Plan Sprinkler System Layout: Minimize # Circles for Max Coverage

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  • Thread starter Thread starter Firewolffzc
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Discussion Overview

The discussion revolves around planning the layout of a sprinkler system to maximize coverage while minimizing the number of sprinklers used. Participants explore geometric shapes that can be inscribed within circles to achieve this goal, considering both theoretical and practical implications.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests using geometric shapes like triangles, squares, or hexagons inscribed within the circle to minimize gaps in coverage.
  • Another participant agrees that hexagons might be the best shape due to their ability to minimize the area between circles.
  • A different viewpoint is presented, indicating that while hexagonal packing may be optimal for covering an infinite plane, irregular packing could be more effective in finite spaces.
  • Concerns are raised about the need to mathematically prove that hexagons provide the smallest area between circles compared to other shapes.

Areas of Agreement / Disagreement

Participants express differing opinions on the optimal geometric shape for sprinkler coverage, with some favoring hexagons while others suggest that irregular packing might be better for finite areas. The discussion remains unresolved regarding the mathematical proof of these claims.

Contextual Notes

Limitations include the lack of specific mathematical methods proposed for proving the area efficiency of different shapes and the dependence on the definitions of coverage and packing in finite versus infinite spaces.

Firewolffzc
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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?
 
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Welcome to PF!

Hi Firewolffzc! Welcome to PF! :wink:

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


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?
 
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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:​
 


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?
 
no, that seems fine, so long as you actually prove …
Firewolffzc said:
… hexagons are the best shape because they make the smallest area between circles.

:wink:
 
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/
 
cool! :cool:
 


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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