## Patterns

A pattern defines some form of correlation between the states of elements within a system. All systems show some form of 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 element’s states. A correlation is a relationship of some kind between any two or more variables in which they change together over a period of time.__1__ A combination of correlations between elements forms a regular or intelligible pattern.__2__
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.__3__ Patterns have an underlying mathematical structure; indeed, mathematics in its modern form may be understood as the science of patterns.__4__ Similarly, in the sciences, theories explain and predict regularities in the world__5__ through modeling the correlated change 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 environment, predict outcomes and act effectively within it.

### Correlations

A correlation is a mutual relationship or connection between two or more things,__6__ where the properties associated with each element in the relation change with respect to each other in some way. The term correlation derives from the Latin cor-‘together’ and -relatio ‘relation’–– as it refers to things that more or go together.__7__
A correlation can describe any related change in elements, change in the density of two materials, changes in the physiological form of two organisms, changes in beliefs and values of a society over time etc. Science proceeds by making a series of empirical observations and then tries to draw relations between things; “this thing is bigger than that,” “these people are more powerful than those,” “all of the springs for the past ten years has been warm” etc. Out of these observed empirical correlations we can develop patterns of organization that help to make the world more intelligible.
Three primary factors to the type of correlation are; whether the relationship is positive or negative, the strength of the relationship and whether it is linear or nonlinear.__8__ 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 earnt, the less you have the less will be earnt. 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 has been traveled; the fuel will go down as the distance traveled increases.
Correlations have varying degrees of strength. A strong correlation means the variables move together exactly and proportionally. A weak correlation means that the relationship between variables is only partial. For example, there is a correlation between age and health, but it is relatively weak. Given someone’s age we can not 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 within 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.__9__

### Linear & Nonlinear

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. A nonlinear correlation describes how the associated change in each may itself change over time, i.e. the proportionality to the change between elements can change over time, thus mapping out a graph that is not a straight line. For example, there is often a linear relationship between the distance one has to travel 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. But there is a nonlinear correlation between the size of a factory and the cost of maintaining it, to operate a plant of one thousand square meters would not cost twice as much as that of five hundred square meters, because of the synergies to economies of scale.

### Pattern Robustness

The robustness of the pattern is then a function of the number of relations and the strength of the correlations within those relations. If all the parts are interconnected and 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’s state is supposed to correlate directly to every other member, making for a very strong pattern that we would identify immediately. These strong linear correlations are much easier to predict because of their manifest, 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 the pattern is low when there are few connections and a weak nonlinear correlation 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 weaker nonlinear patterns, the correlations are not manifest – they may be intermediated by many different elements – and they may be nonlinear. We do not know all of 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

Symmetry is probably the most fundamental organizing principle to patterns. Symmetry describes what is similar about two or more elements. Symmetry helps us to capture the fundamental concepts of “sameness” and difference. Symmetry – in the abstract – 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 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 phenomenon 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. Symmetry is at the heart of modern mathematics being studied in the area called group theory. Symmetry has become fundamental to our understanding of the basic laws of physics for almost a century now. 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.__10__ Nobel laureate PW Anderson wrote in his 1972 article More is Different that “it is only slightly overstating the case to say that physics is the study of symmetry.”__11__
A symmetry – in the 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 have just applied some transformation to it to derive another related; element if we took the transformation away we would return back to the original element. Symmetry is an important property of both physical and abstract systems, and it may be displayed in precise mathematical terms or in more aesthetic terms.__12__

### Asymmetry

Asymmetry is the absence of, or a violation of, symmetry.__13__ Asymmetry may be understood as a lack of perceived transformation that will map one element in the relation to another. Asymmetry, in it generalized sense, describes how things are different within some frame of reference. For example if we take a tree that is asymmetrical, having more branches on one side than another, then unlike a symmetrical tree – where we could simply perform a flip operation on one side to get the other – 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.
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, i.e. without transformation to map between them, now on a higher level of abstraction have symmetry.

### Order

Symmetry can be used to define the level of order within a system. Order means the arrangement or disposition of people or things in relation to each other according to a particular sequence, pattern, or method.__14__ This particular sequence that defines order can be understood as some transformation or symmetry between the elements states or over time. If we look at an object like an isosceles triangle or a square they will appear much more orderly than an irregular triangle because the isosceles triangle and square have more symmetry to them. Symmetries help us to grasp the world around us and to find order in it by compressing information.

### 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 we could just state the first element and then the transformation of doubling that would generate all the elements in the pattern after this. Thus we could generate the whole set 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. 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 it. Inversely, 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 and 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 it.

### Complex Patterns

Whereas symmetry describes simple 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. Asymmetry 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, it would require a detailed description of the entire pattern to understand it fully. Complexity can be understood as some interaction between symmetry and asymmetry to create a pattern that has order but is also somewhat random and chaotic; it is this interplay between the two that is a defining feature to complexity patterns of all kind.