Mathematics of Dominoes
Many mathematicians have been fascinated by the chain reaction of a falling domino. After all even the simple fact that a small domino is able to knock over a much larger one suggests there’s something quite intersting with the chain reaction that is occuring. It’s obviously due to the momentum of the first domino but how can we work out the mathematics involved in this process. After all just look at the momentum produced in this famous domino toppling scene – there’s another one on the BBC Iplayer, check this post and how to access it from the States – https://mathsurvey.org/how-to-watch-bbc-iplayer-in-america.html
So how does this chain reaction work and how much bigger can each domino get to maintain the reaction. There have been lots of guesses and arguments and commonly people tend to believe that each domino can topple one 1.5 times the size. This is of course also dependent on spacing as well as the momentum generated.
In a physics context it’s quite simple – when you lay a domino on it’s side it stores a certain amount of potential energy. This energy is released when it’s pushed over. The force required to knock the domino over is less than that generated when it falls over. Therefore there is an amplification of the force which means a little domino can knock over a bigger one.
Well a mathematician from the Netherlands has simplified the issue for us all in a mathematical analysis. His name is Van Leeuwen and makes a number of simplifications. Things like the friction involved is infinite so there can be no sliding. No elasticity in the contact between dominos and that they always stay in contact with each other.
Van Leeuwen then demonstrates that when the dominos are spaced optimally that the maximum growth size of each domino is a factor of two. So my wife was wrong, I don’t waste too much of my time browsing on the internet – I’ve now discovered this evening how to change to a fake ip address from here http://www.theninjaproxy.org/tv/a-fake-uk-ip-address/ and also figured out how many dominos I’d need before I could harness enough energy to start knocking over tower blocks using this chain reaction !!
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