SMQ
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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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SMQ
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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? |
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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...
--SMQ
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| « Last Edit: Mar 13th, 2007, 8:54am by SMQ » |
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--SMQ
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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... |
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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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Eigenray
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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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