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| Set of Spheres (Posted on 2005-11-03) |
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Let A, B, and C be spheres that are tangent pairwise and whose points of tangency are distinct. Let {D1, D2, ..., Dn} be a set of spheres each of which is tangent to spheres A, B, and C. For i = 1 to n, Di is externally tangent to Di+1 (where Dn+1 = D1).
What is the value of n?
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Submitted by Bractals
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Rating: 4.0000 (1 votes)
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Solution:
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Consider the special case where two of the spheres A, B, or C have infinite radii and are therefore parallel planes. The third sphere lies between them and is tangent to both. Clearly, there are six D-spheres that surround this third sphere and are tangent to the parallel planes. Each of these D-spheres is externally tangent to its two neighbors. This "chain" of six spheres can be rotated about the third sphere into an infinite number of positions (think of six golf balls on a table surrounding a seventh).
Every other case can be converted into this special case by inversion about one of the pairwise tangent points and then the six spheres can be inverted back. Since there are three points that we can use for inversion, there are three chains of six D-spheres with an infinite number of positions. Whether these chains are disjoint I am not sure.
If you are familiar with Steiner Chains you know that you can have three or more circles in the chain and in most cases the last circle in the chain is not tangent to the first. For our three dimensional problem the last sphere is always tangent to the first and we always get six spheres.
Do a google on "Soddy Hexlet" to find information on this fascinating subject. Especially the one at members.ozemail.com for a beautiful Java application. |
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