M&Ms and Mathematics

[Brown]

CNN and Reuters report that oblate spheroids, such as plain M&M's, pack more densely than regular spheres.

Although the report headlines this a "Physics Discovery," it seems to me to be more of a "Mathematics Discovery." Problems associated with "sphere packing" (and other shapes that pack three dimensional space) are well known in mathematics. Martin Gardner has written about the subject often.

In a way, this discovery is not much of a surprise. It is well known that there are several shapes that pack better than spheres. Small cubes, for example, can pack nearly 100% of the available space. The cubes, however, must be arranged. If you dump them all into a container and shake the container, you are unlikely to obtain the optimum packing arrangement.

It is possible that there are several other round shapes, such as super-ellipsoids, that may pack space more efficiently than M&M-shaped objects. (If this turns out to be correct, I will be happy to accept the Nobel Prize in Physics.)

 

[sorgoth]

No Nobel Prize for you! After all, "Genius is 1% inspiration and 99% perseveration." ...or something like that.

Anyway, you don't do the work, you don't get the prize!

Seriously, though, would there be a way to mathematically deduce which shapes pack densest?

 

[Small Town Jesus]

Maybe that why 'Smarties' are called 'Smarties'.

STJ

 

[Brown]

No Nobel Prize for you!
...
Seriously, though, would there be a way to mathematically deduce which shapes pack densest?

No soup for me, either!

It seems to me that the question is not merely what shapes pack the densest, but what shapes pack the densest when dropped randomly into a space. There are all sorts of box-like shapes that pack together without any gaps, but if you dropped them randomly into a container, they would probably not orient themselves to eliminate gaps.

I wonder what would happen with tetrahedrons or with half-octahedrons (similar to the Egyptian pyramids)? I wonder what would happen with capsule shapes (like rice or Tylenol capsules)? I wonder whether snub shapes are better packers than shapes with all edges being round?

It seems to me you ought to be able to mathematically model these shapes and randomly pack them, and then "shake" them, and measure the volume of the "gaps." I'm not sure that this would yield an algorithm for finding which shapes have the greatest packing density, but it could be used to compare one shape to another.

 

[Matabiri]

Packing problems are Hard.

It still hasn't been proved that CCP and HCP are the densest possible packing arrangements for spheres, for example. Hence rigorous solutions in this area would probably get you known, at least.

 

[Soapy Sam]

What if they carry a charge or magnetic field? A random pile of cubes might end up as a big cube...in fact, why do the words "Sodium chloride crystal" keep coming to mind?

 

[sorgoth]

What if they carry a charge or magnetic field? A random pile of cubes might end up as a big cube...in fact, why do the words "Sodium chloride crystal" keep coming to mind?

You want to pack them, not stick them all together.

 

[BTox]

You can pack those M&Ms to 100% of available space by simply heating and melting them.

 

[NoZed Avenger]

Jeee-zuss, people.

Just buy the bigger bag. Problem solved.

Just slip the prize under the door if I don't answer.


N/A

 

[Tez]

Packing problems are Hard.

It still hasn't been proved that CCP and HCP are the densest possible packing arrangements for spheres, for example. Hence rigorous solutions in this area would probably get you known, at least.

I think most mathematicians expect Hales' proof to hold up...

 

[Skeptoid]

I think most mathematicians expect Hales' proof to hold up...

http://mathworld.wolfram.com/KeplerConjecture.html

Unfortunately, Hales' proof has proved extremely difficult to verify. While it has been submitted to the Annals of Mathematics and will likely be published some time in 2003 or 2004, when it appears, it will carry an unusual editorial note stating that parts of the paper have not been possible to check, despite the fact that a team of 12 reviewers worked on verifying the proof for more than four years and that the reviewers as 99% certain that it is correct (Holden 2003, Szpiro 2003).

In response to the difficulties in verifying his proof, in January of 2003, Hales launched the "Flyspeck project" ("Formal Proof of Kepler") in an attempt to use computers to automatically verify every step of the proof. Unfortunately, Hales expects the project is likely to take 20 person-years of labor (Holden 2003, Szpiro 2003).

 

[Tez]

http://mathworld.wolfram.com/KeplerConjecture.html

Heck, I've accepted papers for publication while being significantly less than 99% certain they were correct!

 

[Skeptoid]

Such is the rigor of mathematics.

 

[TeaBag420]

CNN and Reuters report that oblate spheroids, such as plain M&M's, pack more densely than regular spheres.

Might there be an appeal to common sense based on the ratio of volume to surface area?

 

[Matabiri]

I think most mathematicians expect Hales' proof to hold up...

Even if it does, it's still Hard Mathematics.

 

[Bluegill]

You can pack those M&Ms to 100% of available space by simply heating and melting them.

But you'd have to do it in your mouth, not in your hand.

 

[CurtC]

It occurred to me once, while getting a bucket of range balls (golf) from a vending machine, that this thing wouldn't work as well if spheres could be tightly packed.

But when I was thinking about this problem several years ago, my thought was that they should pack tightly, in this manner: arrange a set of spheres on a flat surface so that each is touching its six neighbors. Now add a second layer, with each resting on the hole formed by three balls in the lower layer. When the second layer is done, each ball will be touching its six neighbors on that layer, plus the three below it. Keep packing on more layers. Each sphere will touch three below, six on the same layer, and three above. Each one of those twelve neighbors will touch two of the other neighbors.

It seems like this would be tightly packed, but I know it can't be right. Can anyone explain what it is about this arrangement that doesn't work?

 

[TillEulenspiegel]

I dunno.... reading some posts on this board I believe one could make a case for the densest packing to be exibited in a skull shape.

 

[Skeptical Greg]

CNN and Reuters report that oblate spheroids, such as plain M&M's, pack more densely than regular spheres.

Although the report headlines this a "Physics Discovery," it seems to me to be more of a "Mathematics Discovery." Problems associated with "sphere packing" (and other shapes that pack three dimensional space) are well known in mathematics. Martin Gardner has written about the subject often.

In a way, this discovery is not much of a surprise. It is well known that there are several shapes that pack better than spheres. Small cubes, for example, can pack nearly 100% of the available space. The cubes, however, must be arranged. If you dump them all into a container and shake the container, you are unlikely to obtain the optimum packing arrangement.

It is possible that there are several other round shapes, such as super-ellipsoids, that may pack space more efficiently than M&M-shaped objects. (If this turns out to be correct, I will be happy to accept the Nobel Prize in Physics.)

If they pack so well, then why is the package still half empty?

 

[Wrath of the Swarm]

CurtC: what you describe is hexagonal close packing, as far as I can tell. Take a look at the Mathworld link.

The question is whether we can prove that's the best packing method.