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 26,041
    Science Advisor
    Homework Helper

    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 26,041
    Science Advisor
    Homework Helper

    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 26,041
    Science Advisor
    Homework Helper

    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 26,041
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?