Posers and Puzzles

Posers and Puzzles

  1. DonationAcolyte
    Now With Added BA
    Loughborough
    Joined
    04 Jul '02
    Moves
    3790
    25 Oct '04 22:48
    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.
  2. Standard memberroyalchicken
    CHAOS GHOST!!!
    Elsewhere
    Joined
    29 Nov '02
    Moves
    17317
    26 Oct '04 12:28
    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?
  3. my head
    Joined
    03 Oct '03
    Moves
    671
    26 Oct '04 20:37
    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)