Tying Shoes: Math May Make Case for How We Lace

Ben Harder
for National Geographic News

December 4, 2002

Math-phobes of the world, take heart! You might not have realized it,
but you've been acing at least one math problem each morning since the
day you gave up Velcro.

The way most people tie their shoes is, in fact, the most effective way to secure footwear to feet, according to mathematician Burkard Polster.

There are two basic approaches to lacing up: "criss-cross" lacing—the method most common in North America—or the mainly European style of "straight" lacing, in which laces run parallel above the eyelets and cross below them.

"The strongest lacings turn out to be the two most commonly used ones," said Polster, who teaches at Monash University in Victoria, Australia. "In some unquantifiable way, the common ways of lacing shoes also seem to be the simplest to execute."

"When you pull on the ends of a shoelace," Polster wrote in the December 5 issue of the journal Nature, "it acts like a pulley" that pulls eyelets toward each other, thereby holding the shoe firmly to the foot, and creates tension along the lace. The ideal lacing should create uniform tension along the length of the lace.

Which of the two techniques is stronger depends on the distance between the two rows of eyelets, and therefore, at least in part, on the design of the shoes.

There are methods of lacing that use a shorter length of lace, such as crossing from one row of eyelets to the other only after every second eyelet, said Polster. Shoes laced with this technique would sport a "bow-tie" pattern along their tongues but wouldn't grip the foot as firmly as traditionally laced shoes, he said.

No Fast Break For Footwear

"I don't expect shoe manufacturers to get terribly excited about all this," admitted Polster. "I am really talking about idealized shoes and shoelaces. Things like friction, the material that shoelaces are made of, and less than perfect alignment of eyelets are not taken into account at all."

What would inspire someone to devote time to studying how we tie our shoes?

"Tying shoelaces is a simple, familiar example of a geometrical optimization problem," said Ian Stewart, a professor of mathematics at the University of Warwick in England.

"The interest of all work in this area lies in the methods and the use of a simple example to illuminate general principles," Stewart said. "We learn nothing of practical value about shoelaces that couldn't be found by experiment—but we learn some interesting things about how to tackle other problems of a similar kind, and those are often of practical value."

All Tied Up In Knots

Using the most efficient lacing technique is only half the battle; then there's what kind of knot to tie.

"Hundreds of years of trial and error have led to the strongest way of lacing our shoes," said Polster. "Unfortunately, the same cannot be said about the way in which most of us tie our shoelaces."

The problem, he said, is that many people use a "granny knot," which they form by passing first one end of the lace around the other and then the second around the first from the same angle. This forms two identical half-knots on top of each other.

Better than the granny knot, Polster said, is the reef knot, which is commonly used by sailors. The reef knot, also called a square knot, is similar to the granny knot except that its two half-knots are oriented in opposite directions. [ Need a diagram? ]

The bows that people make while tying shoelaces have nothing to do with making the knot stronger, and serve only to make the knot easier to untie. They can be ignored for the purpose of comparing different knots.

"I have a better way of tying shoes," said John H. Halton, a professor of computer science at the University of North Carolina at Chapel Hill whose earlier paper on the subject inspired Polster's latest work.

Halton, now 71, learned a memorable trick decades ago, when he was a student at Oxford University. His instructor for the lesson was no luminary of academia; just a salesman at a local shoe store, who patiently demonstrated to a curious Halton how to tie laces with a surgeon's knot.

That knot resembles a reef knot, but includes one extra loop: "The rabbit goes back down the hole and comes up again," as Halton put it. Halton swears by the surgeon's knot and has been using it ever since he learned it. He's even taught the trick to friends and relatives who will humor him. Despite his efforts, he laments, "it simply has not caught on."

"Some people teach Christianity or Islam; I teach shoe tying," jokes the computer science prof. That, and, oh yeah, perhaps a few university-level computation problems that would make the average mortal's brain go numb.