In a fascinating insight, physicists have derived a simple relationship between the body mass and wing area to the frequency with which animals flap their wings or flippers.

The universal equation has been shown to accurately predict the flapping frequency of birds, insects and even long-extinct prehistoric creatures like the flying reptiles, pterosaurs. It even translates to the flapping flippers of swimming creatures like whales and penguins.

The study is published in the open-access journal *PLOS ONE*.

Scientists have long thought that the frequency of flaps should relate to the natural resonance frequency of the wing to save energy. But finding the relationship between wing and body shape to the rate of flapping has proven difficult.

Danish physicists from Roskilde University derived the new formula from basic concepts in physics, including dimensional analysis. It shows that flapping frequency is proportional to the square root of the animal’s body mass divided by the wing’s surface area.

This simple relationship was tested by plotting its predictions against published data on wingbeat frequencies for bees, dragonflies, beetles, mosquitos, bats and a variety of different birds.

The equation was even tested against fin stroke frequencies for penguins and several whale species.

“Data for birds, insects, bats, and even a robotic bird—supplemented by data for whales and penguins that must swim to stay submerged—show that the constant of proportionality is to a good approximation the same across all species; thus the equation is universal,” the authors write.

“Differing by almost a factor 10,000 in wing/fin-beat frequency, data for 414 animals from the blue whale to mosquitoes fall on the same line. As physicists, we were surprised to see how well our simple prediction of the wing-beat formula works for such a diverse collection of animals,” they add.

The physicists applied their formula to the largest known flying animal, *Quetzalcoatlus northropi*– an extinct pterosaur which lived at the time of the dinosaurs. *Quetzalcoatlus *stood as tall as a giraffe when on the ground and had a wingspan of more than 10 metres.

Applying the formula to the ancient beast, they found it would have beaten its wings with a frequency of about 0.7 Hertz, or about 42 times a minute.