How to place circles that there will be no gap. Use least number of circles possible.

  1. I have to plan the layout of a sprinkler system. Basically, each sprinkler shoots a radius of 7.5 feet water, and I want every part of the floor covered with water. How can I use the least number of sprinklers?
     
    Last edited: Apr 21, 2012
  2. jcsd
  3. tiny-tim

    tiny-tim 26,054
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    Welcome to PF!

    Hi Firewolffzc! Welcome to PF! :wink:

    Tell us what you think, and then we'll comment! :smile:
     
  4. Re: How to place circles that there will be no gap. Use least number of circles possi

    Well, I think I figured one way to approach. Is it better to imagine not the circle, but a triangle, square or hexagon inscribed within the circle, because these shapes can link without gaps. Then once done with these shapes, I draw a circle circumscribing the shape. I was wondering if this was an effective method. If it is, which shape is best?
     
    Last edited: Apr 21, 2012
  5. tiny-tim

    tiny-tim 26,054
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    yup, that seems a good idea!

    i'd guess it's the hexagon, but you'd better do the maths! :wink:
     
  6. Re: How to place circles that there will be no gap. Use least number of circles possi

    Well, see, I don't know what math to do! I figured out that the reason we can use triangles, squares and hexagons is because the measurement of their angles are multiples of 360. (triangle)60*6=360, (square)90*4=360, (hexagon)120*3= 360. Well, it is not possible to have a shape with a 180 or 360 degrees angle, therefore hexagons are the best shape because they make the smallest area between circles. Wow, I can't believe I figured that out myself. Anyway, is there more to it or is this the final solution?
     
  7. tiny-tim

    tiny-tim 26,054
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    no, that seems fine, so long as you actually prove
    :wink:
     
  8. Re: How to place circles that there will be no gap. Use least number of circles possi

    A hexagonal packing might be the best for covering the entire plane (not sure), but for finite size rooms, an irregular packing can do better

    This is a page with the best known results for covering a square with circles

    http://www2.stetson.edu/~efriedma/circovsqu/
     
  9. tiny-tim

    tiny-tim 26,054
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    cool! :cool:
     
  10. Re: How to place circles that there will be no gap. Use least number of circles possi

    Thanks for all your help. Btw, I don't know how to prove something like the area formed between circles circumscribing hexagons is the smallest compared to triangles and squares.
     
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