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 September 12th, 2009, 10:12 PM #1 Newbie   Joined: Jan 2009 Posts: 14 Thanks: 0 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. September 13th, 2009, 01:00 AM #2 Senior Member   Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0 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. September 13th, 2009, 10:24 AM #3 Newbie   Joined: Jan 2009 Posts: 14 Thanks: 0 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. September 13th, 2009, 04:31 PM #4 Senior Member   Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0 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. September 14th, 2009, 11:58 AM #5 Newbie   Joined: Jan 2009 Posts: 14 Thanks: 0 Re: Finding the distances between 2 points sphere. got it, thanks man! Tags distances, finding, points, sphere Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jokerthief Algebra 4 November 7th, 2018 10:10 PM u0362565 Algebra 4 December 16th, 2013 03:22 PM ido596 Abstract Algebra 3 June 13th, 2012 11:01 PM Philipe Algebra 3 June 10th, 2009 12:08 PM julien Math Events 3 August 6th, 2007 05:21 PM

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