The mathematics of pelotons

The front row of le peloton take a quick fag break to discuss the niceties of aerodynamics, flocking behaviour, Game Theory and steroid abuse

We’re “back on the chain gang”, as they say, and looking at a central concept in cycling and the Tour de France: le peloton.

The peloton refers to a large, densely packed bunch of riders, a grouping which seems to flow as smoothly and naturally as a flock of birds. In fact the way the peloton behaves involves a fair amount of physics and maths…

The word “peloton” is related to the English word “pellet”, and comes from a French term meaning either “platoon” or “rolled up in a ball” .

Much of the peloton’s raison d’etre is to do with aerodynamics – how the individual units within this rolled up ball try to shelter in somebody else’s draft or slipstream.

The energy you save from the reduction in drag when you slipstream near/behind other riders can be dramatic. Some say it’s as much as 40% in the middle of a well-developed group.

The elastic band effect

But it’s usually better to be closer to the front of the peloton, to avoid the “elastic band effect”. This is where a change in speed becomes amplified as it “ripples” to the back of the peloton. The guy behind a rider who is changing his speed must adjust slightly faster to avoid collisions.

Being closer to the front also means you can react to attacks and changes in position with less effort. And when you’re nearer the front, there’s less chance of ending up in a crash. Being right at the front  means dictating the tempo to some degree, but  also means  far more hard work.

The peloton’s tipping point

The peloton’s overall shape changes according to many factors:

• A strong headwind or a hard effort on a mountain climb can string out the riders
• A slow tempo or tailwind tends to bunch them up
• Or the width of the road or a crosswind forces them to form into several echelons (an echelon is where riders form a diagonal line across the road, again to minimise wind resistance)
• Or somebody in the middle of it crashes over a small stone, and there’s, er, a butterfly effect

Yet it still retains its overall shape and travels as an integrated unit. Like birds in formation, each “unit” of the peloton obeys fairly simple rules and makes slight adjustments in response to the units around, beside, and particularly in front of it.

Some fairly simple rules generate a complex and very fluid overall shape to the peloton. It has something to do with several branches of mathematics, including the science of complex and self-organising systems.

The mathematician asks:

• What conditions lead to the emergence of flocking behaviour in the peloton?
• How does it emerge?
• What happens when racing uphill, downhill, in crosswinds etc?
• Is there a “tipping point”? A point where the fragile equilibrium breaks down? A level at which the momentum for change becomes unstoppable?

Mathematical papers are written on such matters. Seriously. See for example “The Mathematics of breaking away and chasing in cycling” (Eur J Appl Physiol (1998) 77: 492 – 497).

The peloton is also studied by a branch of mathematics called Game Theory, particularly the game in it called “the prisoner’s dilemma”, but more of that anon.