Michael Phelps swims the butterfly, a stroke whose arm and leg motions were developed by different people at different times. PHOTOGRAPH BY CHRISTOPHE SIMON / AFP / GETTY

When an amateur swimmer tries to learn the butterfly, a couple of questions might come to mind in between gasps for air: Who invented this flummoxing stroke, and why? Professionals such as Michael Phelps make the butterfly looks effortless, an act of coördination, grace, and endurance; for beginners, it can look and feel like a wild, flailing doggy paddle. But these questions are as difficult to answer as the stroke is to master. As with other paradigm-shifting inventions, like jazz music and the croissant, the butterfly stroke is the result of a series of small innovations rather than of any single big one. Because of that, tracing the stroke’s origin is difficult, an exercise in weighing disparate accounts.

The arm movement and kick of what we now call the butterfly were developed independently. The dramatic arm stroke can be traced back to around 1930, when most competitive swimmers still used the traditional breaststroke. The International Swimming Hall of Fame credits an Australian, Sydney Cavill, as the inventor of the butterfly armstroke, while others credit a German, Erich Rademacher, and still others say it was an American, Henry Myers. What we can say is that, at around the same time, these swimmers were all experimenting with recovering their arms out of water instead of in it. (During roughly the same period, others were experimenting with a different out-of-water motion: the front crawl.) The butterfly arm motion expended more energy than the breaststroke and required considerable coördination, but it paid off with faster times. Rademacher tried out the arm motion, paired with the traditional breaststroke kick, in races as early as 1927, during a visit to the United States. He insisted that he was still operating within the rules of breaststroke swimming. In 1933, Myers used the new arm stroke at a Brooklyn Y.M.C.A. swim meet, where he beat the American medley champion Wallace Spence. According to the swimming coach and historian Cecil Colwin’s 2002 book, “Breakthrough Swimming,” race officials were surprised by Myers’s technique, but they didn’t disqualify him.

During the same years that Rademacher, Myers, and Cavill were experimenting with their arms, a young physicist named Volney Wilson was spending much of his free time gazing through the glass at the Shedd Aquarium, in Chicago. Later a collaborator on the Manhattan Project, Wilson, whom most people called Bill, was an avid swimmer. In the summer of 1934, he began to seriously think about the different ways that animals propelled themselves through water. At the aquarium, he noticed that fish moved their tails from side to side, while mammalian aquatic animals, such as whales and dolphins, moved their tails up and down. In the pool at the Chicago Athletic Club, he began to experiment with a new kind of leg movement for human swimmers: the dolphin kick. He quickly mastered it, pairing the sine-wave legs with the traditional arm movement of the breaststroke. Wilson then tried to popularize his new kick through demonstrations at swim meets, and even won an Olympic trial, in 1938, before being disqualified for using an illegal technique, an anecdote noted in passing in Richard Rhodes’s seminal 1986 book, “The Making of the Atomic Bomb.” But, in his day, Wilson’s contribution to the sport was mostly overlooked. He remained an avid swimmer until his death, in 2006, at the age of ninety-six. “When Bill was in his nineties, I took him to the Y.M.C.A. to demonstrate the stroke to me,” David Schrader, Wilson’s biographer, told me some time ago. “He couldn’t do it then, and neither could I.”

Instead of Wilson, credit for the dolphin kick usually goes to David Armbruster, who coached swimming at the University of Iowa from 1917 to 1958. The story, according to the International Swimming Hall of Fame, is that Armbruster’s interest in the dolphin kick had been piqued as early as 1911, after he witnessed a demonstration of the movement by George Corsan, the man credited with popularizing swimming on a mass scale through his work as the head instructor for the Y.M.C.A. In the nineteen-thirties, Armbruster, with his swimmer Jack Sieg, began to experiment with a dolphin-like kick. Before long, Armbruster had encouraged all of his swimmers to use it in place of the breaststroke’s traditional frog kick.

It was only a matter of time before arms met legs—but when exactly it occurred is difficult to say. The International Swimming Hall of Fame considers the Japanese swimmer Jiro Nagasawa among the first to combine the butterfly arm stroke with the dolphin leg kick, and he’s given credit for helping to popularize the stroke. In 1945, Nagasawa set a world record using the butterfly. By 1954, the stroke was officially recognized as the “butterfly” by the International Swimming Federation. Two years after that, the stroke was accepted into the Olympics as a separate category for competition.

In recent years, Michael Phelps’s mastery of the underwater dolphin kick has helped attract renewed attention to the hydrodynamics of the movement. In 2003, Rajat Mittal, now a mechanical-engineering professor at Johns Hopkins University, began working with a graduate student, Alfred von Loebbecke, and others to apply their research—analyzing the physics of complex flows using numerical simulations—to human swimmers. “We contacted U.S.A. Swimming to see if they were interested, and the timing was good,” Mittal told me. “The dolphin kick was a big deal at that time, around the 2004 Olympics. Phelps and [Natalie] Coughlin were rising up the ranks, and they were doing that partially because they were very good at the dolphin kick. The idea was to understand more about that kick.”

Hearing Mittal tell it, the kick’s effectiveness is directly tied to its visual beauty. “It’s a tough kick to master because it works really well if you can pass that wave down your body very smoothly—the smoother that wave is as it passes down your body, the faster you will cut through water,” Mittal said. “Everything has to operate in unison in order to do this properly.” Perhaps not surprisingly, the secret to a good dolphin kick lies not only in technique but in physiology. “We looked at a number of underwater videos of Natalie Coughlin and Michael Phelps, and they had an incredible amount of fluidity in the way they were essentially able to flip and flop their legs back and forth in the dolphin kick,” Mittal said.

Many swimmers and coaches agree that the butterfly is the hardest stroke to learn, an act of coördination that calls to mind the grade-school challenge of simultaneously rubbing your stomach and patting your head. Once mastered, however, that changes. To watch Olympic swimmers do the butterfly is to witness a metamorphosis: half-human, half-fish, wholly mesmerizing.