Because I obviously don't have useful things to do with my time, this idea popped into my head the other day, and I've been puzzling over it.

Start with a marble, any size, and build a box around it (in this case it would be a perfect cube) that touches the marble on all six sides. That marble fills a bit over 52% of the volume of the cube.

Now build the smallest rectangular box that will hold two marbles so the marbles touch each other and all sides of the box are touched by at least one of them. I guess each marble would touch five sides of the box.

Repeat this for larger and larger boxes, always putting the maximum number of marbles that will fit in the box, stacking them vertically as necessary.

Here's the question: Would the volume of the marbles always be 52% of the volume of the box? Would there be an optimum shape for the box to maximize the relative volume of the marbles? How about non-rectangular boxes, i.e. spherical or cylindrical?

If this were a two-dimensional matrix (that is, the marbles all in one plane, not stacked vertically) I have the math to figure it out. But adding that vertical stacking makes things complicated when it comes to calculating the volume of the air gaps as the marbles nest into each other.

Hmmm... I wonder if the Nobel Committee has a prize for geometry?

tanstaafl.

This is a very old problem (called sphere packing) that's been studied (at least) since the late 1500's. If you have some novel solution for higher-order dimensional packing, you could very much be in the running for a Nobel prize -- this packing problem has all kinds of interesting applications from communications to big data.

Even the significantly simpler 2D equivalent, packing circles into a circle, is surprisingly subtle.

Peter