[math-fun] Loose packing problem
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). Jim Propp
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. 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/> 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. 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. erich
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).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Is it known (or even suspected) that there are nonrigid solutions besides 11? Guess: 16. --rwg On 2016-03-04 10:07, James Propp 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).
Jim Propp ____________________________________
the short answer is yes to suspected and no to known: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1a1 <http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1a1>
On Mar 4, 2016, at 2:31 PM, rwg <rwg@sdf.org> wrote:
Is it known (or even suspected) that there are nonrigid solutions besides 11? Guess: 16. --rwg
On 2016-03-04 10:07, James Propp 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).
Jim Propp ____________________________________
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This reminds me -- what is the smallest n so that the smallest square that can accommodate n^2 non-overlapping unit circles has edge length < n? On Fri, Mar 4, 2016 at 2:38 PM, Erich Friedman <erichfriedman68@gmail.com> wrote:
the short answer is yes to suspected and no to known: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1a1 < http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1a1>
On Mar 4, 2016, at 2:31 PM, rwg <rwg@sdf.org> wrote:
Is it known (or even suspected) that there are nonrigid solutions besides 11? Guess: 16. --rwg
On 2016-03-04 10:07, James Propp 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).
Jim Propp ____________________________________
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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empirically, n=7 works, and n=6 doesn't seem to. see packomania.com <http://packomania.com/>.
On Mar 4, 2016, at 2:54 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This reminds me -- what is the smallest n so that the smallest square that can accommodate n^2 non-overlapping unit circles has edge length < n?
On Fri, Mar 4, 2016 at 2:38 PM, Erich Friedman <erichfriedman68@gmail.com> wrote:
the short answer is yes to suspected and no to known: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1a1 < http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1a1>
On Mar 4, 2016, at 2:31 PM, rwg <rwg@sdf.org> wrote:
Is it known (or even suspected) that there are nonrigid solutions besides 11? Guess: 16. --rwg
On 2016-03-04 10:07, James Propp 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).
Jim Propp ____________________________________
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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10, 15, fiddlesticks. Let's ask this nice problem for any triangular number T_n of balls and the associated tight exact triangle. This must be the same as asking the question unconditionally for a triangle just epsilon smaller, for small enough epsilon, as Erich has pointed out previously. For n = 2, I think the maximum is 2. ((( Or, a closely related problem without edge effects: Consider a rack in the shape of a 60-120 rhombus with the edges identified to make a 60-120 torus* that just fits n^2 balls nestled against each other just as in a tight triangular rack . . . and ask the same question: How many balls can fit without touching (i.e., in a very slightly smaller toral rack) ? The side of the rhombus would be nD, where D is a ball's diameter. I like the torus shape since for any two points x, y of it, there is an isometry of the whole thing taking x to y. And the 60-120 torus comprises, along with the square torus, the two most symmetrical tori. It is like any flat torus a Lie group, so acts on itself by translation, but also has the dihedral group of order 12 acting as isometries as well. ))) —Dan —————————————————————————————————————————————————————————————————————— * A 60-120 rhombic torus can also take the guise of a regular hexagon, with opposite edges identified in each case. (Puzzle: Show how to dissect the rhombic torus into the hexagonal one.) If the side of the rhombus is L, then the side of the equivalent regular hexagon would be L/sqrt(3).
On Mar 4, 2016, at 4:03 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 15 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).
participants (7)
-
Allan Wechsler -
Dan Asimov -
Dan Asimov -
Erich Friedman -
James Propp -
Mike Stay -
rwg