Tangled headphones are a problem we all suffer from, but there is actually a mathematical formula that describes exactly how they will tangle. Even when you take proper care of the headphones and coil them before you put them in your bag, they are still likely to be tangled when you take them out. Physicists from the University of California at San Diego have just published a paper on the formulas behind headphone knotting. They studied mainly the probability of a headphone tangling based on length of the cord. They ran 3,415 trials of shaking a headphone in a box and found some interesting results.
[Image Source: PNAS]
"It is well known that a jostled string tends to become knotted; yet the factors governing the “spontaneous” formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots." ~ PNAS
When a pair of headphones is less than 46 centimeters, it is highly unlikely that they will become tangled, but the higher in length they get, the probability increases exponentially. You can see the probability curve in the image below.
"Above a critical string length, the probability P of knotting at first increased sharply with length but then saturated below 100%. This behavior differs from that of mathematical self-avoiding random walks, where Phas been proven to approach 100%. Finite agitation time and jamming of the string due to its stiffness result in lower probability, but P approaches 100% with long, flexible strings." ~ PNAS
[Image Source: PNAS]
You can notice that the probability tops out at around 55 percent. A pair of iPhone headphones is 139 cm, according to Business Insider, which puts them right in the 50% range of being tangled. You can measure and plot your own headphones on this graph to see if you are doomed to a life of headphone entanglement as well. The physicists also found that the tangling happens when the stray ends are allowed to freely lay. When a bag or container is shaken, these free ends tend to wander and intertwine with the loops around them, causing tangles.