Calculating Packing Ratios for 3-Dimensional Objects

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Raynuva

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Hello,

This is a personal project I am working on. This is not homework in any form or fashion. I am attempting to calculate a theoretical maximum of objects that can fit within a perfect Sphere. The only thing that I can find that is even remotely close is the calculation for determining how many perfect Circles can fit within a larger perfect circle: (Larger Radius / Smaller Radius) ^ 2 x .9069 ± 1%.

My specific problem is this: How many 3 meter tall, 0.5 meter radius Cylinders can fit within a 50 meter radius Sphere?

On a cursory calculation, the Sphere has a volume of 523,599.7756 meters cubed, and the Cylinder has a volume of 2.3562 meters cubed, and simple division leads to (a little surprising and humorous) 222,222 Cylinders fitting within the sphere.

The problem I am running into is the Cylinders need to maintain their own unique shape, so a Packing Ratio is required to account for this. How would I even begin to approach this calculation to get closer to an answer?

And, as a side question, the Circles-within-a-Circle calculation accounts for margin of error due to different packing methods with the final ±1% in the equation. What would be the margin of error for a calculation of this nature?
 
Consider slices through the sphere, formed with parallel planes that are 3m apart...

slice.png

If you look at an individual slice it can be approximated as a cylinder with 3m depth into which the smaller cylinders can be packed. This enables you to use your "calculation for determining how many perfect Circles can fit within a larger perfect circle" for each slice. Obviously you need to use the smaller radius for each slice.

I'm not saying that this will be the optimal packing strategy, but it will provide a decent lower-bound estimate. (There will be two 1m "caps" left at both ends if you divide the sphere into 16 slices of 3m width=48m. It should be possible to pack some "lying down" cylinders into those shapes!)

EDIT: Whoops, I confused radius with diameter in that slice/cap calculation. But you probably get the idea!
 
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Thanks for the advice on this, Cubist. I do want to ask two questions as a follow-up.

A) Given that the sphere is 3-Dimensional in nature, how would I calculate the smaller "Circle" of the slices of the circle? I'm long out of any mathematics classes for this to be readily available to me.

B) Similar to the above question, but how would I calculate the area of the dome-shaped caps in which I could pack further cylinders?
 
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