Mathematics of climbing
The Mathematics of climbing encompasses various mathematical concepts and principles that enhance the understanding and performance of climbing activities. Climbing can be categorized into styles such as bouldering, ice, and free climbing, each requiring different techniques and physical attributes. Notably, anthropometry plays a significant role in climbing, where body measurements—like the ape index—are analyzed to understand climber efficiency and success. The fall factor, a critical metric derived from mathematical equations, quantifies the potential impact of falls, taking into account variables such as rope length, climber mass, and gravitational forces. Additionally, mathematical modeling is applied to climbing, drawing parallels between tackling climbing challenges and solving complex mathematical problems. Competitive climbing also employs grading systems that blend objective measurements with subjective feedback on difficulty. Overall, mathematics serves as a vital tool in optimizing safety, strategy, and performance in climbing pursuits, providing insights that go beyond mere physical strength and skill.
Mathematics of climbing
Summary: Effective climbing relies on mathematical principles, and there are connections between climbing and mathematical problem solving.
Climbing is the use of the human body and assisting equipment to ascend or descend steep surfaces. Climbing can be done professionally, such as for construction or in the military, for exercise or competition, or for performance—in the case of parkour. There are different styles of climbing depending on the object, such as bouldering, ice, tree, and rope climbing. If the weight of the climber is supported by equipment, it is called aid climbing; when the weight is supported only by the climber’s muscles, it is called free climbing. Mathematics plays a role in successful climbing and in analyzing various aspects of the discipline. Mathematician Skip Garibaldi said, “Climbing has a lot of puzzles that have to be solved. It’s not just strength or skill.”
Anthropometry in Climbing
Anthropometry is the mathematical study of body measurements in order to understand human variability. For example, studies show that elite climbers, on average, tend to have small stature, low body mass, and a high handgrip-to-mass ratio compared to the population as a whole. Compared to nonclimber athletes with similar physical conditioning, they are frequently linear, with narrow shoulders relative to hips. Ape index is the ratio of a climber’s arm span to height. In adults, it is usually close to one, as illustrated in Leonardo da Vinci’s “Vitruvian Man.” An ape index greater than one is reputedly advantageous for climbing, and some researchers have found ape index to be a statistically significant predictor of climbing success.
Fall Factor and Impact
Fall factor quantifies how hurtful a fall may be to a roped climber. Mathematicians such as Dan Curtis have derived the fall factor (Fmax) using differential equations. It is a function of the ratio of the total distance the climber falls (DT) to the length of the unstretched rope (L) between the climber and belayer or anchor at the rope’s other end. It is also a function of the climber’s mass (m), the elasticity or “stretchiness” of the rope (k), and gravity (g). Algebraically, it is represented as
Climbing ropes must pass a statistically designed drop test to be certified for sale and use. Other critical safety equipment is also designed using mathematics. One example is the curve of cams used in the “friend” devices that secure ropes to crevices in rock walls, which may be optimized using systems differential equations, sometimes with polar coordinates. The devices themselves are an application of logarithmic spirals.
Climbing Theories and Modeling
Many people have drawn parallels between climbing mountains and solving mathematical problems, especially great challenges like summiting Mount Everest and solving a problem like the Riemann hypothesis, first proposed by mathematician Bernhard Riemann. Analyses have shown that Everest climbers engage in multistep problem solving with altitude changes, rates, percentages, conversions, approximations, and division of large numbers. Mathematician-climber John Gill said that problems in both mathematics and climbing are often solved by “quantum jumps of intuition.” Patterns found in the natural features of some popular climbing locations can very mathematical. The Navajo Sandstone formation includes rounded domes and saddle shapes with remarkably precise-looking contour lines.
At the same time, the geometric diversity and complexities of climbing surfaces and the variety of techniques used by climbers have made developing a single theory of optimal climbing strategy difficult. However, several methods are used to quantify characteristics of different climbs and probabilistic models can be used to make decisions. Competitive climbers assign climbing grades to climbing routes, using objective and subjective criteria, to describe their difficulty. Other systems assess the technical difficulty of required moves, the stamina necessary, exposure to the elements, or the frequency of difficult moves. Mathematician Alan Tucker demonstrated using graph theory that the classic Parallel Climbers mathematical puzzle has a solution for any mountain range.
Further Reading
Curtis, Dan. “Taking a Whipper: The Fall-Factor Concept in Rock-Climbing.” The College Mathematics Journal 36, no. 2 (2005).
Garlick, S. Flakes, Jugs, and Splitters: A Rock Climber’s Guide to Geology. Kingwood, TX: Falcon, 2009.
Tucker, Alan. “The Parallel Climbers Puzzle.” Math Horizons 3 (November 1995).