Plan Sprinkler System Layout: Minimize # Circles for Max Coverage

[Firewolffzc]

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?

 

[tiny-tim]

Welcome to PF!

Hi Firewolffzc! Welcome to PF!

Tell us what you think, and then we'll comment!

 

[Firewolffzc]

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?

 

[tiny-tim]

yup, that seems a good idea!

i'd guess it's the hexagon, but you'd better do the maths! ​

 

[Firewolffzc]

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?

 

[tiny-tim]

no, that seems fine, so long as you actually prove …

… hexagons are the best shape because they make the smallest area between circles.

 

[willem2]

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/

 

[tiny-tim]

cool!

 

[Firewolffzc]

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.