 Have you ever wondered why, when we look around us in the natural world, we see very few straight lines, but the straight line is ubiquitous in the systems we engineer. From buildings to circuit boards it appears to be the default position. One way to understand this is that the systems we engineer are based upon our scientific and mathematical understanding of the world, which has inevitably started by describing the simplest and most orderly systems, that is to say those that are composed of linear forms and relations. From Euclid to Newton and on, science has been focused upon the orderly systems of perfect squares, triangles and linear relations of cause and effect, that can be encoded in beautifully compact equations. Thus we describe the real world as a kind of approximation to these perfect linear forms. But let's take a look at some of the basic principles underlining the theory of linear systems. Linear is basically a fancy word for line, and a line is often understood as the shortest or most direct path from one point to another. To say there is a linear relationship between two things is to say that there is a direct relationship of cause and effect between them. Let's take an example of this. Say I'm playing baseball and I hit a ball with a bat. If we make a model of this system, we see that the energy inputted by my swinging of the bat will be directly proportional to the output of the ball's momentum as it travels off in the opposite direction. This is a simplified model, but it illustrates the direct or linear relationship between cause and effect, my swinging of the bat and the ball's motion in response. There are of course many examples of linear systems, particularly within physics, but we can capture the underlining logic of linear systems in general with a model that has only two simple rules called the superposition principles. Each states firstly that the output to the system will always be directly proportional to the input. So if there is a linear relationship between the amount of fuel I put in my car and how far it will go, well then if I put twice as much fuel in the car it will go twice as far. And secondly that if we add two or more systems together, then the output to this combined system will simply be the sum of the two outputs of the original systems. So say we have two tractor factories, each producing a million tractors a year. Well if we merge them then we will get a factory that will produce two million tractors a year. Linear systems are deterministic, meaning that if we know their present state we can fully determine their past and future states. This can be seen by plotting a linear system as a graph where it will always be depicted as a straight line. Although linear systems modelling has proven highly successful in many areas and is often a very good approximation, the reality is that we live in a world with ecosystems economies societies and physical systems that are not governed by the superposition principles and thus are what we call non-linear. An example of non-linearity might be listening to two of your favorite pieces of music at the same time. Because there is a relationship of interference between them, the result of the experience will not be a simple equation of adding the enjoyment from listening to each independently. This illustrates how non-linearity arises whenever there is some relationship between elements within the system that can be either synergistic making the output of the system greater than the sum of its parts or one of interference making the output less than the sum of its individual components. To illustrate this further let's take an example of four workers producing clothing. In isolation each seamstress can sew a given amount of clothes within a day. Now if we put them together we might get one of three results, firstly they might not interact with each other very much meaning the whole would simply be the sum of its individual parts. But equally likely they might form some cooperative relationship which lets them each specialise in a particular function making them more efficient as a whole and thus the output of the system would be greater than the sum of its parts due to these synergistic relations. Or inversely they might start all talking with each other, getting little work done and thus the total output would be less than the sum of the individual outputs due to these relations of interference. Non-linearity can also arise from feedback loops whereby the same process is iterated with the output fed back as the input to the next cycle. A classical example of this is compound interest where at the end of each period the balance plus interest is fed back into the formula to compute the next cycle of interest accumulation. Non-linear functions are an important concept within non-linear science and have been used to create a whole new type of geometry called fractal geometry whereby iterating a simple function generates irregular organic looking patterns that can model many of the geometric forms we see in nature from the structure of seashells to the rugged formation of mountains. In these non-linear systems superposition fails meaning one cannot break the system down into smaller sub-problems and then add their solutions. We must consider a non-linear problem in total. It is this need to approach non-linear systems as a whole that is giving rise to new, more holistic approaches to science that are developing under the canopy of the term complex system science. Non-linearity in all its shapes and forms is at the heart of many of the 21st century challenges to science as we try to extend the scientific framework beyond its dependency upon linear systems theory to finding new ways to embrace the complex world we live in on its own irregular and imperfect terms.