# Mathematicians Create Entirely New Dive with 5 Twists and 1.5 Somersaults

Theorists use mathematical modelling to invent a dive that’s never been attempted, but could be soon.

The Olympic sport of diving combines athleticism and agility with power, grace, and precision. Dives are judged according to the takeoff, flight, and entry into the water. But the final score is then multiplied by the dive’s degree of difficulty. So a simple dive perfectly executed often scores less than a difficult dive that’s partly botched.

For this reason, dives have become increasingly complex. At the Beijing Olympics in 2008, the most complex dive had a degree of difficulty rated at 3.8; this was a reverse 2½ somersault with 2½ twists. Today the most difficult dive is a reverse 4½ somersault in the pike position rated at 4.8. More difficult dives are anticipated by FINA, the sport’s world governing body.

So divers are constantly on the lookout for ways to improve. And that raises an interesting question—just how many somersaults and twists can be combined in a 10-meter dive?

Today we get an answer of sorts thanks to the work of William Tong and Holger Dullin at the University of Sydney, Australia, who have built a mathematical model of the way the human body can twist and turn in the air. They’ve used this to propose an entirely new sequence of body shape changes that can convert pure somersaulting motion into pure twisting motion and back again.

This sequence of moves allows the body to twist faster than ever before. Tong and Dullin say that with this new technique, it is possible to perform dives of previously unheard of complexity.

To show off their approach, they have designed a never-before-attempted dive consisting of 1.5 somersaults with five twists. They call this the “513XD dive” (following FINA’s diving classification code) and say they think it will be achievable in the near future.

First some background. The laws of physics ultimately limit how labyrinthine a dive can ever be. The most significant limit is gravity, which determines how long a diver can spend in the air before hitting the water. From a 10-meter platform, it takes 1.43 seconds to fall, a time that can be increased to about 1.6 seconds with a good jump.

The number of somersaults and twists that can be completed in this time is also limited. Diving rules prevent divers from twisting as they jump. Instead, twisting can only be achieved by converting somersault motion in midair by changing the shape of the body.

The amount of angular momentum available to the diver is constant during the flight and cannot be changed midair. So the amount of angular momentum that the diver generates during the takeoff is crucial, because it also determines how many twists and somersaults will be possible.

Divers can convert somersaults into twists by moving their arms as they rotate. Starting with both arms raised, bringing one arm down causes the body to twist while raising it again stops the twisting motion. The speed at which the arms move determines the speed of the twist. Snappy movements create more impulse and so lead to faster twists, allowing the diver to twist further during the fall.

The new move from Tong and Dullin uses a longer sequence of arm movements to generate even more twisting motion. The somersaulting diver starts with both arms extended and drops the left arm to the side, as before.

But the next motion is entirely new. The diver next raises the left arm while at the same time lowering the right arm. This increases the rate of twist. Next, the diver raises the right arm while at the same time lowering the left. Finally, the diver raises the left arm so that both are above the head again and this stops the twisting motion and ends the dive. Of course all this has to happen while the diver is somersaulting through 1½ turns.

Dullin and Tong use a biomathematical model of the body to simulate how all this can take place. In particular, they calculate how long it takes to make five twists and 1½ somersaults and show it can be done in 1.8 seconds , assuming the diver generates only moderate levels of angular momentum during takeoff.

This is longer than divers have in the air. But the pair say that there are various ways to make gains. An obvious way is to increase the amount of angular momentum during takeoff. Also, the diver spends a significant amount of time—0.4 seconds—with arms and legs stretched out to achieve the full 1½ somersaults. This could be reduced by taking a tucked or piked position (although their model is as yet unable to incorporate these positions).

These changes should be achievable for a world class diver, say Dullin and Tong. “This leads us to conclude that real world athletes can in principle execute the 513XD dive.”

“This would revolutionize the sport of diving if successfully performed in competition,” they say. A dive that is theoretically engineered using a mathematical model paves the way for other changes as the model begins to incorporate other body shape changes such as tucks and pikes.

Of course, no diver has attempted the 513XD dive yet. “By simulating the 513XD dive we hope to provide coaches and athletes with insight and motivation so that the dive may one day be executed in competition,” say Dullin and Tong.

The work has applications in other sports, too, such as aerial skiing and snowboarding. “Also, the conversion from pure somersault to pure twist (and vice-versa) has applications in space maneuverability where airborne time is not a factor,” say the team.

That’s interesting work that uses mathematical modeling and engineering principles to change the nature of sport. And if any divers out there fancy their chances at the 513XD, let us know. We’d love to see a video of your attempts.

Ref: arxiv.org/abs/1612.06455 : A New Twisting Somersault—513XD