The Kissing Number of an n-dimensional sphere is the number of identically-sized spheres which can touch it simultaneously without any of the spheres overlapping. The kissing number of a 3-dimensional sphere is 12. But, there is a fair bit of play involved. In fact, if you had some sticky spheres, you'd almost think there might be some way that you could get a 13th one in there.
In fact, if you cluster nine of the touching spheres together as closely as you can, then you can rotate the cluster of the remaining three. In this way, you can move spheres around the surface of the center sphere. You form a three-cluster, rotate it a bit, separate out one ball into a different three-cluster, rotate it a bit, until you get the one ball into the place you wanted it.
Now, I'm not sure you have to do the moving in clusters of three like that. You may be able to rotate a two-cluster. And, I think you can just worm one ball around independently by backing obstructions out of the way momentarily to let the one pass.
But, for the purposes of this dream, assume that you have to rotate them in clusters of three.
I was dreaming about a self-organizing system of spheres that fought off invaders. Twelve spheres could band together to kill off (crush, vaporize) any sphere they were all touching. The spheres also gave off a small radio signal. Each sphere gave off the same exact signal as each other sphere.
The spheres were fairly well packed. They were well packed enough in the area that they had to perform these three-cluster rotations for a sphere to move about. Because of these three-cluster rotations, there is a certain parity to the system at all times. To move one item by one unit, you need to move two other items in a very particular way.
By virtue of the radio signal and/or by how many spheres are in close proximity to another sphere, the sphere could tell if it were at an edge. If it were at an edge, it could use a series of movements designed particularly to bring food particles between the spheres in toward the center of the pack.
Suppose that the radio signal given off by these spheres affects a holographic memory within the spheres. The parity from the three-cluster rotations would serve as a checksum in each sphere's holographic memory.
An invading sphere would not be party to that holographic memory (at least not any historical parts of it). Every now and again, a dozen spheres will "kiss" the sphere between them. They will ask it some question about the group memory. If seven or more of the kissing spheres feel the kissed sphere answered incorrectly, they blotto it.
And, then Isaac woke me up.... so I'm not sure what happened next.