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CTK Exchange
ranjitr303
Member since Nov-23-10
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Nov-23-10, 10:01 AM (EST) |
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"circle in circle"
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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 ?
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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
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>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
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Nov-23-10, 12:13 PM (EST) |
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2. "RE: circle in circle"
In response to message #0
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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...
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jmolokach
Member since Aug-17-10
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Nov-23-10, 12:33 PM (EST) |
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4. "RE: circle in circle"
In response to message #2
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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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jmolokach
Member since Aug-17-10
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Nov-28-10, 08:54 AM (EST) |
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7. "RE: circle in circle"
In response to message #4
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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
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Nov-23-10, 12:13 PM (EST) |
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3. "RE: circle in circle"
In response to message #0
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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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