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Subject: "circle in circle"     Previous Topic | Next Topic
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Conferences The CTK Exchange This and that Topic #964
Reading Topic #964
ranjitr303
Member since Nov-23-10
Nov-23-10, 10:01 AM (EST)
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"circle in circle"
 
   how many circles of radius r can fit in a bigger circle of radius R ?

ans: let n denote the maximum number of circles that can fit in the bigger circle

n = 180/(sin^-1)(4*(r^2)/(R^2)) (approximately)

where sin^-1 denotes sine inverse
r^2 denote r squared
R^2 denote R squared


IS THE FORMULA CORRECT ?
IF NO PLEASE SEND THE CORRECT ANSWER ?

ranjitr


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  Subject     Author     Message Date     ID  
circle in circle ranjitr303 Nov-23-10 TOP
  RE: circle in circle alexb Nov-23-10 1
  RE: circle in circle jmolokach Nov-23-10 2
     RE: circle in circle jmolokach Nov-23-10 4
         RE: circle in circle alexb Nov-23-10 5
         RE: circle in circle jmolokach Nov-28-10 7
  RE: circle in circle jmolokach Nov-23-10 3
  RE: circle in circle jmolokach Nov-28-10 6

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alexb
Charter Member
2668 posts
Nov-23-10, 10:06 AM (EST)
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1. "RE: circle in circle"
In response to message #0
 
   >IS THE FORMULA CORRECT ?

Honestly? I've no idea. I do not understand what does "approximately" mean. For R = 3r, there seems to be a rather noticeable discrepancy.


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jmolokach
Member since Aug-17-10
Nov-23-10, 12:13 PM (EST)
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2. "RE: circle in circle"
In response to message #0
 
   I got n = 180 / , but this would only work for convenient values of (r,R) where the ratios of r:R are likely not integers. I have a feeling that my formula must be embedded into some greatest integer function? Or perhaps a GIF somewhere embedded in the formula...not sure

Here is a table of values for r, R, n using the formula I have and r = 1 ...

r R n

1 2 2
1 3 6
1 4 9.244412735
1 5 12.43307536
1 6 15.60203165
1 7 18.76159266
1 8 21.91591164
1 9 25.06699928
1 10 28.21595144

I would love to see someone write an applet for this sort of thing...

molokach


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jmolokach
Member since Aug-17-10
Nov-23-10, 12:33 PM (EST)
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4. "RE: circle in circle"
In response to message #2
 
   OK let me try this again since it didn't post correctly..

n = 180 / arcsin (r/(R - r))

That was my original intent and it apparently got left out of the message somehow...

molokach


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alexb
Charter Member
2668 posts
Nov-23-10, 12:35 PM (EST)
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5. "RE: circle in circle"
In response to message #4
 
   Just do not use square brackets. In this software, they play the role of the HTML's angular brackets.


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jmolokach
Member since Aug-17-10
Nov-28-10, 08:54 AM (EST)
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7. "RE: circle in circle"
In response to message #4
 
   Never mind. The formula I gave appears to be incorrect as I conjecture that if R = 5 you can only fit 12 units circles... and according the to wikipedia page I referenced, you should be able to fit 19.

I would be interested in the correct formula. There is yet another table here:

https://mathworld.wolfram.com/CirclePacking.html

...but again no formula is given. Although the nested radicals make me think trig identity somewhere...

Good luck!

molokach


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jmolokach
Member since Aug-17-10
Nov-23-10, 12:13 PM (EST)
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3. "RE: circle in circle"
In response to message #0
 
   It'should also be noted that R must be greater than or equal to 2r in order for the formula to work... any r < R < 2r would give the answer "1" for n I suppose....

So maybe even perhaps the formula should be given piecewise?

molokach


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jmolokach
Member since Aug-17-10
Nov-28-10, 08:54 AM (EST)
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6. "RE: circle in circle"
In response to message #0
 
   Not exactly sure if my formula is correct, but here is a page you might want to check out. It does not give citation or a formula, so beware...

https://en.wikipedia.org/wiki/Packing_problem#Circles_in_circle

I believe the table here is an attempt to make R a function of n where r = 1 (a unit circle). It'seems that it is consistent with my table but I attempted (like you) to solve for n in terms of R (and also r).

molokach


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