Math of the Penguins

Animals have evolved to protect against the cold in myriad ways. Whales insulate with blubber. Bison congregate near geothermal springs. Black bears shelter in caves. And emperor penguins, facing Antarctica’s subzero temperatures and gale-force winds, huddle.   

“A penguin huddle looks like organized chaos,” said François Blanchette, a  mathematician at the University of California, Merced. “Every penguin acts individually, but the end result is an equitable heat distribution for the whole community.”

It turns out that penguins execute their huddles with a high degree of mathematical efficiency, as Blanchette and his team discovered. More recently, Daniel Zitterbart, a physicist at Woods Hole Oceanographic Institution, helped develop and install high-resolution cameras to observe undisturbed huddling behavior. Zitterbart’s team recently discovered which conditions cause penguins to huddle, and they are investigating the possibility that the penguins’ mathematical behavior may reveal secrets about colony health over time.

At the bottom of the world, hundreds of thousands of emperor penguins emerge from the sea each April to trek over 50 miles to their inland colonies. After breeding, the females return to the sea for food and the males stay behind, each incubating a solitary egg in a pouch above their feet. Without nests or food, they brave the elements by huddling together on stable pack ice to maximize ambient heat and minimize exposure. 

Though dominant winds can appear to push a huddle along the ice, the truth is more nuanced. Blanchette and his team’s model made clear that the birds do not move in unison. Penguins in the huddle’s center, where temperatures reach a sweltering 100 degrees Fahrenheit, mostly stand still. A bird who finds himself on the huddle’s windward side is soon driven to relocate to its warmer, leeward side. As more birds leave the windward side, penguins in the center soon find themselves exposed. In due course, these penguins also depart for the leeward side. 

Huddles typically last a few hours, during which the penguins may cycle through multiple rotations from the huddle’s cold exterior to its warm interior. In the process, each individual prioritizes his own warmth, yet the huddle’s heat is shared by all. 

Penguins seem to know what mathematicians learned long ago: The densest packing of shapes on a plane is a hexagonal grid. According to Blanchette’s model, the birds arrange themselves as if they were each standing on their own hexagon in a grid. Most huddles start off as misshapen blobs. Wind flow and temperature around the huddle prompt a first penguin — typically the coldest on the windward side — to relocate. This penguin, known as the mover, waddles in search of new neighbors in the relative warmth of the huddle’s leeward side.

Ali Kaya


Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.