"How many dominoes does one need to topple a domino as tall as the Domtoren?" was a question in the Dutch Science Quiz 2012 and the inspiration for a quirky bit of mathematical physics from J M J van Leeuwen of Leiden University in the Netherlands.

The Domtoren is a 112 m-tall cathedral tower in Utrecht and the idea is to begin with a standard-sized domino, which topples a larger domino. This then topples an even larger domino and so on until a Domtoren-sized domino can be felled. The process is called "domino multiplication" because a tiny tap on the first domino can, in principle, topple a huge monolith. Now, Van Leeuwen has calculated the upper limit on how much larger each successive domino can be. In principle, his calculations suggest that the maximum ratio of successive domino heights can be about 30% larger than the widely accepted value of 1.5.

The underlying principles governing the domino effect are simple: each brick is given potential energy when it is raised against gravity and stood on end. Given the slightest push a brick will fall, releasing this energy. It is capable of toppling a larger neighbour because the energy required to tip the bigger brick over is much less than the potential energy the smaller brick releases on falling.

### Real-world effects

But in the real world not all that energy is channelled into bringing down the next domino in line. First off, the dominoes bounce a little as they strike one another. Next, they have a tendency to slip along the surface they are stood on as they are nudged, lessening the chance of a fall or causing them to fall back towards the striking domino. And finally, once in contact, they drag against one another as they fall.

In his model, Van Leeuwen simplifies the situation by assuming that the collisions are completely inelastic, that the friction between the dominoes and the surface they stand on is infinite, and that the dominoes, once touching, experience zero friction and simply slide over one another.

### Chain reaction

Van Leeuwen's idealized model shows that – assuming optimal spacing between successive dominos, a fixed density and thickness-to-height and width-to-height ratios for all dominoes – the maximum theoretical "growth factor" is two. In other words, each domino can be no more than twice the height of the one that strikes it, if the chain is to continue. Until now, that limit was widely thought to be 1.5.

"In real life, [these assumptions] are not realized," Van Leeuwen concedes. "But in principle you could reach two," by standing the bricks on a very high friction surface and lubricating their upright surfaces.

### Energy amplification

Even with a more modest ratio, the effect in numbers is impressive, as demonstrated in the above video by Stephen Morris at the University of Toronto. Morris topples a series of 13 dominoes with a growth factor of 1.5. He claims that the energy needed to tip the first fingernail-sized domino is amplified two-billion times by the end of the chain reaction – when a 45 kg block crashes to the floor. "If I had 29 dominoes," says Morris, "the last domino would be as tall as the Empire State Building."

So, starting with a standard domino 4.8 cm tall, how many are needed to topple the Domtoren? Assuming a growth factor of 1.5, it is 20 dominoes, but pushing the growth factor to two, it should easily be done with just 12.

The work is described in a preprint on the arXiv server.