Sphere packing (well . . . circle packing)

Got home late last night from Boston and we all slept in.  Had a fun probability problem planned but a few other morning activities didn’t leave us with time for today’s Family Math project, so I switched gears.

The problem of finding the most efficient way to pack spheres was only solved recently.  It is one of those “really easy to understand, but super hard” math problems that I think lots of folks will find interesting.  I actually don’t know much about it, sadly, but thought the kids would have fun playing around with a simplified version of the problem.

First, though, if you want to see the history of the problem Wikipedia’s Sphere Packing page is a good starting point:

Next we tried to find some other ways to pack the circles.  My younger son thought about the idea of packing the circles in a line.   I’d not thought about that idea so I was happy to hear it.  My older son noticed that you could chop up our existing packing into three rectangles to make the linear shape, so the area of this new linear shape would be the same.    I was pretty excited to see this geometric reasoning rather than having to figure out how to fit the 9 disc in a row on camera!

After considering packing the circles in a line we looked for a new shape.   There is a non-symmetric way to pack the circles that minimizes the “wasted” area on the inside, but pushes some of that waste to the outside.  For our example with 9 circles, the two packing shapes had really similar areas – 625 square inches before vs 638 square inches now.  I wish I had 16 of these golf discs to check these two arrangements for the next larger square!

So, a hastily put together project but hopefully an exciting one.  I like showing the boys easy to understand unsolved (or recently solved) problems from pure math.  This one has the extra nice advantage of actually being able to hold it in our hand!  Fun morning.