Seeing the earth from a distance has changed my perception of the solar system as well. Ever since Copernicus’ theory (that the earth was a satellite of the sun, instead of vice versa) gained wide acceptance, men have considered it an irrefutable truth; yet I submit that we still cling emotionally to the pre-Copernican, or Ptolemaic, notion that the earth is the center of everything. The sun comes up at dawn and goes down at dusk, right? Or as the radio commercial describes sunset: ‘When the sun just goes away from the sky …’ Baloney. The sun doesn’t rise or fall: it doesn’t move, it just sits there, and we rotate in front of it. … Everyone knows that, but I really see it now. No longer do I drive down a highway and wish the blinding sun would set; instead I wish we could speed up our rotation a bit and wing around into the shadows more quickly. I do not have to force myself to call this image to mind; it is there, and occasionally, I use it for other things, although admittedly I have to stretch a bit. ‘What a pretty day’ makes me think that it’s always a pretty day somewhere; if not here, then we just happen to be standing in the wrong place. ‘My watch is fast’ translated into no, it’s not, it’s just that you should be standing farther to the east.

— Astronaut Michael Collins, Carrying the Fire, 1975

If a lion hunts a herd of antelope, what rules govern the herd’s behavior? One intriguing possibility is known as selfish herd theory: Rather than acting to benefit the group as a whole, each member positions itself so that there’s at least one other animal between it and the predator. This produces a pattern known as a Voronoi tesselation — if each dot in the diagram above is an antelope, then the surrounding colored region is the area that’s closer to that antelope than to any of its neighbors. If a lion enters your cell, then you’re the antelope that’s going to get eaten.

This understanding helps to explain some herd behavior. Each animal wants to make its “domain of danger” as small as possible and to be as far as possible from the predator. Dominant animals tend to get prime positions near the center, subordinate animals get pushed to the fringes, and the whole formation evolves continuously as predator and prey move about.

Studies have shown that groups of fiddler crabs tend to take up Voronoi patterns fairly quickly when a predator first appears, and to huddle together when the danger increases as each tries to reduce its surrounding polygon. This actually leads some to move toward the predator as they try to reach the center and put others between the hunter and themselves. Those that violate the movement rules tend to get picked off, which reinforces the evolutionary strength of the strategy.

A family of sets is said to be union-closed if, given any two sets in the family, their union is as well. Here’s an example:

{}, {1,3}, {2}, {1,2,3}, {1,2,3,4}

Combine any two of those sets and you’ll get a member of the same family.

Now, provided our family is finite and consists of more than just the empty set, is there always an element that’s present in at least half of the sets? In the example above, the answer is yes: The element 2 appears in three of the five sets.

Will this always be the case? Strange to say, no one knows. Though the problem is almost absurdly simple to state, it has remained unsolved since Péter Frankl first posed it in 1979.

Henning Bruhn and Oliver Schaudt survey the state of the inquiry here. “The union-closed sets conjecture still has a bit of a journey ahead of it,” they conclude. “We hope it will be an exciting trip.”

(Thanks, Drake.)