Saturday, 29 January 2011

Mathematical Problems 4B: Close-packed spheres

In a previous post, I looked at close-packed circles - arranging circles hexagonally and calculating how much of the available area they fill.  That was fairly straightforward, and gave me an idea on how to calculate the volume occupancy of close-packed spheres.  Put it another way - how many Maltesers could I fit in a box if I could fill it to capacity (minimum spare volume left over)?

Let's start by putting them in a square arrangement - so that they're all in columns and rows.  How much space will they fill?  The easiest way to solve this is to think of one sphere inside its cube-shaped box.

The volume of a sphere S = 4/3 π r3
And the volume of the cube it sits in is 2r x 2r x 2r (since the cube has to be twice the radius of the sphere in height, width and depth)  C = 8 r3

Therefore, the ratio of the two volumes is S/C which is 4 π / 24 (notice that the r cancels - it doesn't matter how big the sphere is) and this is equal to 52.36% - only half of the available volume.

Next, let's look at hexagonal packing - arranging the circles so that they form hexagon patterns, instead of squares.

If we consider just one of the spheres, enclosed in a regular hexagonal prism, then we have this arrangement:

The volume of a sphere is...

And the volume of the hexagonal prism it occupies is found by multiplying the area of the base by the height.  The height is 2r (it's twice the radius of the sphere in height) and we can look at the base as being made up of six equilateral triangles...

So the volume of the hexagon, H, is 

And now that we have the volume of the hexagonal prism, H, and the volume of the sphere, S, we can work out how much of the prism is being filled by the sphere.

So, if we want to maximise the number of Maltesers in a box of chocolates, it makes sense to arrange them hexagonally, and not cubically.  80% volume coverage, compared to just 52% for the cubic arrangement, is definitely worth having!

Hexagonally arranged Maltesers (a Christmas present!)

On a more theoretical note, science textbooks regularly quote that close-packed spheres fill 80% of their volume, but I've never noticed any of them prove it.  So, this blog post counts as closure from a figure that's been drifting around since my A-level chemistry days, and which has recurred frequently since then.  None of the books seemed bothered enough to spend time on it - perhaps it's too much like maths and not enough like chemistry!

Next time - something different.  It might be projectiles, or escape velocity (conveniently related to each other) or it might be something about chemistry - in which case it'll be a rant (consider this advance notice!).

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