## Math and MEMS: More Exponentials

Yesterday, Allyson Hartzell, the Director of Failure Analysis, Test, and Reliability at Pixtronix, Inc., visited Girls’ Angle to tell us about how she uses math in her work.

She works with Microelectricomechanical Systems, or MEMS. MEMS are tiny machines. They are so small that a hundred of them could fit across the width of a human hair. Even though these machines are so tiny, they can be very complex. Allyson showed us pictures of MEMS that contained several working gears. She even showed us a steam engine. According to Allyson, if you have ever used a Wii remote, you have relied on a MEMS, because it is a MEMS that enables the Wii to know how you’re moving the Wii remote around.

Last Wednesday, I blogged about exponential growth. At one point in her presentation, Allyson showed us this graph:

D.M. Tanner, et al. (2000) 'MEMS Reliability: Infrastructure, Test Structures, Experiments, and Failure Modes'. Sandia Report SAND2000-0091, p. 78 (Courtesy Sandia National Laboratories, Radiation and Reliability Physics Dept., http://www.mems.sandia.gov).

This graph gives a perfect example of why understanding exponential growth is so useful. It shows the number of revolutions that a tiny gear can make before one can expect it to break down and no longer work. If you look at the vertical axis, the equally spaced marks are labeled by powers of 10, starting at $10^3$ and ending at $10^{10}$. Having grasped exponential growth, we know that $10^{10}$ is much bigger than $10^3$. Each time you go up a mark along the vertical axis, you are looking at an increase in the longevity of the gear not by a little bit, but by a whole factor of 10. So, for example, one thing this graph shows is that about 70% of the gears will last 1,000,000 revolutions, and of the gears that survive 1,000,000 revolutions, half will last not just a little bit longer, but 100 times longer (assuming that the number of revolutions per unit time is constant).

You might be wondering, “Why bother with all these exponentials? Why not just redo this graph using axes that are labeled so that evenly spaced marks represents evenly spaced numbers?” The best way to answer this question is to go ahead and try to convert the graph to one that uses such axes. You will find that it is practically impossible to do. Don’t take my word for it. Try it yourself! If you succeed, please send us your graph; we’d love to see it!