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September 12th, 2009, 11:12 PM   #1
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Finding the distances between 2 points sphere.

The problem is as follow:

"We pick 9 random points on a sphere of radius 1. Prove that there exist two points whose distance between them is no more than sqrt(2)."

Note: the distance wanted is not the arc length but the magnitude of a straight line connecting a pair of point.
In tackling this problem, I was able to prove the above case for 2D circle but not sphere (with just 4 points would be sufficient to yield a distance of no more than sqrt(2) for a circle). I am still working and trying to expand the 2D prove to accommodate for the 3D sphere, however I have made no progress after spending a few hours on it. Perhaps there are other more effective method of approach. Any help would be greatly appreciated.
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September 13th, 2009, 02:00 AM   #2
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Re: Finding the distances between 2 points sphere.

Start with the pigeonhole principle:
Center a sphere of radius 1 at the origin.
Since there are 9 points, you must have a set of (at least) 5 with x coordinate >= 0 OR you must have a set of (at least) 5 with x coordinate < 0.
If you repeat the process with y and z, you can find a segment of the sphere which must contain (at least) 2 points and proceed from there.
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September 13th, 2009, 11:24 AM   #3
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Re: Finding the distances between 2 points sphere.

Initially, I imagined instead that the surface area of the sphere is being evenly divided by the total number of point n (i.e. the sphere with radius 1 has an area of 4pi, divided by the number of points there are, say, n =2, then 4pi/2 = 2pi; which means that two points on the sphere are 2pi (m^2) area away from each other). But that leaves me with a problem more or less similar to he pigeonhole approach: how do you determine the exact (x,y,z) position vectors of the pair of point on a sphere (and the angle between the two vectors)? Because only with the knowledge of their coordinate value can I then apply the distance formula.
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September 13th, 2009, 05:31 PM   #4
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Re: Finding the distances between 2 points sphere.

With the pigeonhole principle, you can isolate two points to the same octant of space. So, for instance, they both lie such that they have all three coordinates >= 0.
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September 14th, 2009, 12:58 PM   #5
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Re: Finding the distances between 2 points sphere.

got it, thanks man!
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