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ecoist
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6 points in a unit circle
« on: Sep 17th, 2007, 2:58pm » |
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Show that, given any 6 points inside a circle of radius 1, some two of the given points are at most 1 unit apart.
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towr
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Re: 6 points in a unit circle
« Reply #1 on: Sep 17th, 2007, 3:22pm » |
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[edit]misread the problem[/edit]
Evenly spaced they're all 1 unit apart. So there isn't enough room for them all to be further apart.
If any are further apart, the total distance going from one neighbour to the next around the circle back to the first even goes down
Hmmm, you probably want something more rigorous. I wonder if holy pigeons have anything to do with it..
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| « Last Edit: Sep 18th, 2007, 12:16am by towr » |
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ecoist
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Re: 6 points in a unit circle
« Reply #2 on: Sep 17th, 2007, 4:50pm » |
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I never thought that this problem was obvious, yet many people believe that it is obvious. I have seen three different rigorous proofs.
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Aryabhatta
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Re: 6 points in a unit circle
« Reply #3 on: Sep 17th, 2007, 8:28pm » |
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on Sep 17th, 2007, 4:50pm, ecoist wrote:| I never thought that this problem was obvious, yet many people believe that it is obvious. I have seen three different rigorous proofs. |
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To be fair, I am pretty sure towr is not claiming that this is obvious.
And I agree with you, many people think things are obvious when they actually are not. For instance, the Jordan Curve Theorem.
omigod. Do i sound like srn347?
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| « Last Edit: Sep 17th, 2007, 8:29pm by Aryabhatta » |
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ecoist
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Re: 6 points in a unit circle
« Reply #4 on: Sep 17th, 2007, 9:32pm » |
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Sorry, left the wrong impression. towr clearly sees that the problem is not obvious. Others thought the problem was obvious, but I couldn't convince them otherwise. And no, Aryabhatta, I doubt if even the ventriloquist who won "America's Got Talent", could sound like srn347!
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towr
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Re: 6 points in a unit circle
« Reply #5 on: Sep 18th, 2007, 12:15am » |
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Oh wait, it says in the circle (on the disk), not on the circle (circumference).
I need to read better.
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mikedagr8
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Re: 6 points in a unit circle
« Reply #6 on: Sep 18th, 2007, 12:32am » |
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Seems like I should be able to do this...
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Grimbal
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Re: 6 points in a unit circle
« Reply #7 on: Sep 18th, 2007, 6:47am » |
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You could consider the radius (line from the center to the border) where each point is and prove that if the distance between 2 points is >=1, the angle between their radii is >=pi/3.
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rmsgrey
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Re: 6 points in a unit circle
« Reply #8 on: Sep 18th, 2007, 8:23am » |
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How about:
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If any of the points are the centre, then the rest of the circle is at most 1 away from it, so done.
Otherwise, pick any of the 6 points and draw the diameter through it, and the two diameters Pi/3 radians away from it, dividing the circle into 6 equal wedges.
Each wedge has the property that any two points in it are at most 1 apart, so if any wedge has two points in, we're done.
But our original point is on the boundary between two wedges (by construction) so counts as being "in" both of them, leaving only 4 other wedges and 5 other points, so at least one wedge must have two points in, so we're done. |
Not sure how obvious that was.
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ecoist
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Re: 6 points in a unit circle
« Reply #9 on: Sep 18th, 2007, 8:44am » |
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Don't know about others, but I would like a proof of the obvious(?)
Each wedge has the property that any two points in it are at most 1 apart.
Omigod! Am I sounding like srn347?
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Grimbal
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Re: 6 points in a unit circle
« Reply #10 on: Sep 18th, 2007, 9:08am » |
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A wedge fits in a Reuleaux Triangle which has constant width 1. Obviously it cannot contain a segment of length>1.
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| « Last Edit: Sep 18th, 2007, 9:08am by Grimbal » |
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Joe Fendel
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Re: 6 points in a unit circle
« Reply #11 on: Sep 18th, 2007, 9:10am » |
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Strikes me that this is equivalent to the question of asking how many non-overlapping disks of radius 0.5 we can fit in the disk of radius 1.5.
Any mileage we can get from that formulation?
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towr
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Re: 6 points in a unit circle
« Reply #12 on: Sep 18th, 2007, 9:28am » |
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on Sep 18th, 2007, 9:10am, joefendel wrote:Strikes me that this is equivalent to the question of asking how many non-overlapping disks of radius 0.5 we can fit in the disk of radius 1.5.
Any mileage we can get from that formulation? |
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You could put 7 such .5 radius circles in the 1.5 radius disc. However, not without them touching (having a non-zero distance between them).
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Michael Dagg
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Re: 6 points in a unit circle
« Reply #13 on: Sep 18th, 2007, 11:02am » |
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Indeed so, otherwise you could place six discs of radius r > 1/2
inside a disc of radius 1+r; equivalently, you could place six unit discs
inside a disk of radius s = (1+r)/r < 3 . Well, you can indeed
place not just six but seven unit disks inside such a disk when s = 3 ;
that's the "kissing pennies" arrangement. But it's a very tight arrangement:
increase the radius just a little and not only do the six pennies around
the outside crash into each other, but they crash into the center one too!
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Regards,
Michael Dagg
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JP05
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Re: 6 points in a unit circle
« Reply #14 on: Sep 18th, 2007, 6:26pm » |
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Isn't this just a circle packing problem?
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