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• #### Probability and Statistical Physics / Connecting Microscopic and Macroscopic 53-58

Cellular Automata

Sublimely Complex

A sandpile grows in an hourglass, one grain at a time. Occasionally the impact of one sand grain will create a minicascade of grains. Even less often, a minicascade will trigger a macrocascade that changes the shape of the whole sand pile. How often do these cascades occur, and do they follow a predictable pattern?

A snowflake grows by attaching one molecule of water at a time. How does that process lead to the lacy, frilly shapes that we see in real snowflakes? What do we get if we grow the snowflake in different ways, using different attachment rules?

A city has two directions of traffic, east and north. At each intersection a car can proceed only if there is no other car already in the intersection. How many cars do you need to have in the city before gridlock ensues? Can the traffic self-organize into moving patterns?

These are three examples of a stylized mathematical model called a cellular automaton. Often inspired by real physical objects (such as sandpiles, snowflakes, and traffic), cellular automata demonstrate that amazingly complicated large-scale effects can arise from very simple local rules. They have been applied to other phenomena as diverse as avalanches and wildfires.

There is a widespread belief that highly complex systems can only be understood through computer simulation. However, mathematicians have found that simulation is far from the only way to discern patterns and phenomenology.

“Packard snowflakes,” invented by Norman Packard in 1984, begin with a hexagonal lattice, like a honeycomb, in which one cell is filled by an ice crystal. One particularly interesting snowflake grows by the following rule: If an open cell is adjacent to 1, 4, 5, or 6 cells that are already filled, then at the next time step that cell fills and freezes. However, nodes that are adjacent to 0, 2, or 3 “frozen” nodes remain unfrozen at the next step.

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