The universal flapping of wings | Polarjournal
Six hundred thousand beats per minute separate the fastest and slowest animals as they flap their wings. But they all have one thing in common. Image: Dørte Mandrup Arkitekter/Mir

Like Leonardo da Vinci in his day, Danish physicists at Roskilde University have been studying the proportions of living creatures with the ability to fly and dive. They found one universal mathematical formula linking these animals.

“Write about movement under water and you will have the flight of birds in the air”, reads Leonardo da Vinci’s thoughts. Never has this phrase been more relevant than since June 5, when Danish physicists from the University of Roskilde published their findings in PLOS ONE. They focused on the architecture of winged animals, whether flying or diving. Intrigued by the proportions of their wings and fins, the researchers mathematically reconstructed the mechanics that enable these animals to rise into the air or dive to depths. Surprisingly, they found themselves face to face with a universal equation. From mosquitoes to penguins, the speed of their movement is linked to the surface area of their wings or fins, and the body mass of the animal carrying them. More than 400 species were included as part of the study.



“We started by solving an equation for bird flight,” says study co-author Tina Hecksher. And, much to the team’s surprise, these feathered aerial species all follow the same line. So why not bring insects into the picture? After all, bees also escape the earth’s gravity. And mammals? Bats stretch their skin to carry their weight above ground… In each case, the equation holds.

Following Da Vinci’s thoughts, the question arises “Does the law apply underwater”? Researchers set out to answer this question. “If you want to dive, you have to push up to go down. Which is the equivalent of what birds do when they push air down to go up”, she compares. Taking into account the difference in density between air and water, the equation is absolute, from the Northern bottlenose whale to the humpback whale, and from the butterfly to a swan.

Yet wouldn’t this formula be a little oversimplifying given the diversity of living forms? Every animal has its own way of flapping its wings. The shape of the wing differs between a penguin and an albatross, as do the angle of attack and the flapping style… Physicists have found a constant for each species to include in the general equation. “Against all odds, our results show that it is common to all species,” Tina Hecksher tells us, “It’s around 11.”

Does evolution follow a strict rule for swimming and flying? “Somehow, this study leaves the question open”, confides the physicist. “I suspect there are energetic reasons for this constancy. It’s all about minimizing the cost of movement.” The equation could help us go back in time to the age of the dinosaurs. According to their estimates, a pterosaur’s 10-meter wings beat slowly, at 35 beats per minute. However, the formula should not work in the infinitely small. Fluids change their properties. This should be studied for the plans of a future flying machine: nanorobots.

Camille Lin, Polar Journal AG

Link to the study : Jensen, J.H., Dyre, J.C., Hecksher, T., 2024. Universal wing- and fin-beat frequency scaling. PLOS ONE 19, e0303834. https://doi.org/10.1371/journal.pone.0303834.

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