Scientific American reference
In the Scientific American article, the worms rules were divided into groups, or fields.
See illustration for choice/field grouping. (not available at this time)
- Field 1 comprised those situations where the worm would arrive at a point with no paths
taken. The choice would thus be either a gentle turn
(60° right) or b sharp (120° right).
- Field 2 encompasses one taken path. The choices were a, b, c, or
d of the remaining paths. Five different permutations are grouped under
each lettered choice.
- Field 3 for two taken paths is subdivided into 4 subgroups, each one requiring a lettered
choice from a, b, or c. Each letter covers 2
- Field 4 covers three taken paths. The remaining two choices is specified by combining
20 permutations into 2 groups of 10, giving choice a or b.
It is interesting to note that despite numerous permutations being grouped under a single letter
choice, no conflicts have been encountered. No pattern has been found that requires rules
outside of these restrictions.
It is not clear to this author why in the above rules the choices were gathered under field
categories. Indeed, the article explains that there are a great many redundant patterns
generated by formulae using this method. For example, the simplest worm path
(9 units) has the formula:
Giving 162 species. The parenthesized letters denote choices which have no apparent
consequence. Using my rule structure and notation reveals only two unique 9-unit
patterns differentiated by the order the paths are taken:
The cloud path, rotated to correspond with the Scientific American
illustration on page 122 has a path length of 10,795 units and is generated by the formula:
Note: The formula given in Scientific American was incorrectly shown as
In summary, my rule structure was based not on an algebraic notation which predetermines worm
behaviour, but rather by observing a worms progress dynamically and applying choices as
they become necessary. In this way, only a minimal set of rules define the pattern uniquely
and sufficiently. The numbers shown are the relative path directions made at each decision
point, in chronological order during the life of the worm.
By exploring every alternative course, backtracking, and applying yet unused selections in an
ordered manner, all possible patterns were generated. The choices of paths
were made in numerical order according to the path numbers illustrated on the
Description & Rules page. In this way it was found that there are
411 unique patterns named #1 to #411. Far less than the articles claimed 1296
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