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Text: Letters to a Young Mathematician. Ian Stewart. Basic Books: 2006. New York. pp 46-48.
"From a distance they look like sheet music, fat little blobs on rows of horizontal lines. There seem to be special places they like to perch, and it’s not at all clear to me why, but one thing stands out. If a lot of birds are perching on a wire, they end up evenly spaced.
"That’s a mathematical pattern, and I think there’s a mathematical explanation. I don’t think the birds 'know' they ought to space themselves out evenly. But each bird has its own 'personal space,' and if another bird gets too close, it will sidle along the wire to leave a bit more room, unless there’s another bird crowding it from the other side.
"When there are just a few birds, they end up randomly spaced. But when there are a lot, they get pushed close together. As each one sidles along to make itself feel more comfortable, the 'population pressure' evens them out. Birds at the edge of denser regions get pushed into less densely populated regions. And since the birds are all of the same species (usually they’re pigeons), they all have much the same idea of what their personal space should be. So they space themselves evenly.
"Not exactly evenly, of course. That would be a Platonic ideal. As such, it helps us to comprehend a more messy reality.
"You could do the math on this problem if you wanted to. Write down some simple rules for how birds move when the neighbors get too close, plonk them down at random, run the rules, and watch the spacing evolve. But there’s an analogy with a common physical system, where that math has already been done, and the analogy tells you what to expect.
"It’s a bird crystal.
"The same process that makes birds space themselves regularly makes the atoms in a solid object line up to form a repetitive lattice. The atoms also have a 'personal space': they repel each other if they’re too close together. In a solid, the atoms are forced to pack fairly tightly, but as they adjust their personal spaces, they arrange themselves in an elegant crystal lattice.
"The bird lattice is one-dimensional, since they’re sitting on a wire. A one-dimensional lattice consists of equally spaced points. When there are just a few birds, arranged at random and not subject to population pressure, it’s not a crystal, it’s a gas.
"This isn’t just a vague analogy. The same mathematical process that creates a regular crystal of salt or calcite also creates my 'bird crystal.'"
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