Friday, July 23, 2010

The math of rock climbing

By Carol Clark

Do you know your ape index? What’s it like to fall 40 feet down a sheer cliff face, while dangling from a rope hundreds of feet from the ground?

Watch the video of Emory mathematician Skip Garibaldi describing his rock climbing experiences on El Capitan in Yosemite National Park. He also explains some basic climbing math, such as the fall factor, used to reduce the risk of injury during a rope climb.

“Climbing has a lot of puzzles that have to be solved,” Garibaldi says. “It’s not just strength or skill. You really have to think about the different ways you can place your body.”
Photo by Craig Clarence.

The sport seems to attract mathematicians, he adds. “When I learned how to climb, in San Diego, Mike Freedman was a professor there. He has the Fields Medal for his work on the Poincaré conjecture, and he helped develop the San Diego climbing scene.”

One of Garibaldi’s collaborators, noted French mathematician Jean-Pierre Serre, “has bouldered at Fontainebleau, near Paris, for decades,” Garibaldi says. And mathematician John Gill, who went to high school in Atlanta, and graduated from the University of Georgia, is considered the father of modern bouldering by many climbers.

Atlanta has a thriving climbing scene, Garibaldi says. Check out the Emory Rock Climbing Arena.

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  1. Climbers know that fall factor is important -- it determines how much impact a falling climber feels when the rope comes tight. But most don't know the reason why. You can find a good explanation in the article

    "Taking a whipper---The fall-factor concept in rock climbing" by Dan Curtis, The College Mathematics Journal, vol. 36, no. 2 (March 2005), pages 135-140.

    The explanation amounts to solving a differential equation, and he walks you through it. One semester of college calculus (in the language of the AP exams, Calculus AB) should give you enough background to read Curtis's article.

    By the way, in case it wasn't clear from the video, the fall factor is always between 0 (best) and 2 (worst or most unpleasant).

  2. A really interesting exercise to do is to calculate the KiloNewtons of force on a piece of gear in a given fall. For example, how much much force is held by your first piece of gear if you are a 190# climber who is 7 feet above your first piece, which is placed 9 feet above the ground.
    The results are very surprising, and are non-intuitive to climbers who think the first piece doesn't have to be "that good" because it is closer to the ground than pieces higher up.
    I built a spreadsheet which takes 4 variables;
    climber weight, rope stretch percentage, rope length, and fall length. That spreadsheet changed the way I climb trad. ;-)