How many circles can be placed inside another circle?

In summary, the conversation discusses the problem of determining the maximum number of circles that can be placed within a larger circle of a certain radius. The solution is based on mathematical principles and involves exploring packing fractions. There are references to websites that discuss similar problems, but it is noted that finding a general solution is difficult and may require extensive trial and error. The conversation also mentions the difficulty of finding proofs for optimization problems, with examples such as packing spheres in 3D and a 1 million x 1 million square.
  • #1
rozan330
2
0
We have a circle of certain radius.How many circles of smaller radius(we are provided with the ratio of radius) can be placed within the larger circle? Help me to determine the least number of the smaller circles that can be filled?? I am trying to generate a generalized solution, and looking for help??
 
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  • #2
I assume you mean the maximal amount, and not the least. Is your general solution supposed to be mathematical or a program?
 
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  • #4
First of all thank you, for showing interest in my problem.

I assume you mean the maximal amount, and not the least.
Yes, i wanted to say the least amount of gap, but it became otherwise.
Is your general solution supposed to be mathematical or a program?
I am working on this problem on mathematical basis and also reviewing the packing fraction concept for this. Till now I have got a solution that gives the certain no. of circles that can be filled but the area of gap unfilled by the circle may exceed the area of 2 or even more circles. Which I wanted to discuss about.


Do you mean something like
http://mathworld.wolfram.com/CirclePacking.html
http://www2.stetson.edu/~efriedma/packing.html
?

Yes I mean to solve something like in the second link. But I wanted a general solution that gives the max no. of circles that can be filled when the radius of two circles are provided.
 
  • #5
At least the link says that you can't write a perfect program, because apparently it takes years of mathematical proofs to find the true optimum.
http://www2.stetson.edu/~efriedma/cirincir/

To write a program is very complicated and you probably can only do it by extensive trial.
 
  • #6
Gerenuk said:
At least the link says that you can't write a perfect program, because apparently it takes years of mathematical proofs to find the true optimum.
http://www2.stetson.edu/~efriedma/cirincir/

To write a program is very complicated and you probably can only do it by extensive trial.

Quite surprising fact.
If it takes mathematicians years to find out solution for a even a particular case then there is no way that a general solution would come plain and simple.
 
  • #7
Of course for very large ratios (many circles), the solution is mostly hexagonal packing. Maybe there is even a limit ratio from which on there is an algorithm which finds optimal packing by putting a large chunk of hexagonal packed circles and filling up the gaps.

I find the circle in square packing interesting, because you see transitions between hexagonal packing and square packing.

Quite often actually proofs of optimization problems are quite difficult. For example in 3D the single case of packing sphere in an infinite space seems to have a natural solution, but its hard to proof that this is really the best:
http://mathworld.wolfram.com/KeplerConjecture.html
After 400 years the proof is still not 100% checked.
 
  • #8
Gerenuk said:
Of course for very large ratios (many circles), the solution is mostly hexagonal packing. Maybe there is even a limit ratio from which on there is an algorithm which finds optimal packing by putting a large chunk of hexagonal packed circles and filling up the gaps.

I don't know if it will be that simple. On one hand, the hexagonal packing is scary-efficient; it's hard to imagine 'improving' on it much. On the other hand, the square packing is even more obvious and hard to improve on, yet I remember an example (sorry, no link!) along these lines: take a 1 000 000 x 1 000 000 square packed with unit squares; clearly the best solution is 1 000 000 000 000 squares lined up the usual way. Now expand the square to size 1000000.01. By tilting squares appropriately, (thousands) of new squares can be accommodated.

I wish I had the actual example... I know nothing of packing problems myself.
 
  • #9
CRGreathouse said:
take a 1 000 000 x 1 000 000 square packed with unit squares; clearly the best solution is 1 000 000 000 000 squares lined up the usual way. Now expand the square to size 1000000.01.
Well no, that's the point. You can use a hexagonal packing to fit in more circles comfortably. There will be 999999 circles horizontally, but there will be 1154700 rows!
I hope I didn't do a mistake, but it seems fairly logical that the boundary mismatch is not enough to make up for the hexagonal tight packing?!
 

1. How do you determine the maximum number of circles that can fit inside another circle?

The maximum number of circles that can fit inside another circle is determined by dividing the area of the larger circle by the area of the smaller circle and rounding down to the nearest whole number. This is known as the "packing density" and can be calculated using the formula n = π/√3, where n is the number of circles that can fit inside the larger circle.

2. Can any size circle fit inside another circle?

No, not all sizes of circles can fit inside another circle. The size of the smaller circle must be less than half the diameter of the larger circle in order for it to fit inside. If the smaller circle is larger than half the diameter, it will overlap with other circles and therefore cannot fit.

3. What is the relationship between the size of the circles and the number of circles that can fit inside?

The bigger the circles, the fewer can fit inside another circle. As the size of the circles increases, the number of circles that can fit inside decreases. This is because as the circles get larger, there is less space for them to fit without overlapping.

4. Is there a limit to the number of circles that can fit inside another circle?

Technically, there is no limit to the number of circles that can fit inside another circle. However, as the size of the circles decreases, the number of circles that can fit inside increases, approaching infinity. In practical terms, there is a limit to the number of circles that can fit inside a certain size of circle due to the limitations of physical space.

5. Can the number of circles that fit inside another circle be calculated for non-circular shapes?

Yes, the number of circles that can fit inside another shape can be calculated for any shape as long as the shape has a known area and the circles are all the same size. The formula for calculating the maximum number of circles that can fit inside a non-circular shape is n = A/πr^2, where n is the number of circles, A is the area of the shape, and r is the radius of the circle.

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