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Math Help - How many circles of diameter d on average in area of size A?

  1. #1
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    How many circles of diameter d on average in area of size A?

    Hi,

    I have an unlimited large plane. I am trying to calculated how many circles with a certain diameter d can fit into any square of size SxS on the plane. I think this should be the maximum density. So the question is NOT "How many circles can I fit into square of size x?", as it does not matter if the circles overlap the boundaries of the square.

    (The setting is: How do I calculate how many people that try to keep distance d between their cores (modelled as circles) can fit into an area of size 1: What is the maximum density?)

    I have looked into circle packing but that seems to calculate slightly different things.


    I think it's either

    2 / sqrt(3 * d * d)

    or

    2 / (sqrt(3) * d * d).

    Any help (even a search word for google!) would be appreciated!
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: How many circles of diameter d on average in area of size A?

    The densest packing is indeed the circle packing problem, and the ratio of circles to total area is \frac {\pi}{\sqrt {12}}

    Both of your two alternative choices have a problem with dimensions. The first has units of 1/length, and the second 1/length^2, whereas the answer should be dimensionless.
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  3. #3
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    Re: How many circles of diameter d on average in area of size A?

    I don't understand?

    The question is : How many circles with diameter d can fit into an area of 1 x 1.The answer should always contain d?

    circle packing example.bmp

    In this situation, how many circles (so that could be 1 + 1 + 1 ... + .5 + .3 + .7) with diameter d fit into the red square of size 1 x 1?
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