You can easily fit fifteen billiard balls into a rack without any of them touching. There's about a half an inch of gap between the balls and the rack when they're packed tightly. https://upload.wikimedia.org/wikipedia/commons/a/a4/Cribbage_pool_rack_close... On Fri, Mar 4, 2016 at 10:07 AM, James Propp <jamespropp@gmail.com> wrote:
Erich,
first of all, i don't know what kind of pool you've been playing, but i can
fit 15 billiard balls in a standard pool rack.
Gack. How embarrassing: I'm mixing up pool with bowling. (I may have misspent parts of my youth, but clearly not in billiards halls.)
second of all, assuming a somewhat smaller rack that only fits 10 balls, it
is easy to fit 8 balls that do not touch. an arrangement can be found here: http://www2.stetson.edu/~efriedma/cirintri/ < http://www2.stetson.edu/~efriedma/cirintri/>
Thanks! I realized after I sent my email that I should have first checked your website to look for a page on packing disks in equilateral triangles.
third of all, the non-touching condition does not add anything new. the question "how many balls of radius r+e will fit inside a shape of size s?" is equivalent to "how many balls of radius r will fit inside a shape of size s-e?", for slightly different e.
Good point.
fourth of all, i don't know off the top of my head melissen's proof that n=8 is the maximum for your problem, but i can probably look it up if it is important. it likely uses a more sophisticated version of the pigeonhole principle, like most packing proofs do.
No, it's not important, at least for the time being.
I find it striking that (a) 13 billiard balls can be placed in a standard billiard frame without touching, but (b) it would be hard to experimentally confirm this, since the amount of slack involved is minuscule (the relative difference between 4 + 2 sqrt(6) / 3 + 10 sqrt(3) / 3 and 8 + 2 sqrt(3) is about 0.5%).
Thanks,
Jim
On Mar 4, 2016, at 7:03 AM, James Propp <jamespropp@gmail.com> wrote:
How many billiard balls can be packed into a standard billiard rack if no two balls are allowed to touch? (Of course ten will fit if they are allowed to touch.)
I can fit six, but I don't see how to prove that seven is impossible. (I can prove that in a smaller rack that could accommodate six touching balls, you can't fit more than four non-touching balls; it's a nice application of the pigeonhole principle.)
Is there literature on this flavor of packing problem?
Note that a question about packing non-touching disks of radius r can be paraphrased as a question of the form "Is it possible, for all epsilon > 0, to pack disks of radius r-plus-epsilon ..." (where the non-touching condition is dropped).
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