Could a domino small enough to hold in your hand cause a chain reaction that could topple something as big as 112-meter tall tower? It sounds like a plot hatched by a kooky domino-themed super villain, but a new mathematical model shows it's theoretically possible.
A typical domino is just under 2 inches tall, 1 inch wide, and about one-quarter of an inch thick. These dimensions create a thin block that's just stable enough to stand upright yet unstable enough to fall over with the slightest nudge.
"If you make them too thick, for instance if you had dominoes like cubes, they would never [fall]," says physicist Hans van Leeuwen of Leiden Univ. in the Netherlands.
Each upright domino is also full of potential energy. When the first domino falls, the force of gravity turns that potential energy into enough kinetic energy to topple a domino larger than itself. That taller, heavier domino stores even more potential energy, and that energy will continue to mount so long as each falling domino’s kinetic energy can overcome the potential energy of their more massive neighbors.
Mathematicians have traditionally assumed that no domino could knock over a neighboring domino more than about one-and-a-half-times its own width, height and thickness, or a "growth factor" of 1.5. But there was no overarching mathematical model. So, when last year's annual Dutch National Science Quiz TV Show, run by public broadcaster VPRO, asked how many dominoes it would take to topple a domino the size of the 112-meter-tall Domtoren – the tallest church tower in the Netherlands – van Leeuwen set out to calculate just how much punch a falling domino packs.
But falling dominoes are deceptively complex. A domino can slide against its neighbor after colliding, losing some energy to the friction. Alternatively, if there's too little friction at the domino's base, its bottom can slip out from underneath, and the domino will lose some of its forward momentum. For van Leeuwen's equation to work simply, he needed to rule out those factors and imagine an ideal, purely mathematical domino that could avoid all these potential complications.
It turned out, van Leeuwen says, that an ideal domino could knock over a domino twice as tall, wide, and thick as itself – a growth factor of about 2 – so long as the dominoes were hollow. That means that while it would take about 20 solid dominoes with a growth factor of 1.5 to knock over a domino the size of the Domtoren, hollow dominoes that avoid friction could knock over a tower-sized brick in merely 12 steps.
Van Leeuwen posted his calculations this month on the prepublication website arXiv.org.
But the science quiz show wanted to set a record, and use van Leeuwen's findings on domino growth factor to topple the biggest domino ever in just 10 steps. So they put van Leeuwen's math to the test and built a series of hollow, wooden dominoes, the largest of them a half-ton, 26-foot-tall monster.
These were not the idealized dominoes of van Leeuwen's model, so building each successive domino twice as big would have seriously risked failure. Instead they made each domino five-thirds the size of the last, a growth factor of 1.67. When they tried it, the 26-footer at the end came tumbling down just as they'd hoped, all from initially knocking over a domino of normal size.
"It made it," says van Leeuwen. "That was fun to see."
It was enjoyable to see the large domino fall, van Leeuwen says, but knocking over an actual tower or skyscraper is not really plausible. It would require dominos that are solid instead of hollow, and a solid 112-meter domino would weigh 80,000 tons. There’s no crane imaginable that could lift that kind of weight, he says.
The model provides an answer to a fun question, according to physicist Michael Johnson of the Univ. of Central Florida in Orlando, who was not involved with van Leeuwen's work.
Johnson says that questions like this help inspire people to become mathematicians and scientists.
"People who do mathematics and do science have to stay inquisitive," Johnson says. "There's kind of a playfulness in that."