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Math Help - Bounds for circles in circle packing

  1. #1
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    Bounds for circles in circle packing

    We would like to pack as many circles of diameter
    d as possible into another large circle of
    diameter
    D. We denote by N(d;D) such number.
    (a) Give the upper bound for
    N(d;D).
    (b) Give in details how we can calculate a lower bound for
    N(d;D). Find an explicit value for

    N
    when d = 9 mm and D = 72 mm.


    ANY IDEAS?!

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  2. #2
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    With a bit of effort you too could have found this site.
    Circle Packing -- from Wolfram MathWorld
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  3. #3
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    eqns

    But, i need equations which gave values on Circle Packing -- from Wolfram MathWorld beacuse part a) deals with variables
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  4. #4
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    Quote Originally Posted by carlosinho View Post
    We would like to pack as many circles of diameter

    d as possible into another large circle of

    diameter D. We denote by N(d;D) such number.
    (a) Give the upper bound for N(d;D).
    (b) Give in details how we can calculate a lower bound for N(d;D). Find an explicit value for

    N
    when d = 9 mm and D = 72 mm.

    ANY IDEAS?!
    Parts a) and b) are so vaguely worded that we can give bounds as follows:

    Clearly if d<D: 1<floor(D/d)<N(d;D)< floor(D^2/d^2)

    but these are not tight.

    RonL
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  5. #5
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    To my knowledge, there is no formula that will compute N(d;D) precisely. However, Eckard Specht has compiled a table of observed packings for N up to 900 (and beyond). From his data, it's clear that D/d varies smoothly as a power function of N.

    There's a javascript calculator at the bottom of Fiber Stuffing Revisited | Saphum that uses Specht's data to estimate N(d;D).

    For D = 72mm and d = 9mm, N = 51.

    -peter
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