Yaneer Bar-Yam

We use the word "pattern" a lot. A formal definition of a pattern is quite abstract and examples that follow the formal definition are needed to explain why this definition is really what we are talking about.

A pattern is

...a set of relationships that are satisfied by observations of a system, or a collection of systems

...a simple kind of emergent property of a system, where a pattern is a property of the system as a whole but is not a property of small parts of the system.

...a property of a system that allows its description to be shortened as compared to a list of the descriptions of its parts.

A simple type of pattern is a repeating structure in space, like periodic patterns often found on wallpaper or tiles. Shifting the view by one repeat length leads to seeing the same thing. Similar to repeating patterns in space, we can also have a repeating pattern in time, like the seasons. Generally, a pattern can have both repetitions in space and time. Examples include persistent behaviors like hurricanes.

We also think of patterns as prototypes or exemplars. This is the sense in which we use it to describe a pattern for making a dress. In this case the pattern is not about the relationships within the dress, but about the possibility of repeating the dress many times. The idea of repetition is still there but it happens in thinking about the collection of dresses made from the same pattern.

The relationship between the pattern as repetition and the pattern as prototype is just like the relationship between two types of emergent properties: emergent properties that arise from relationships between parts of a system, and emergent properties that arise from relationships between a system and its environment (the larger system of which it is a part).

The idea of patterns is used more generally by recognizing that repetition of tiles or cycles of seasons is a form of relationship. It is a relationship between what happens when you look at different places or times. More general types of relationships are also possible, and we still consider what we see to be a pattern.

In a description of a system, repetition corresponds to a redundancy: saying the same thing again and again. When there is redundancy we can shorten the description by saying the repeating part only once and saying how many times it is to be repeated. This makes it possible to shorten the overall description.

Understanding how patterns form (pattern formation) and how patterns are recognized by the brain (pattern recognition) are two important areas of study in complex systems.

Related concepts: relationships, pattern formation, pattern recognition, emergence, scale