Geometry problem

Suppose you try to cover the real plane with non-overlapping disks as follows: first you pack unit disks in as closely as you can (ie hexagonal packing), and then in each subsequent iteration you put in each the largest disk that will fit into each gap left by the previous iteration (ie you put one disk into each gap).

Give a formula (possibly recursive) for the 'proportion' of the plane that is left uncovered after the nth iteration.

Originally posted by Acolyte
Suppose you try to cover the real plane with non-overlapping disks as follows: first you pack unit disks in as closely as you can (ie hexagonal packing), and then in each subsequent iteration you put in each the largest disk that will fit into each gap left by the previous iteration (ie you put one disk into each gap).

Give a formula (possibly recursive) for the 'proportion' of the plane that is left uncovered after the nth iteration.

Hmmm. I've worked out that the proportion after the first iteration (ie when all the disks are the same size) is pi*3^(1/2)/6, but I haven't done any more than that, although I'll see what I can do later.

What motivated this one, by the way?

after the first iteration, an equal precentage of the remaining space will be filled with each itteration, if that's any help (and not blatanly obvious)