Emergence is a natural system characteristic. The agents interaction in the system are difficult to understand. We are able to understand the initial rules because they are clear defined, for example if we have 3 live agents, the closest agent to the right will die. At the end of the process, the outcome or the system behavior it's also well defined, because we can see it and we can measure it.

Complex systems have their first appearance in late 19th century, when Poincaré showed that a system consisting of three bodies like the Sun, Earth and Moon, couldn't be solved by analytical methods (linear equations). 100 years later, Alan Turing published the Morphogenesis to explain mathematically how works cells subdivision. The foundations of artificial intelligence, systems dynamics, fractals, chaos theory, cellular automaton, emerged. Complex system application are being studied in all humans fields. And since early 80s, its applications to architecture have been studied.

##### PATTERNS

In an complex system, the order is not imposed by a controller outside the system. There is no architect with the instructions to follow. The Behavior and pattern emerge from the system rules. It's very difficult to predict or study a system based on a single agent and their rules. It's not possible to understand a city studying a person.

Patterns and rules organize the environment: Birds flocking, ant colonies, Zebras skin patterns, etc. Agents in the same complex system share similar training mechanisms, for example the bees in the colony. The bees, following simple rules are self-organized and are able to resolve in a very efficient way different kind of problems. The hexagonal pattern in the honeycomb emerge from thousands of bees continuously contributing to build it.

The human brain is not able to predict the behavior of thousands of bees, but is good at identifying and creating geometric and arithmetic rules. Our brain, for example, is built by a network of neurons established from the patterns that we have learned. For example, once we learned where the light switch is in the wall, we turn on and off the light without to think where the switch is located. If we try a different room, Our brain will check for the light switch as if we were in the first wall, Because our neuronal network has been established a pattern connection that is activated every time we have to turn on the light switch in a wall.

##### CELLULAR AUTOMATA

The zebras stripes alternate in contrast between black and white areas, this can be translated to `1`

when is black and `0`

when the area is white. The simple way to start is define a list of 1s and 0s, each number is represented by a square, where `1`

represents a black square and the `0`

a white square. Beginning from the second square in the list to the second last square in the list, every square will have 2 neighbors one at the left and one to the right. The first list is the generation 0.
The next square list (`Generation t`

) its defined from the previous generation (`Generation t - 1`

). The value (`1`

or `0`

) of each square (cell) in the new list is calculated in relation to the neighbors in the previous list. Since each cell has two neighbors all possible combinations for 3 continuos cells of 1s and 0s is 8, therefore it make sense to restrict to 8 the number of rules.

###### Generation Rule.

Examples for Rules A and B. the above rule is represented by: `[ 0, 1, 0, 1, 1, 0, 1, 0 ]`

- Rule A: the new cell is set to
`0`

(White). - Rule B: the new cell is set to
`1`

(Black).

##### GAME OF LIFE.

John Conway's game of life is another cellular automata. Is defined on a two-dimensional square lattice, composed of a central cell and the eight cells which surround it, 9 neighbors in total. The state of the squares in the lattice change according to a time step.

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