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Like annoying (yeah, I said it) people at ball games, emperor penguins do the wave. Like so:

The waves happen when the penguins huddle together, a behavior that helps them stay warm in an environment that can reach -50 degrees Celsius (-58 degrees Fahrenheit) with winds up to 200 kilometers per hour (about 120 mph). Thousands of individuals can gather together in a huddle. And once in a while, someone moves. That triggers shifts throughout the system that researchers see as waves. Now, a team of researchers from Germany and France has figured out the mathematics of these waves, as you do.

Penguins react to the movements of the birds immediately around them, the researchers found. That is, a penguin will step forward when he sees the bird in front of him step forward. Meanwhile, that step entices the penguins behind him to step forward. A shifting penguin will also make the individuals in front of him step forward and so on until the front of the line. Multiply these adjustments by thousands of penguins, and you get the penguin wave.

Other researchers have found that flocks of (flying) birds and schools of fish work similarly. Although human observers may see complex patterns in the flocks and schools that flash by, any individual animal doesn’t have an overall idea of what the group looks like; he’s just focused on reacting to the movements of his neighbors.

To test their ideas, the penguin researchers wrote a mathematical model of huddles, then compared the model’s results with timelapse recordings they made of emperor penguin colonies living near the French Antarctic research base Dumot d’Urville, and its German counterpart, Neumayer III.

Here’s what they found:

  • Any penguin anywhere in the huddle can start a wave. It only takes a step, though the speed at which the wave propagates depends on how big of a step the instigator takes.
  • Penguins like to stay about two centimeters apart from each other. Each penguin’s insulating feather layer is about 1.2 centimeters thick, so the researchers think it’s likely that penguins only like to touch each other slightly in a huddle, so that they’re close together, but they don’t crush their natural insulation.
  • Two nearby huddles with waves propagating through them can merge, a phenomenon researchers found both in their model and in their penguin videos.

There are still some mysteries to penguin huddles. For example:

  • Why do wave-initiators initiate waves in the first place? (Is he just shifting his weight? Who knows?)
  • Why is there usually a 30-second or longer delay between waves, during which everyone stands still?

The researchers published their work today in the New Journal of Physics.