The Curious Mathematics of Domino Chain Reactions

You’ve probably seen the domino effect in action where a row of standing slabs topple successively. Usually, the dominos are all the same size but a toppling domino actually has enough momentum to push over a bigger one. So it’s possible to set up a row of successively larger dominos that can be toppled by the push of a tiny slab at the beginning–a domino chain reaction.

So here’s an interesting question. How much bigger can each succeeding domino be?

Today, J M J van Leeuwen at Leiden University in The Netherlands takes this problem by the scruff of the neck and gives it a good mathematical shaking. It turns out that the answer–the maximum growth factor– isn’t quite as simple as the problem might suggest.

There are various videos, such as this one, on the internet that give a good demonstration of the chain reaction effect. The standard thinking is that a domino can topple another about 1.5 times as big, provided that the spacing between them is optimal.

The basic physics is straightforward. Standing a domino on its end stores a certain amount of potential energy which is released by pushing it over. However, the force required to topple the domino is smaller than the force it generates when it falls. It is this “force amplification” that can be used to topple bigger dominoes.

But the devil is in the detail since there are various ways that the dominoes lose energy as they topple. For example, a toppling domino comes to rest on its neighbour. So the collisions are inelastic, which is the main source of lost energy.  And in practice, the dominos can slide along the floor when they are hit and this can seriously impede toppling.

So van Leeuwen makes a series of simplifications in his mathematical analysis. He assumes that the friction between the ground and the dominoes is effectively infinite so that they cannot slide. He assumes the collisions are fully inelastic so the dominoes stay in contact with each other when they collide. He also assumes that once in contact with each other, the dominoes slide frictionlessly over each other. 

Given these assumptions, he then shows that with optimal spacing, each succeeding domino can be no more than about twice as big as the previous one, which is maximum growth factor of no more than about 2. 

That’s significantly more than has been assumed in the past. He admits that achieving this limit is probably unrealistic in practice because the assumptions can never hold perfectly. For example, dominoes will always slip by a small amount. 

Nevertheless, even a growth factor of 1.5 leads to some extraordinary chain reactions. A series of 13 dominoes that grow at this rate will amplify the force needed to push the smallest by a factor of 2 billion. And it doesn’t need a particularly long series before the largest dominoes are the size of sky scrapers.

Entertaining mathematics!

Ref: arxiv.org/abs/1301.0615: Domino Magnification