There is a lamp in space.(Consider lamp a point) Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)
Problem
Source: Iran 2003
Tags: geometry, 3D geometry, sphere, prism, combinatorics proposed, combinatorics
26.08.2004 09:02
i post 4 month ago but no one gave a soln
26.08.2004 21:37
Maybe it's time to post a solution? In plane, I think it's not true, but it space it seems true. What about other dimensions? I mean , I know a problem like this: Given a system of a) Circles b) Spheres is it always possible to move far away from the system one of the pieces ?
26.08.2004 22:06
Well, it's clear that we can block a lamp in the plane with circles, but the spacial analog seems really tricky.
14.09.2004 09:11
Well consider all lattice points in 2n∗2n∗2n cube with center the lamp. Now consider a sphere of radius 13 of center of each lattice point. Now use Minkowski theorem for big n
14.09.2004 12:09
what's Minovski theorem for big n?
16.09.2004 12:21
If A is a convex set of points in space with volume more than 8 and if (x,y,z)∈A then (−x,−y,−z)∈A then A has a lattice point except (0,0,0)
17.09.2004 01:12
Now that you said, I remember a solution.... The great idea is to take all the lattice points, then build circles of equal radiuses R in those latice points and with Minkowski you show that for a quite big radius , at the distance 1R there will be no light. Am I right? Or maybe left? Just kidding ....
05.10.2010 02:06
Sorry to bump such an old problem but I have a comment here. I encountered this problem(labeled as Iran 2003) in the form: Prove that there does not exist a finite set of spheres of equal radius such that no two intersect and that the ray never goes to infinity. I have been trying to solve this problem for a while with no success. Does anyone have a solution?
05.10.2010 07:42
Without using Minkowski, it's enough to use compactness. The claim can be also generalized to Rd, for any finite d. Place the light in the origin, and place equal open d-discs of any radius r<1/2 centered at every lattice point except the origin. Then put a (d−1)-sphere S of the same radius r centered at the origin, and consider the projections of all discs on S. Every disc projects to an open subset of S, while the projections of all lattice points on S form a dense subset of S. Therefore the projections form a cover of S and, by compactness of S, there's a finite subset of projections which still covers it. Keep the corresponding discs, and remove all the others. Now you're blocking every ray of light with a finite number of arbitrarily small equal discs in Rd. @RaduSorici: either the above proof is flawed, or you misread the problem. Maybe there were more constraints on the placement of your spheres?
05.10.2010 13:14
MindFlyer wrote: ... the projections of all lattice points on S form a dense subset of S. Therefore the projections form a cover of S and ... Take the interval [0,1]; enumerate the rationals in it by (rn)n≥1. Choose ε>0 and open intervals In=(rn−ε/2n+1,rn+ε/2n+1). Then the union of the In has measure at most ε, so it does not cover [0,1]. Thus just qualitative density and compactness arguments are not enough.
05.10.2010 15:12
Oh right, I don't know why I said that. However the proof can be fixed quite straightforwardly. Any straight line through the origin passes arbitrarly close to some other lattice point, so the projections of the disks indeed cover the sphere.
06.10.2010 13:52
A similar problem appears in Engel and he claims six sphere is sufficient. Consider a rectangular prism with three different sized faces and the lamp is the center point of the prism. Then consider 6 infinite cones that orgininate at the center of the prism and each contain one face of the prism. Then these cones intersect and cover the light source. We can inscribe spheres into these cones. If the spheres intersect then we can vary the dimensions of the prism so that they no longer intersect.
01.06.2019 15:08
Any complete solution to this problem? I am unable to understand the application of Minkowski's theorem mentioned in one of the posts above.Also, the initial constraint includes the fact that no two spheres touch each other. I am pretty sure that the problem can be translated in terms of circles instead of spheres and the plane R2. Any complete solutions using Minkowski's theorem??