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Math Help - calculate the number of best fit golden rectangles in a given area

  1. #1
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    calculate the number of best fit golden rectangles in a given area

    Hello all :-)

    Here's an interesting one that is not as simple as it seems at first.

    Lets say my daughter has a piece of paper. It has a height (H) and width (W).

    She wants to draw an arial view of a number of farmers fields (N). The thing is she's been learning about the golden ratio in art, and wants all of the fields such that they are Golden Rectangles. All of the fields are the same size and have the width (A) and Length (B). She also wants a gap (G) between the fields so the farmer can walk around them without getting amongst the crops and livestock. She also wants the same gap around the perimeter of the farm plot.

    The question is this. For a given number of fields (N), what is the formula to calculate the individual field length (B) and width (A) expressed in terms of the Gap (G), paper height (H) and paper width (W) assuming that all fields are Golden Rectangles, and that there is also a gap equal to (G) all the way round the perimeter of the farm?

    Have fun
    DK

    Approved by CB
    Last edited by CaptainBlack; November 7th 2011 at 08:21 PM.
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  2. #2
    Grand Panjandrum
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    Re: calculate the number of best fit golden rectangles in a given area

    Quote Originally Posted by daviskites View Post
    Hello all :-)

    Here's an interesting one that is not as simple as it seems at first.

    Lets say my daughter has a piece of paper. It has a height (H) and width (W).

    She wants to draw an arial view of a number of farmers fields (N). The thing is she's been learning about the golden ratio in art, and wants all of the fields such that they are Golden Rectangles. All of the fields are the same size and have the width (A) and Length (B). She also wants a gap (G) between the fields so the farmer can walk around them without getting amongst the crops and livestock. She also wants the same gap around the perimeter of the farm plot.

    The question is this. For a given number of fields (N), what is the formula to calculate the individual field length (B) and width (A) expressed in terms of the Gap (G), paper height (H) and paper width (W) assuming that all fields are Golden Rectangles, and that there is also a gap equal to (G) all the way round the perimeter of the farm?

    Have fun
    DK
    Is this a problem for which you want a solution or a challenge to other members of MHF?

    CB
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  3. #3
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    Re: calculate the number of best fit golden rectangles in a given area

    It's a problem I was pondering, and came up with a near solution. So I guess it's a challenge. I am new to the forum - are challenges OK?
    Thank you
    DK
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  4. #4
    Grand Panjandrum
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    Re: calculate the number of best fit golden rectangles in a given area

    Quote Originally Posted by daviskites View Post
    It's a problem I was pondering, and came up with a near solution. So I guess it's a challenge. I am new to the forum - are challenges OK?
    Thank you
    DK
    They should go in the challenges area, but first you should seek approval from the moderators before posting them there.

    CB
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  5. #5
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    Re: calculate the number of best fit golden rectangles in a given area

    came up with a near solution.



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