Exponential growth as plotted by my son

 

IMG_20160619_134418072With my son, my niece, sister and her boyfriend I visited the science museum in Mannheim about two months ago. There where many nice experiments, and as a (former ?) computer scientist I was particularly impressed by Leibniz’ mechanical digital calculator, completed in the 17th century and able to do all four basic arithmetic operations. This of course is another example for the superiority of Leibniz over Newton
One piece was a pure exhibit with no experiments: a system of cogwheels connected sequentially, where each added wheel took ten times longer for one complete cycle compared to the one before. While the first wheel needed five seconds, the 17th and last one did not seem to move, and indeed it completes a cycle about every 1.6 billion years (5*10^{16}\mathrm{sec}).
This is an example of exponential growth of course, so I told him the rice on chessboard story, e.g. the idea that starting with one grain of rice on the first square of a chessboard and doubling it on the next leads to loads and loads of rice. When he asked for rice this weekend I was rather unsuspecting, until he spilled some of it and asked for our help cleaning it up. He started the first four squares (though with linear growth as he didn’t remember completely) and we completed the first row together – after which I started simply weighing the rice, as counting took about a second per grain and already for the 256 grains on h2 I was too impatient, let alone  the 1024 for f2, which would have taken about 16 minutes to count. If this little exercise doesn’t drive home what exponential growth means, I don’t know what would.

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