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September 27th, 2009, 12:09 AM   #1
 
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how many apples

I was wondering if it is somehow possible to mathematically calculate how many apples can you fit in a 1m squared box.
I study maths in uni but we never actually tried solving real problems like this.. so I dont really know how to approach such a problem.
First of all the first problem that comes in mind, is how do i actually calculate the exact dimensions of an apple, because it is not circle to say the least.
Secondly, What is the best way to organize the apples in a box so I can fit the most apples inside and there will be the least number of "empty" spaces.
Then thirdly, how do i calculate how many apples are there?
(Lets say that all apples are the same size and are at an average size).
Does anyone have a clue?
Thanks, Jorden.
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September 27th, 2009, 05:17 AM   #2
 
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Re: how many apples

Start off with using some known smaller dimensions [You will need to define the size of the apple], and do so in 2 dimensions first [circles filling a square or rectangle.] Look at the case when they are aligned vertically and when aligned on an angle [slip one row over and down to allow one to sit between the two below.] If you do it enough times you will see a pattern you can generalise.

Now [and only after the first experiments] look at the problem in 3D. You will have spheres sitting one on the other vertically, then in the second case shifted over and down to have one sit within three at its base. Again try looking at smaller specific examples and build a pattern to allow you to generalise.

In the first, 2D case you will be looking at the properties of the equilateral triangle and the square as you consider the orientation of the centers of the circles, figuring area of circle and enclosing spaces [difference of square and sectors.] In the second, 3D case, you will be looking at the properties of the tetrahedron and the cube. You might study those first if you still don't have a clue.

One clue is that, after you do the study, you will see that the second case in each study will allow for slightly more objects to fill the same space, as generally more space is gained by the shift than is lost. I have no suitable drawing program for the iMac yet or I'd send an image to clarify.
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