Paterson's worms are a family of cellular automata devised in 1971 by Mike Paterson and John Horton Conway to model the behaviour and feeding patterns of certain prehistoric worms. In the model, a worm moves between points on a triangular grid along line segments, representing food. Its turnings are determined by the configuration of eaten and uneaten line segments adjacent to the point at which the worm currently is. Despite being governed by simple rules the behaviour of the worms can be extremely complex, and the ultimate fate of one variant is still unknown. The worms were studied in the early 1970s by Paterson, Conway and Michael Beeler, described by Beeler in June 1973, and presented in November 1973 in Martin Gardner's "Mathematical Games" column in Scientific American. Electronic Arts' 1983 game Worms? is an interactive implementation of Paterson's worms, where each time a worm has to turn in a way that it lacks a rule for, it stops and lets the user choose a direction, which sets that rule for that worm.
History
Paterson's worms are an attempt to simulate the behaviour of prehistoric worms. These creatures fed upon sediment at the bottom of ponds and avoided retracing paths they had already travelled because food would be scarce there but, because food occurred in patches, it was in the worm's interest to stay near previous trails. Different species of worm had different innate rules regarding how close to travelled paths to stay, when to turn, and how sharp a turn to make. In 1969 Raup and Seilacher created computer simulations of the fossilized worm trails, and these simulations inspired Paterson and Conway to develop a simple set of rules to study idealized worms on regular grids. Conway's original model was a worm on an orthogonal grid but this produced only three different species of worm, all with rather uninteresting behaviour. Paterson considered worms on a triangular grid. Paterson's worms were described by Beeler in a Massachusetts Institute of Technology AI Memo (#[1]) and were presented in November 1973 in Martin Gardner's "Mathematical Games" column in Scientific American, and later reprinted in Gardner 1986. These simulations differed in approach from other cellular automata developed around the same time, which focused on cells and the relationships between them. Simple computer models such as these are too abstract to accurately describe the behaviour of the real creatures, but they do demonstrate that even very simple rules can give rise to patterns resembling their tracks.
Rules The worm starts at some point of an infinite triangular grid. It starts moving along one of the six gridlines that meet at each point and, once it has travelled one unit of distance, it arrives at a new point. The worm then decides, based on the distribution of traversed and untraversed gridlines, what direction it will take. The directions are relative to the worm's point of view. If the worm has not encountered this exact distribution before it may leave along any untraversed gridline. From then on, if it encounters that distribution again, it must move in the same way. If there are no untraversed gridlines available, the worm dies and the simulation ends.
Discussion There are many different types of worm depending on which direction they turn when encountering a new type of intersection. The different varieties of worm can be classified systematically by assigning every direction a number and listing the choice made every time a new type of intersection is encountered. The six directions are numbered as follows:
Direction 0 indicates the worm continues to travel straight ahead, direction 1 indicates the worm will make a right turn of 60° and similarly for the other directions. The worm cannot travel in direction 3 because that is the gridline it has just traversed. Thus a worm with rule {1,0,5,1} decides to travel in direction 1 the first time it has to make a choice, in direction 0 the next time it has to make a choice and so on. If there is only one available gridline, the worm has no choice but to take it and this is usually not explicitly listed.
A worm whose ruleset begins with 0 continues in a straight line forever. This is a trivial case, so it is usually stipulated that the worm must turn when it encounters a point with only uneaten gridlines. Furthermore, to avoid mirror-image symmetrical duplicates, the worm's first turn must be a right hand turn. A worm dies if it returns to its origin a third time, because there are then no untraversed edges available. Only the origin can be lethal to the worm. There are 1,296 possible combinations of worm rules. This can be seen by the following argument:
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