 The study of the process of emergence can be understood as the study of pattern formation. So before we can go much further into the subject, we need to firstly define what we mean by this idea of a pattern as it's clearly central to the whole enterprise of studying emergence. The term pattern is an integral part of many different areas from art and design where it's understood in more aesthetic terms to mathematics and computing where it's understood in more structural terms. The first thing for us to note is that the term pattern is a highly abstract concept. Indeed, it is probably one of the most abstract concepts in our entire vocabulary. But this does not mean we cannot define it with precise and careful terminology. Here we will define a pattern as any form of correlation between the states of elements within a system. All systems exhibit some form of a pattern either in space between their parts or over time within some process and these patterns can be understood as the product of some form of correlation between the constituent elements states. A correlation is a structured relationship of some kind between variables. A combination of correlations between elements forms a regular or intelligent pattern. If there is no correlation between parts then they are randomly associated. Randomness can be understood as the absence of organization and thus the opposite of a relational pattern. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. All patterns have an underlining mathematical structure. Indeed, mathematics in its modern form may be understood as the science of patterns. Similarly, in the sciences theories explain and predict regularities in the world through modeling the correlated changes in properties between things. In the everyday world it is the aggregation of correlated phenomena into composite patterns that enables us to make sense of our environments, predict outcomes and act effectively within it. A correlation is a mutual relationship or connection between two or more things where the properties associated with each element in the relation change in respect to each other in some way. The term correlation derives from two Latin words meaning together and relation as it reflects two things that move or go together. Thus a correlation can be described as any related change in elements change in the density of two materials, changes in the physiological form of two organisms changes in the beliefs and values of the society over time etc. Science proceeds by making a series of empirical observations and then tries to draw relations between these things. This thing is bigger than that. These people are more powerful than those. All of the springs for the past 10 years have been warm etc. Out of these observed empirical correlations we can develop patterns of organization that help to make the world more intelligible to us. Three primary factors to the type of correlation are whether the relationship is positive or negative, the strength the relationship and whether it is linear or non-linear. A positive correlation means the variables move in the same direction both rising or falling together. For example, the amount of money one has in a bank account will be positively correlated with the total amount of interest you earn on that account. The more money you have in the account, the more interest that will be earned. The less that you have, the less that will be earned. This is a positive correlation. In a negative correlation, both variables move in the opposite direction like the relationship between the amount of fuel in one's car and the distance that one has traveled. The fuel will go down as the distance traveled increases. A second factor that defines a correlation is its varying degrees of strength. A strong correlation means the variables move together exactly and proportionately. A weak correlation means that the relationship between variables is only partial. For example, there is a correlation between age and health but in terms of its strength it is relatively weak. Given someone's age, we cannot predict their health. Some young people are often ill while some senior citizens are very healthy. There is a general correlation but it is not direct and thus relatively weak. This relation then connects the elements into some combined form of organization where a change in one element will at least partially be associated with the modification of another, thus creating a pattern that represents the combined organization. The term correlation does not imply causation highlights the fact that this connection does not have to be direct. The connection may be intermediated or generated by one or many other variables. A third primary factor determining a correlation is whether it's linear or non-linear. A linear correlation describes an association where the ratio of change between the variables stays constant over time, thus creating a straight line when plotted on a graph. A non-linear correlation describes how the associated change in each may itself change over time, that is to say the proportionality to the change between elements can also change, thus mapping out a graph that is not a straight line and thus we call it non-linear. For example, there is often a linear relationship between the distance one has traveled and how long it takes to get to the destination. If the destination is twice as far it will take twice as long to drive there. This is a linear correlation but there is a non-linear correlation between the size of a factory and the cost of maintaining it. To operate a manufacturing plant of 1000 square meters would not cost twice as much as that of 500 square meters because there is a synergy of economics of scale. The robustness of a pattern is then a function of the number of relations and the strength of the correlation within those relations. If all the parts are interconnected and their variables change exactly with all the others then we have a strong or robust pattern. An example of this would be an army troop marching together. Every member state is supposed to correlate directly to every other member making for a strong pattern that we would identify immediately. These strong linear correlations are much easier to predict because of their explicit direct relations and proportionality. We could for example easily predict what one of the members of the army troop will do when the others move. Likewise the robustness of a pattern is though when there are few connections and weak non-linear correlations between them. For example, there might be a weak pattern between the price of rice in Thailand and the price of eggs in Norway. With these weak non-linear patterns the correlations are not manifest or explicit. They may be intermediated by many different elements and they may be non-linear. We do not know all the factors that might connect the price of rice in Thailand and that of eggs in Norway and these connections will likely change with varying degrees over time. Symmetry is probably the most fundamental organizing principle to patterns. Symmetry helps us to capture the fundamental concepts of sameness and difference. Symmetry in its abstract sense defines how two things are the same under some transformation. The term symmetry comes from the Greek word meaning to measure together. Geometrically symmetry means that one shape becomes exactly like another when it is moved in some way. When a turn, flip or slide transformation is performed. This concept can then be generalized to describe how two or more things are the same under some transformation. Symmetry is a fundamental feature of pattern formation, a pervasive phenomena in our world found in the spatial and geometric relations between forms as can be seen in architecture, in how events take place over time, in the composition of music or of a sculpture or painting. Symmetry likewise is at the heart of modern mathematics being studied in the area called group theory. For over a century now, symmetry has become fundamental to our understanding of the basic laws of physics. This concept has become one of the most powerful tools in theoretical physics because it has become evident that practically all laws of nature originate in symmetries. The Nobel laureate Philip Anderson wrote in his 1972 article More is Different that quote, it is only slightly overstating the case to say that physics is the study of symmetry. A symmetry in its most abstract sense describes a rule that will map or transform one element in a relation to another. For example, a snowflake has a geometric symmetry to its form, what is called a reflection symmetry, where one side can be transformed into the other by applying a reflection transformation. In this way the original element has not changed, we've just applied some transformation to it to derive another related elements. If we then took the transformation away, we would return back to the original elements. A symmetry is the absence of or violation of symmetry. A symmetry may be understood as a lack of perceived transformation that will map one element in the relation to another. A symmetry in its generalized sense can be used to describe how things are different within some frame of reference. For example, if we take a tree that is asymmetric having more branches on one side than another then unlike a symmetrical tree where we could simply perform a flip transformation on one side to generate the other side with this asymmetric tree there is no transformation that will map one side to the other. Thus it is asymmetric and we would say that one side of the tree is different to the other because of this asymmetry and lack of perceived transformation that would integrate the two. Symmetry and asymmetry can be understood to be relative to frame of reference or information. As we go to higher levels of abstraction we see that things that before appeared different that is to say without transformation to map between them now on a higher level of abstraction come to have a symmetry. Symmetry can also be used to define the level of order within a system. Here the term order is being used to mean the arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern or method. This particular sequence that defines order can be understood as some transformation or symmetry between the element states or over time. If we look at an object like an isosceles triangle or a square they would appear much more orderly than an irregular triangle because the isosceles triangle and square have many more symmetries to them. Symmetries then help us to grasp the world around us and to find order in it by compressing information. Patterns that are symmetric can be defined in terms of some subset of the entire pattern and a transformation that when performed will generate the other forms within the pattern. For example, if we had a number pattern of say 2, 4, 8, 16, 32 we would not need to itemize each element in the set as we could just state the first elements and then the transformation of doubling that would generate all the elements in the pattern after this. Thus we could generate the whole pattern with just one piece of data and one rule. Because of this symmetry within the pattern we can now represent or describe the whole pattern with only a very limited amount of information and the same goes for any ordered system. Because symmetries define order we can describe an ordered system in terms of some small set of data and transformations in so doing compress the amount of information needed to describe the pattern. Conversely, because asymmetry in the generalized sense means a breaking of a rule for every asymmetry we will need to add more information. If our pattern was 2, 4, 8, 16, 18, 36, 72 our original rule of doubling each time has now been broken and we would have to add an extra rule to account for this broken symmetry thus more information to describe the system. The same is true for any broken symmetry. If a car had a big dent on one side of it, a broken symmetry, we would have to add an extra piece of information to describe that. Whereas symmetry describes simple orderly patterns in that there is a small set of rules governing the difference between the parts that can be used to generate the whole pattern. Once we understand those rules the pattern will appear relatively simple. A symmetry describes complicated patterns in that it requires significantly more information and rules to generate the whole pattern. A tangled ball of string is complicated because there are no symmetries to the pattern. This complicated pattern would require a detailed description of the entire system to understand it fully. Complexity then can be understood as some interaction between symmetry and asymmetry in creating a pattern that has both order but is also somewhat random and chaotic. It is this interplay between the two that is a defining feature to complex patterns of all kind.