Why do flat cords tangle less in my pocket?

We've all experienced this: put a pair of earbuds in your pocket, and when they come out, they're a tangled, knotted mess, cf. e.g. Why do earphone wires always get tangled up in pocket?

However, you can also buy versions with flat, ribbon style cables rather than the typical round ones. The flat versions are often billed as "tangle free" or "tangle proof". While claims of being immune to knotting are exaggerations, they do indeed seem to tangle less. Why?

• Ernie's answer is a good hint concerning the physical root cause for this difference, but concerning the following conclusions it is a problem for geometry/topology. You should repost the question with Ernie's explanation on the math forum. The outcome might be interesting. May 17, 2015 at 23:54
• Mathematical explanations of physical phenomena are on topic here. Reposting this on one of the math sites is poor advice, because even incorporating Ernie's remarks, this isn't a math question. An answer about tangle mechanics would have to consider flex and other material properties. That's physics. It would get bounced here or closed. May 18, 2015 at 12:24

Earbud cords form coils in your pocket. Even if you crumple the cord, it eventually finds its way into coils as you move and the cord is agitated. The coils lie in planes that are stacked.

It's been found that agitated coils spontaneously braid and knot themselves. See this paper: http://www.pnas.org/content/104/42/16432.full, which found that formation of knots in an agitated string is dependent on (1) the agitation time, (2) the length of the string, (3) the type of agitation, and (3) the stiffness of the string. A long, flexible string tends toward theoretical predictions of knotting, whereas a stiff string resists self-knotting.

Flat earbud cords are extremely stiff on two of their four faces. They do not bend easily through the narrow faces. Only the wide faces can easily bend. The wide surfaces fold, but they resist bending in other directions, across their narrow sides. Also, they resist twisting, and will only twist in a corkscrew with a relatively wide radius. Think of a flat cord lying in a coil in the x,y plane. With its wide faces perpendicular to the plane, but parallel to the z axis, the flat cords exhibit stiffness along the z axis, and resistance to leaving the x,y plane. In order to self-braid, the cord would need to flex above and below the x,y plane. Their shape resists this.

Round cords, on the other hand, bend and twist in all directions, so there is little resistance to the loops of a coil flattening into ellipses of various major and minor axes which then shift lateral position and send bights above and below the planes of other ellipses. They have equal flexibility in all three planes, so their propensity to self-braid by sending loops above and below the x,y plane is uninhibited.

Some manufacturers have reduced self-braiding and self-knotting by stiffening their round earbud cords. But early models were notoriously flimsy, and braided with very little agitation.

• You managed to pinpoint the probable root cause. But we still do not know, why a cord which refuses to bend in one direction is less prone to tangling. I think from this point this is more likely a math question than a physics question. May 17, 2015 at 23:50
• @Ariser: Physicists have found the cause of "spontaneous knotting of an agitated string", which seems to be random braiding of parallel strands. Stiffness of the cord reduces braiding, so major stiffness along the two faces of a flat cord probably prevents or at least reduces spontaneous braiding. See this: blogs.discovermagazine.com/seriouslyscience/2014/06/18/… May 18, 2015 at 1:22
• Can you elaborate on why the flat structure exhibits those properties? I also completely fail to understand the relevance of the Family Guy reference. How does that contribute to an answer to my question? May 18, 2015 at 12:29
• @Esoteric Screen Name: In order to self-braid and self-knot, a cord needs to loop above and below the x,y planes of its coils. Flat cords resist flexing and sending loops into the z dimension, so they do not knot as easily as round cords, which easily enter all three dimensions. I've edited my answer, and included a link to a paper that explores this through topological and mathematical analysis. The Family Guy reference is a literary device to point out the propensity of round cords to knot. I added clarification to the reference. May 18, 2015 at 16:54
• This answer was more or less my first thought. I would add that to form a knot requires a loop, and a loop in a long string forms by twisting a 180 turn - a "bight" - so that it closes, which is why twisting resistance is important. May 18, 2015 at 19:42