Video: Champion formula
There are many ways to row a boat, but it took a physicist to figure out which should work best
IT IS 7am on a cloudy Monday morning and the banks of the river Thames in west London are humming with activity in preparation for an experiment. At the , the men’s rowing eight are about to take to the water in an unusual boat.
The placement of rowers in a boat – its “rig” – conventionally has the oars arranged alternately to the left and right. In this boat, however, the order is seemingly random. As the crew gingerly takes to the water, the first two men are pulling to the left, the next four to the right and the final two to the left.
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The crew is testing a rig suggested by , a mathematical physicist at the University of Cambridge. According to Barrow’s calculations, this configuration should outperform the standard rig in one very important way.
The puzzle is that this configuration has never been used in a top-class competitive race, as far as Barrow can tell. “The rig seems entirely new to rowing,” he says. So this morning’s paddle, at żěè¶ĚĘÓƵ’s suggestion, is a rough test of whether his calculations have the ring of truth.
The force generated by an oar being pulled through water has been . Though the size and direction of this force varies throughout the stroke, it can always be resolved into two components: one that pushes the boat forward and another that pushes it to the side. Obviously, any sideways motion of the boat is wasted energy and the ideal rig would eliminate this. But what should this rig be?
This is just the kind of symmetry problem that mathematicians find irresistible, which is perhaps why it came to fascinate Barrow, who is better known for his work on cosmology and popularising science than his prowess on the water. “I’ve never rowed competitively but I’m fascinated by the links between mathematics and sport,” he says.
Barrow began by modelling the problem using an idealised crew. His rowers are identical automatons that are equally spaced and which move in perfect unison, untroubled by the vagaries of biomechanics or oar design. The forces generated as they pull through perfect symmetrical arcs are identical, varying in time in exactly the same way. This variation, or force-curve, is important. That’s because the sideways component of the force changes direction halfway through each stroke: during the first half, it is directed towards the boat, while in the second half, it is directed away (see diagram).
It is easy to imagine that a symmetrical arrangement of such idealised rowers will cancel the sideways force at every point during their stroke. Yet Barrow’s analysis shows that a conventionally rigged boat should constantly wiggle from side to side as it moves through the water. “This takes extra energy from the rowers and slows the forward progress of the boat,” he says.
“A rowing boat with a conventional set up should constantly wiggle from side to side, wasting rowers’ energy”
The right moment
To understand why, Barrow looked at one measure of the effect of the sideways force – its “moment”. This is its tendency to rotate an object about some point, and the combined rotational effect of all the forces on an object is the sum of all their moments about a point. In , to appear in the , Barrow chose to take moments about the stern of a conventionally rigged racing four. If the rowers sit 1 metre apart and the distance from the stern to the first rower is 1 metre, then that rower creates a moment about the stern of 1F, the second rower contributes -2F (because he is 2 metres from the stern and is pulling on the opposite side to the first rower), the third rower 3F, and the fourth -4F. The total moment about the stern during the first half of a stroke is (1 – 2 + 3 – 4)F, or -2F. For convenience we can simply set F equal to 1, making the moment about the stern -2. During the second half of the stroke, the sideways forces are reversed, so the moment about the stern becomes +2. This is why boats using the conventional rig should wiggle as they move – they constantly experience a twisting influence that changes direction at the midpoint of the rowers’ stroke.
Then Barrow asked the obvious question: are there configurations of rowers for which the moments of the sideways forces cancel?
For a crew of four, the number of possible configurations is pleasingly small. There is the conventional rig – which we can represent as lrlr – that produces a moment about the stern of ±2. There are also two others: llrr and lrrl (and their mirror images, of course.) For the llrr rig, the moment about the stern is ±4, clearly a disaster. But for the lrrl rig, the moment of forces is 0, exactly what Barrow was looking for. This rig has actually appeared in competition. Now known as the Italian rig, it was first used in 1956 by a rowing team from the Moto Guzzi motorcycle manufacturer, based on the shores of Lake Como in Italy. The team went on to win gold for Italy in the men’s coxed fours at the Olympic games in Melbourne, Australia, that year.
What of the eights? Barrow showed that there are four arrangements where the moments exactly cancel (see diagram). One of these is lrrllrrl, in which the Italian rig is simply repeated. Indeed, this rig was used by Italian teams in the 1950s and 60s.
Such innovation wasn’t lost on Karl Adam, one of the rowing world’s most famous coaches, who worked at a rowing club in Ratzeburg, Germany. Adam and his crews began to experiment with another arrangement, lrlrrlrl, with much success in the 50s and 60s. Now known as the German rig, this also turns out to have zero moment and it was used by the Canadian team which won gold at the 2008 Beijing Olympics.
Barrow has also found two other rigs with zero moments and which, as far as he knows, haven’t appeared in competition. One of these is lrrlrllr, which is the Italian rig followed by its mirror image. The second is more exotic: the llrrrrll rig which was put through its paces by the Imperial College crew. So why have these arrangements never been adopted in a race?
One clue comes from our test on the Thames. The Imperial College crew noticed that the llrrrrll rig is affected by an unexpected ergonomic issue. In their usual rig, the rowers use the person two in front of them to synchronise their movements. However, you can’t do that with the llrrrrll rig.
Mind the puddles
There is a more serious problem, says Volker Nolte, professor of biomechanics at the University of Western Ontario in Canada and former coach of the Canadian national rowing team. When a rower finishes a stroke and lifts the oar blade out of the water, it leaves behind a vortex, known in rowing as a puddle, that creates a swell, raising the surface of the water by several centimetres.
In a conventional rig, when rowers on the same side are two apart, there is enough time for the swell to die down before the next oar passes over it. But when the rowers on the same side are directly behind each other, the swell from the oar in front remains high enough to catch the one behind. With four rowers behind each other in Barrow’s rig, this is a problem crews can do without and may help to explain why the conventional rig has prevailed.
In fact, says Nolte, Adam tried all these rigs in the 1960s. His problem was that boats of the time were relatively slow, so even with a conventional rig, rowers towards the stern of the boat ended up putting their oars into the turbulent water left by rowers at the bow. This turbulence reduced the thrust generated.
Adam found that both the German and llrrrrll rigs minimised this problem. But when he discovered the limitations of the latter, he settled on the German rig. Today’s boats go fast enough to outpace their own turbulence so this is no longer a consideration, says Nolte.
So why did the Canadian rowers use the German rig in Beijing? It turns out this had little to do with turbulence or Barrow’s moment analysis. According to Nolte, the Canadians chose this rig because, when everything else had been taken into account, it allowed the team’s lighter rowers to sit nearer the bow. This made the bow lift out of the water causing the boat to “surf” along when travelling at speed, so reducing friction with the water. So much for moments of force.
Although it’s not clear whether competitive rowers will benefit from Barrow’s approach, it may yet find uses in biology. Barrow has looked at how these ideas could be applied to the gaits of animals with a lot of legs – in particular the way millipedes and centipedes cope with the forces their legs generate. It turns out that centipedes generate a sideways wiggle when they walk, while millipedes generate an up-and-down wiggle. Unfortunately, the number of legs such creatures have varies throughout their lives and this has made it hard for Barrow to establish a connection with his moments of forces approach. “It’s still an unsolved problem,” he says.
Meanwhile, the 1, -2, 3, -4… sequence of numbers thrown up by Barrow’s calculations has attracted the attention of that rare breed of mathematicians who study integer sequences – chains of whole numbers where each entry is determined by a rule, often involving the preceding numbers. One such mathematician is of the University of Waterloo in Ontario, Canada. Editor-in-chief of the , Shallit says he has to cast his mind back to 1929 to find another example of an integer sequence applied to sport: a set of numbers known as the Thue-Morse sequence, used to answer a question about an infinite game of chess. He was struck by Barrow’s approach. “The mathematics is not ground-breaking, but the application is fascinating.”
