Sometimes we have to be reminded: all of our mathematics and science are ways for us to explain (to ourselves) how the natural world behaves.  By these tools we give the observed world some order and some predictability.  But it is critical to remember that these tools are not laws of nature that we have deduced, but rather they are human approximations to a complex natural world.  Sometimes these approximations work quite well.  But sometimes they fail spectacularly.  And most of these failures have to do with non-linearities and with ignorance of Hamming's Law.


As human beings we tend to think linearly.  Things are proportional to each other.  The more we eat, the more we weigh.  The faster we run the quicker we get someplace.  A heavier weight hung from a spring produces a proportionately larger displacement of the spring (Hooke's Law).  There are many, many other examples.  In fact, most of the mathematics underlying our physics and engineering explanations of nature are linear in this fashion.  There are at least three reasons for this: first, linear mathematics is generally tractable and we can produce useful solutions, and, second, these linear solutions work quite acceptably over quite a wide range of applications.  Finally, linear relationships are intuitive -- they "feel" right to us.


It is the purpose of this section to applaud the usefulness of linearity, but then to warn that it is not nature itself which is really linear, and that every so often our everyday experience runs up against something which seems non-intuitive and unexplainable and which is caused by intrinsic non-linearity.  Sometimes it is for physical reasons: pull that Hooke's Law spring too far and the elastic limit of the material out of which the spring is made will be exceeded and the spring will not elongate linearly and will likely not return to its original shape when released.


But sometimes the non-linearities we confront are intrinsic to the situation.  The most common example is in the flow of everyday, urban traffic.  Have you been stuck in a traffic tie-up that proceeded slowly until, when you came out the other side of the congestion, you found that there was no apparent cause?  This happens to all of us quite often.  Indeed, there was no need for a "cause" such as an accident or construction.  A branch of mathematics called quadratic queuing describes the flow of traffic.  A queue is the waiting line that builds up at a ticket window, or at a workstation in a factory.  When the ticket window or workstation cannot process the arriving stream of requests as fast as they arrive, a queue builds up.  If, during some period of time, six requests for service come in and only five can be serviced, one will be left over.  If the same thing happens during the next period of time, another request will be left over, making two.  And, indeed, the queue of waiting requests will build up linearly with time at the ticket window or at the factory workstation.  All this seems intuitive enough and we are not bothered by this behavior.


But in the case of traffic flow, while queues form in this same fashion, the queue forms in the line of approach rather than off to the side, so that the next vehicles have less far to come to join the queue.  And, in fact, this queue does not build up linearly, but builds up as the square of time and the difference in the request rate and the service rate.  This is because the queue extends into the approaching vehicles that, in turn, join the queue at earlier and earlier positions.  This quadratic behavior is what gives the name to quadratic queuing and is the reason why traffic is often non-intuitive.  When traffic is heavy-- that is the flow is approaching the limit of what the road can handle -- any small fluctuation can cause a quadratic queue to start to grow right in the middle of the traffic, even though there really was no specific, external cause.  Stories are told of queues that build up at toll booths in high-speed highways, where the queue at the toll booth during heavy traffic grows fast enough so that the end of the queue is approaching oncoming motorists at high speed (perhaps 60 miles per hour) thus causing major crashes and pileups.


Quadratic queuing is only an example of non-linearities occurring in our usually linear world, though it is the one that comes closest to our everyday experience.  But there are many others and they are equally non-intuitive.  Non-linear oscillators make interesting mathematical studies -- rather than oscillating nicely in the way we expect (e.g. pendulums, linear springs etc.) they exhibit really odd, non-intuitive behavior, for example jumping suddenly from one mode of oscillation to another.


To all this we have to add chaos theory.  This theory (oversimplified) says that there are regimes in nature (often those areas governed by non-linearities) where the outcome of a system's behavior is dependent on the very fine detail of its initial conditions.  Thus simply knowing the rough initial conditions which, in a linear world, would permit calculating the system behavior doesn't allow meaningful predictions to be made because of the systems high sensitivity to the details (rather than the generalities) of its initial conditions.  This gives behavior that appears chaotic -- that is, highly unrelated to the general initial parameters -- but which may still be tractable when the details are taken into account.  Unfortunately, sometimes it is impossible to measure the initial conditions well enough to really be able to predict behavior -- weather is a prime example of this.


I have often thought that no engineer should be licensed to practice without having first taken a course on non-intuitive/non-linear phenomena.  The Tacoma Narrows Bridge blew down because of non-linear oscillations that developed in the structure in high winds.  Every mechanical and civil engineer should have to watch the film of that disaster.  Engines fell off the Lockheed Electra aircraft because the high vibration levels in the aircraft caused the engine mounts to enter a regime of non-linear oscillations that eventually broke the engine mounts.  Economic models containing multiple feedback loops do not behave in the ways we are used to -- multiple feedback loops cause a high level of non-linearity, and our linear intuition and experience are poor indicators of what the model will really do.  The non-linear behavior of atmospheric conditions makes the analysis of global warming very difficult and contributes to the disagreements and uncertainties that characterize this important topic.


It is critical that you always ask whether the project in which you are involved lies well within the areas of (usually linear) practical experience or whether the project is trying to move outside of this realm and may then be at risk of encountering unexpected and potentially dangerous behavior.  Too often we assume that our linear world continues as an extrapolation of our experience.  While this often works, a good technical manager is sensitive to this issue and will probe to be sure that the assumption is valid.