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Math Help - Non computational proof sought

  1. #1
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    Non computational proof sought

    Suppose you seek the smallest circle that can contain 11 non-overlapping unit disks. The solution to this problem is known, but the curious thing is that there are two different arrangements that give exactly the same radius for the circumscribing circle. Here is a picture of them:

    It is far from obvious to me that the circumscribing circles have exactly the same radius. I was able to prove it using Maple with a tedious brute force computation. It seems to me there should be some insightful geometric proof, possibly involving some tessellation of the plane, but I have yet to find one. Can anyone help?

    There is more information at this link:
    11 circles in a circle

    I have a similar question for 13 circles. There is a link about that at the above site.
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  2. #2
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    Quote Originally Posted by jbuddenh View Post
    Suppose you seek the smallest circle that can contain 11 non-overlapping unit disks. The solution to this problem is known, but the curious thing is that there are two different arrangements that give exactly the same radius for the circumscribing circle. Here is a picture of them:

    It is far from obvious to me that the circumscribing circles have exactly the same radius. I was able to prove it using Maple with a tedious brute force computation. It seems to me there should be some insightful geometric proof, possibly involving some tessellation of the plane, but I have yet to find one. Can anyone help?

    There is more information at this link:
    11 circles in a circle

    I have a similar question for 13 circles. There is a link about that at the above site.
    I don't really understand what you mean here.
    Are you saying the two circumscribing circles are not of the same radius?

    They look the same to me.
    The 10 small circles in both large circles remain or are fixed in their positions. Only the shaded small circle moved.
    First, or in the left large circle, the shaded small circle is to the "right" of the two small circles attached to the shaded one.
    Then, in the right large circle, the shaded small circle is to the "left" of the same two small circles mentioned above.
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  3. #3
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    Thanks for replying, Ticbol. Let me state it slightly differently to make it clearer. In the left hand large circle all the circles that appear to be tangent really are tangent. Let us keep them all fixed except the shaded circle. Now let us move that shaded circle and try to place it as shown it the large right hand circle.

    There are three possiblities

    (1) It doesn't quite fit, but overlaps a tiny bit, maybe only by say 10^(-12), far to little to be detectable in the illustration. In this case the left hand large circle is optimal, and to avoid the the overlap the right hand large circle would have to be made slightly larger.

    (2) It fits, but in fact there is a little slack around it in the right hand figure position. Perhaps the slack is tiny, say 10^(-12). In this case the circles near the shaded circle can be pulled closer a bit and the circumscribing circle made slightly smaller. In this situation the right hand large circle would be optimal, and the left hand arrangement slightly worse.

    (3) It fits exactly, with all apparent tangencies exact so the arrangements are equally optimal with the left and right hand circumscribing circles exactly the same size.

    To me, it is not at all obvious which of these three possibilities is the correct one. But, by a messy computation I was able to prove that possibility (3) is the correct one, and the two arrangements are equally optimal.

    What I am hoping is that there is some geometric argument with a minimum of computation that will allow one to conclude that (3) is the correct alternative and that the arrangements are in fact equally optimal.
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