Clocks are amazing. They tell us time. In this post I want to talk about working of analog clocks. In case you haven’t seen an analog clock, this is how it looks like:




A wall clock, it’s working part in the back and inside of the working part.

But, what clocks have to do with mathematics? As I have mentioned several times, one major part of mathematics is about counting and clocks “count”! Clocks are amazing counting device, they perform mod 12 and mod 60 calculations (that’s why modular arithmetic is also called clock arithmetic).

Unfortunately, the ideal cases exist only in our abstract world of mathematics. In real world, whatever we build has some error percentage and our motive to minimize this error. A mathematical construction, called Stern-Brocot tree, was created to help build timepieces and understand number theory.


By Aaron Rotenberg (Own work) [GFDL or CC-BY-SA-3.0], via Wikimedia Commons

This “tree” gives an exceptionally elegant way to enumerate the positive rational numbers and is a surprisingly useful tool for constructing clocks.  For more information about this construction read this feature column article by David Austin.


Just like continued fractions, this tree gives us good rational approximations of a given real number. Clocks typically have a source of energy–such as a spring, a suspended weight, or a battery–that using gears turns a shaft at a fixed rate. We can increase the precision by using more number of gears of different teeth count in appropriate combination.

I will end this post with an example from Austin’s article:

Suppose we place a pinion on a shaft that rotates once every hour and ask to drive a wheel that rotates once in a mean tropical year, which is 365 days, 5 hours, 49 minutes. Converting both periods to minutes, we see that we need the ratio 720 / 525,949. The problem here is that the denominator 525,949 is prime so we cannot factor it. To obtain this ratio exactly, we cannot use gears with a smaller number of teeth. It is likewise impossible to find a multi-stage gear train to obtain this ratio. But, as we slide down the “tree” toward 720 / 525,949, the rationals we meet along the way will give good approximations with relatively small numerators and denominators. As we descend the Stern-Brocot tree towards 720 / 525,949, we find the fraction 196 / 143,175, which may be factored into four rational factors, 2/3, 2/25, 7/23 and 7/83. We can therefore construct a four-stage gear train and can get a pretty accurate clock.

I hope I have been able to convince you that clocks are much more interesting than they would appear and you should read the article by David Austin for further references.


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