Knotty Cords

You're done with your iPod, so you carefully coil the headphone cord around the player and stuff it in your pocket. The next time you take it out, however, you find that you have to unravel the cord and undo a knot before you can go back to listening to your music. In fact, it's pretty amazing how easily knots can form by themselves—not only in headphone cords, but also in necklaces, coiled ropes, strings of holiday lights, hanks of yarn, or garden hoses.


A coiled headphone cord, once unraveled, is likely to contain a knot.

Two physicists at the University of California, San Diego recently decided to do some experiments to try to find out why knots form so easily in coiled strings. They looked at what happens when a long string is coiled into a box and the box is then tumbled.

The researchers, Dorian Raymer and Douglas E. Smith, found that complex knots form within seconds—if the string is long and flexible enough. So, for a given stiffness, a string has to be a certain length before a knot will form. Moreover, the longer and more flexible the string, the better a chance it has of becoming knotted, especially when the string is tumbled or shaken for a long time.


This illustration represents the knot experiment, in which knots form in a tumbled string. Dorian Raymer, UCSD.

The researchers did the experiment over and over again: 3,415 times in all. Knots formed in the strings about one-third of the time. The biggest surprise came when the physicists used mathematics to identify the types of knots that formed in the strings. Mathematicians called knot theorists have described many different kinds of knots, according to features such as crossings (the number of times a string crosses over itself). The physicists found that their real-life model produced 120 different knots. In fact, their strings formed all of the possible kinds of knots with up to seven crossings—and seven crossings is quite a tangle.


Digital photos of knots are combined here with computer-generated drawings based on mathematical calculations. Dorian Raymer, UCSD.

Raymer and Smith say that knotting tends to start from one end of a string. As a string tumbles, this free end braids itself with strands lying next to it, weaving over and under adjacent segments to create a knot. The more braiding, the more complex the knot.

So, if your headphone cord is long, thin, and flexible (if?!), the chances of it becoming knotted are annoyingly high. Sigh.

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Muse, March 2008, p. 18.

Border States

Take a look at a map of the United States and locate Colorado and Wyoming. On many maps, these states look like perfect rectangles.

The laws that created these two states specify that each one lies between certain lines of latitude and longitude. (Latitude lines are drawn side-to-side on a globe; longitude lines go top-to-bottom.) Wyoming stretches from 41°N to 45°N latitude and from 104°W to 111°W longitude. Colorado's borders, meanwhile, are defined by 37°N and 41°N latitude, and 102°W to 109°W longitude. If Earth were flat, both states would be rectangles with parallel opposite sides.

But Earth is a sphere, and, on the surface a sphere, although lines of latitude are parallel, lines of longitude get closer together as you travel northward from the equator. So, the northern border of each state is a little shorter than its southern border. For instance, in Colorado, the difference is about 21 miles. The reality is that both states are shaped like trapezoids (four-sided figures with just one set of parallel sides), but on a curved surface.

There's another complication. When the states were created, surveyors had to go out into the wilderness to map the boundaries, using a compass and the stars, along with a few other tools. They followed the appropriate lines of latitude and longitude the best they could, marking the boundaries mile by mile.

The boundary between Utah and Colorado, for example, runs 276 miles from Four Corners (the only place in the United States where four states share a point) to the Wyoming border. Later surveys showed that surveying were made between mileposts 81 and 89 (northward from Four Corners) and between mileposts 100 and 110. These errors put kinks in what should have been a straight line. If you look at a highly detailed map of Colorado, you can see the kinks. There are similar errors along other borders that are supposed to be straight lines, including those of Wyoming.

Interestingly, once a border is defined on the ground and accepted by the government, it becomes official, even if it doesn't follow the written description.

Perhaps it's best to describe Colorado and Wyoming as polygons—geometric figures with many straight sides, even though they are also curved over the surface of a sphere.

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Muse, February 2008, p. 15.

Ways to Lay Track

When you're a little kid, one of the joys of having a train set is assembling all the pieces into the longest possible loop, then operating a train that follows the railroad's twists and turns. And, as your collection of tracks—straight pieces, curves, bridges, tunnels, switches, and crossings—gets larger, your designs get more and more elaborate.

Mathematician Mark Snavely of Carthage College in Kenosha, Wisconsin, is one of those people who never outgrew train sets. When young Brian Snavely, Mark's son and a fan of Thomas the Tank Engine, got his own train set, his dad decided to explore an interesting math problem: finding the number of different ways in which sets of tracks with switches can be laid out to create looped paths.

If you start with only curved and straight tracks, there's really just one choice. You can join the ends to make a loop. If you have enough pieces, the loop might be pretty twisty rather than oval, but it's still a single loop.

Switches, which allow branching paths, make things a lot more interesting. For example, a train reaching a two-way switch (a "Y") can choose either one of two possible paths as it continues on its way. With two two-way switches, you can create an oval from which another loop extends outward.


One example of a railroad layout made with two two-way switches.

Mark, with help from two students, figured out that, with two two-way switches, you can create exactly five different types of layouts. Of course, you might need lots of curved and straight pieces to complete the different loops. All five arrangements have two loops, but they differ in the way the loops are connected and in the sorts of paths that a train can follow. And some of the arrangements aren't very practical. A train can get stuck—unable to leave the loop on which it finds itself. You can always add more pieces of track, but as long as you have just two switches and no free ends, there are still only five fundamentally different patterns.


Here are the remaining four railroad layouts using two two-way switches.

It doesn't stop there with switches. There are three-way switches (Mark says there are seven layouts you can create using two of these) and four-way switches (. . . ? Mark won't tell!).


This railroad layout uses just one three-way switch.

And, you can use more than two of each type of switch. The possibilities are endless, not only for creating interesting layouts for trains (the expertise of Brian), but also for mathematical study (Mark's department).

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Muse, January 2008, p. 36.