Scientific American reference

In the Scientific American article, the worm’s rules were divided into groups, or fields.
  1. 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).
  2. 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.
  3. 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 permutations.
  4. 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.
See illustration for choice/field grouping. (not available at this time)

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:

1b2(ac)3(abc)(abc)(abc)(abc)4a 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: 2 0 0      and      2 4 4 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: 1a2c3abaa4a      which under my notation becomes:      1 0 4 0 5 2 4 Note: The formula given in Scientific American was incorrectly shown as  1a2abaa3c4

     In summary, my rule structure was based not on an algebraic notation which predetermines worm behaviour, but rather by observing a worm’s 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 article’s claimed 1296 species.

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