Math Is Fun Forum

sirsosay

Member

Registered: 2005-12-15

Posts: 11

How do you calculate how many spheres can fit in a larger sphere?

There is a competition where you have to "guess" how many gumballs are in the gumball machine.

There gumball machine is spherical so I want to calculate a thoretical amount of gumballs inside.

I wan to keep away from subtracting volume from volume and figure out a more precise calculation that takes into account wasted space.

If someone could help me figure this out that would be great.  Thanks

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: How do you calculate how many spheres can fit in a larger sphere?

Actually your question is very famous and hard. Such questins are known as spere-packing problems.
Here's a start:
http://en.wikipedia.org/wiki/Sphere_packing
If you want more information, i'm sure google will help.

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IPBLE:  Increasing Performance By Lowering Expectations.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: How do you calculate how many spheres can fit in a larger sphere?

In a real world case the sphere packing will be far from ideal - lots of waste space.

Start by working out the volume of the gumball machine.

Then get some gumballs and see how they fill the volume of a bowl or glass.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: How do you calculate how many spheres can fit in a larger sphere?

Hey guys!

Why don't we make it simpler? After all the auther said "guessing" instead of exact number.

If the ball placement can be monotonous when in a large amount, we can calculate the limit of the ratio of occupying space and waste space when the amount goes to "infinity", thus we can give an approximate answer.

I can start with the 2-D case-Circles instead of Balls
Like this pattern:
.....................
...OOOOOOO...
...OOOOOOO...
...OOOOOOO...
......................
Focus at one black circle in the center and its surrounding 6 balls, (in fact all touching) and we can find a sexangle outbound it as its ideal area in the whole  area consumed by balls. Think of a bee-nest's side, and imagine you can place circles in to each cell, and that if your each circle is big enough to tough 6 sides of the cell, they are in the space-economy allignment. You cannot make the circle consume lesser space.

O  O
O ● O
  O O

And the circles area percentage of the whole is just the ratio of area of each circle to that of the cell.

            Pi*r*r
___________________ = 90.7%
       
  6*(1/2*r*(rSec60°))

And if you know the whole area, guage the circled area using the percentage, and divided by each circle's area, you may find an approximate answer for their amount.

How about 3-D case? The ball case? I don't know.

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X'(y-Xβ)=0

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: How do you calculate how many spheres can fit in a larger sphere?

Oh the machine transform my pattern of OOO, forget about it, go to the bee-nest illustration.

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X'(y-Xβ)=0