M&Ms and Mathematics

[_Q_]

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've seen claims from manufacturers of ice machines that their cubes are shaped for "high displacement" to pack more densely into a liquor glass (higher ice/liquor ratio). I'm pretty sure that some of these vendors also use terms like, "higher profits" in their product literature.

Of course, the ice cubes still need to be cost-effective to make in the machine, and they can't be so small that they cease to be perceived as "ice cubes" by the customers.


_Q_

 

[Dorian Gray]

I am applying for a grant to experiment with other candies as well. This is important research that must not be cast aside and forgotten!

 

[CurtC]

Isn't it true that for spheres, if you put 12 spheres packed around a central one, that they will not be tight, that they will be able to wiggle around, but not so much as to fit a 13th?

What I was asking earlier, with what turns out to be hexagonal close packing, is why those won't be tight? I mean, each neighbor sphere is touching two other neighbors, right? It seems like it would, but I'm pretty sure it can't, because then those 12 spheres' centers would make a dodecahedron with triangle sides, which can't be the case.

 

[Brown]

Isn't it true that for spheres, if you put 12 spheres packed around a central one, that they will not be tight, that they will be able to wiggle around, but not so much as to fit a 13th?

This is true. The 12 spheres do not touch one another, and moreover, the "extra room" between spheres, if added up, would be enough to accommodate a 13th sphere.

However, I understand it has been proven that there is no physical way to arrange the 12 spheres so as to make room for a 13th.

 

[Psi Baba]

Do all of the objects have to be the same shape? By that I mean could you have a mix of 50% oblate speroids and 50% hyperboloids (I made that word up: I'm thinking of something with an hourglass shape and rounded ends). Those should pack rather tightly I should think.

Small Town Jesus: many Americans might not know that what you know as Smarties are like our M&Ms. In the U.S. there is a candy called Smarties (Rockets in Canada) that is made of powdered dextrose pressed into small concave pellets. I'm quite addicted to them myself. The chocolate Smarties sold everywhere else in the world are made by Nestlé.

 

[JSFolk]

Psi, I'm glad you brought that up. I moved up here to Canada right before Halloween and had the damnedest time trying to convince fiancee and her kids that Smarties in the US where little double-concave thingies, exactly matching their Canadian "Rockets".

On a side note, would Rockets and Smarties (international versions) pack more tightly since the curves would match up?
I'm not willing to do the math on this, but if somebody does, I'll accept 1/10th of the prize money and a mention in his/her acceptance speech.