Forget Dark Energy: MIT Physicists Have Finally Cracked Overhand Knots

A big rock climbing fall is called a whipper. A climber may be protected from "hitting the deck" by a rope, but there are many routine occasions in which the rope accumulates slack. Any slack translates directly to a corresponding amount of freefall for the wayward climber. In some cases, this can wind up being 20 or 30 feet or more of slack, which is a lot space for a human body to build up some momentum.

So, when the rope finally does catch the falling climber, the result is some possibly rather harsh whiplash. Climbing ropes are amazingly tough (and stretchy), but that doesn't matter a bit if the climber isn't attached to one of those ropes by a bombproof knot capable of taking several thousand newtons worth of sudden impact. It's a peculiar combination of high-tech (ropes are pretty advanced these days) and the amazing, ancient low-technology of a knot.

It's safe to say that most people don't think very much about knots. Sailors, stagehands, surgeons, fishermen—yeah, OK. That's still a pretty limited subset of the population. Everyone else just ties their shoes while fudging the occasional granny knot where needed.

Knots are indeed a relatively ancient art, a technology developed across centuries of trial and error and some very old, intuitive notions of symmetry and elegance. (The more "ugly" or random a knot looks, the less likely it is to function well.) The basic physics and mechanics of knots are, however, relatively unstudied scientifically. If a knot works then it works—what more is there to ask?

Quite a bit, it turns out. In a study recently accepted for publication in the Physical Review Letters, engineers at MIT and Pierre et Marie Curie University in Paris offer a new fundamental theory of knots based on relationships between topology, the mathematics of spatial relationships, and the basic mechanics of friction and pliability.

"The simplest type of knots that we tie everyday is known as the overhand knots," explains study co-author Khalid Jawed. "The topology of overhand knots is defined by the unknotting number, n (number of times the knot must be passed through itself to untie it). Shoelaces are commonly tied using the reef knot, which comprises two overhand knots each with n=1. We combine experiments and theory to study the mechanics of overhand knots, and essentially answer the question how much force has to be applied to tighten the knot."

The force, which can be interpreted as the "strength" of a given knot, is given by a couple of different but related factors. The first is friction. As we wrap rope around rope more and more times, you're adding more and more surface contact and, thus, friction. Friction is then implicated in the knot's other dominating forces, which come from the tension and general bendability of the rope.

The researchers were able to quantify the strength of a knot by using a pair of mechanical arms to tighten it down. You can imagine the arms functioning as something like horizontal scales, measuring mass as resistance instead of weight. What they found was that by wrapping the rope around itself 10 times (in the overhand knot configuration) it was possible to up the knot strength 1,000 times from a single-wrap configuration..

Pedro Reis, an engineering professor at MIT and the lead investigator behind the current study, originally sought to build on the work of the French theoretician Basile Audoly. Audoly had previously developed a theory relating the mechanical forces involved in overhand knots based on experimental observations of only single and double twists, assuming that the same theory could be used to predict the forces involved in knots of increasing twists. Turns out there's a bit more to it.

"When Pedro Reis showed me his experiments on knots with as much as 10 twists, and told me that they could resist such a high force, this first appeared to me to be far beyond what simple equations can capture," Audoly explained in an MIT statement. "Then, I thought it was a nice challenge."

The relationship between twists and knot strength turned out to be non-linear: What goes into the knot (twists) is not directly proportional to what comes out of the knot (strength).

"This theory helps us predict the mechanical response of knots of different topologies," Reis said. "We're describing the force it requires to close a loop, which is an indicator of the stiffness of the knot. This might help us to understand something as simple as how your headphones get tangled, and how to better tie your shoes, to how the configuration of knots can help in surgical procedures."