Conference rusure::math

This is actually an old trick question.  a 12X10X8 foot office is of
course a 144 x 120 x 96 inch office, which (without packing) is about
          13 x  10 x  8 = 1040 11-inch balloons, though you could probably
pack a hundred more in.

Dividing volumes doesn't give you the answer you want, since it ignores
the empty space between the balloons.

  John

    A sphere's volume is  4/3 * pi * r^3.  Here  r = 5.5" = .458', and
    
    each balloon's volume is  .403 cu. ft.  The office is  960 cu. ft.,
    
    so you might get 2380 balloons into it.  This ignores space between

    balloons, but I think .0 asked for that, and who knows what happens
    
    when all the balloons start to squish each other?  So how are the

    first 144 balloons doing?
    
    John

    You can pack over 1 million 11" diameter balloons
    into that office, provided you deflate them carefully and
    stuff'em in reasonably large cardboard boxes.

I assume it's a 10x12-foot office with 8-foot ceilings. Along the 12-foot wall
you can put 13 baloons, then a row of 12, then 13, then 12... You can get
6 double rows of 25 on the floor with no sweat. Pile 8 of those and you have
1200 with a lot of loose space. There are more efficient packings; I think you
might be able to get 1400 in there, snugly.

Lynn Yarbrough 

    I believe the problem of finding an optimal packing for spheres is
    unsolved, so we will probably have to settle for estimates.
    
    
    				-- edp 

>I assume it's a 10x12-foot office with 8-foot ceilings. Along the 12-foot wall
>you can put 13 baloons, then a row of 12, then 13, then 12... You can get
>6 double rows of 25 on the floor with no sweat. Pile 8 of those and you have
>1200 with a lot of loose space. There are more efficient packings; I think you
>might be able to get 1400 in there, snugly.
>
If you alternate layers, e.g. 13-12-13-12 with 12-13-12-13, you can get
ten layers, or 1500 balloons, and still have a little slack space.

Lynn Yarbrough 

    I believe the packing where each sphere touches 12 neighbors is
    the best space-filling packing (i.e., when there are no boundary
    conditions).  It's unimaginable that this could be improved.
    Has this been proved optimal?  I don't know.

    BTW, for this packing, what is the ratio of volume-of-spheres to
    volume-of-space?  What is the ratio for the square-like packing?

>    I believe the packing where each sphere touches 12 neighbors is
>    the best space-filling packing (i.e., when there are no boundary
>    conditions).  It's unimaginable that this could be improved.
>    Has this been proved optimal?  I don't know.
>
>    BTW, for this packing, what is the ratio of volume-of-spheres to
>    volume-of-space?  What is the ratio for the square-like packing?

As I recall, there is a packing in which one can very nearly get a 13th
tangent sphere, and the optimality of the 12-packing is in doubt. The
12-packing is composed of tetrahedral 4-sphere groups. I haven't figured out 
what the packing density is yet...

A sphere occupies Pi/6 ~ .5236 of its cube.

In the spirit of Doug's Sci Am. article about using real physics
to solve NP-complete problems, I offer the following method of deciding
what the best packing for spheres is:

	Take a bunch of well-greased round ball bearings, and stuff them
	into a balloon such that the balloon is stretched over them.
	They will naturally position themselves in the best packing.
	Dump the wad into a full bucket of water and measure how much
	spills out.

	Of course, they may not actually arrange into the best
	packing configuration.  So do it several thousand times and
	see which time you spill the LEAST water.

	(Oh yeah, don't dump the wad into the water, place it in slowly
	so it doesn't splash)

/Eric

>	Take a bunch of well-greased round ball bearings, and stuff them
    
    (Eric would have suggested marbles, but he lost his :-)
    
    Can one ascertain the packing algorithm by looking at the bumps
    on the balloon surface?  How many marbles (bearings) would one need?
    Does the balloon have to be spherical?  Does it help to "massage"
    the stuffed balloon in hopes of obtaining an optimal packing in
    relatively few trials?
    
      John

    I'm reminded of a Rolex (?) watch commercial from years ago.
    While the narrator described the precision fit of the pieces,
    the ad showed the pieces being tossed into a paper bag, and
    the bag shaken.  Voila!  Out slides a beautiful Rolex watch.