It's Not Just Guesswork Anymore: There Is Mathematics to Zigzagging

March 11, 2008  

We all know that a straight line is the shortest distance between two points. But when it comes to traversing uneven terrain, the answer is to zig and zag, say two researchers.

According to a mathematical model developed by archaeologist Marcos Llobera (University of Washington) and mathematician T.J. Sluckin (University of Southampton, U.K.), a zigzag course provides the most efficient way for humans to go up or down steep slopes.

“I think zigzagging is something people do intuitively,” said Llobera. “”People recognize that zigzagging, or switchbacks, help but they don’t realize why they came about.”” One might "expect a similar process on any landscape," said Llobera, "but when you have changes in elevation it makes things more complicated,” he noted. ”There is a point, or critical slope, where it becomes metabolically too costly to go straight ahead, so people move at an angle, cutting into the slope. Eventually they need to go back toward the direction they were originally headed and this creates zigzags. The steeper the slope, the more important it is that you tackle it at the right angle,” he said.

“You get a different pattern if people are going up or down and this may lead to the emergence of shortcuts. Walking downhill generally takes less energy except for braking. We would expect to see different paths going up and down, but what we end up with is a compromise and shortcuts aren’t as apparent,” said Llobera.

The researchers discovered that both hairpin bends (switchbacks) and shortcuts are "efficient strategies for downhill walkers, while uphill walkers retain switchbacks. For weakly inclined slopes, the best strategy involves walking directly uphill or downhill." For sufficiently steep slopes, however, they suggest that the best strategy should undergo a "transition to a broken symmetry solution corresponding to the switchback trail patterns typical of rugged environments. The critical slope at which this transition takes place should be less steep for uphill and downhill walkers. The theory should be amenable to empirical investigation."

Among other applications, their mathematical model "will enable us to generalize the work of previous authors to real landscapes, eventually permitting the reconstruction of ancient patterns of movement in archaeological landscapes."

The study, titled "Zigzagging: Theoretical Insights on Climbing Strategies," appeared in the Journal of Theoretical Biology (November, 2007).

Source: Newswise