Symmetries of block replacement L-systems

"Symmetry is a complexity-reducing concept; seek it everywhere." Alan Perlis

What are the possible symmetries of an L-system defined by an initial symbol (or block of symbols) and a set of block replacement rules? What are the possible symmetries of a set of block replacement rules? What are the possible symmetries of the pattern that is generated by such an L-system? How are these symmetries related to each other?

I can answer some of these questions, but only in a semi-coherent way. I'll give examples that illustrate some aspects of symmetry. These examples use 2x2 block replacement, but they can be generalized to larger or non-square block replacement rules.

Possible symmetries of block replacement L-systems

What is a symmetry of a rule, a set of rules, a pattern, or an L-system? An object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations, and vice versa. In the context of these block replacement L-systems, the operations of interest are 1) rotations and mirrorings that preserve a square, and 2) permutations of rules that preserve the system.

A single 2x2 rule can have a subset of the symmetries of a square: four-fold or two-fold rotational symmetry, horizontal or vertical mirror, two diagonal symmetries.

asymmetric

single diagonal mirror symmetry

vertical mirror symmetry

vertical mirror symmetry

Rule with mirror symmetry across two diagonals

mirror symmetry across two diagonals, or pi radian rotational symmetry

fully symmetric (mirror across any axis, or any n*pi/2 rotation)

Some of the symmetries of a pattern generated by a system can be inferred from the symmetries of the component rules. For example if all rules share one of the above symmetries, the pattern will also have that symmetry.

The relationship between rules can be symmetric -- a set of rules can have an equivalent relationship to another set of rules. For many systems these relationships are easily sketched as a directed graph. [To Do: Make a directed graph of the Binary tally system.] In this case rules 0 and 2 (black and white) are complements -- swapping the replacement rule results in an identical pattern after swapping these symbols. The symmetry between rules is manifested in the pattern the system generates.

The four rule Reflected binary (gray code) system is only slightly more complicated. In this case the initial symbol is assymetric with respect to the graph's bilateral symmetry.

It is not always easy to recognize the symmetry of systems. [To Do: Example of an obfuscated symmetry.] How do we find the symmetries between rules?

The symmetry and self-symmetries of the pattern generated by an L-system is determined by the symmetries of the rules, and the symmetry between rules. To summarize these symmetries, the L-system program generates a diagram showing the overall symmetry of the pattern and where self-symmetries occur within the pattern.

[To Do: What is a "by generation" symmetry, with respect to the period of a dynamical system.]

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