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:

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,836
251
Welcome to PF!

Hi Firewolffzc! Welcome to PF! :wink:

Tell us what you think, and then we'll comment! :smile:
 
  • #3


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:
  • #4
tiny-tim
Science Advisor
Homework Helper
25,836
251
yup, that seems a good idea!

i'd guess it's the hexagon, but you'd better do the maths! :wink:
 
  • #5


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?
 
  • #6
tiny-tim
Science Advisor
Homework Helper
25,836
251
no, that seems fine, so long as you actually prove
… hexagons are the best shape because they make the smallest area between circles.

:wink:
 
  • #7
2,034
305


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/
 
  • #8
tiny-tim
Science Advisor
Homework Helper
25,836
251
cool! :cool:
 
  • #9


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.
 

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

  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
1
Views
8K
Replies
2
Views
2K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
0
Views
3K
Replies
2
Views
2K
Top