L-systems generate weevils, pizza toppings, and matriarchal lineages

In Western mathematics, the Fibonacci sequence is famously linked to a puzzle presented in Fibonacci's 1202 publication, Liber Abaci.

The problem asks us to imagine a rabbit colony where a pair of newborns takes one month to reach maturity, and every adult pair produces a new pair of offspring every month thereafter.

The Mechanics of Growth

To model this, we can assign symbols to the rabbits:

• a represents an adult pair
• b represents a baby pair

The growth rules can be expressed as: $\text{Rule 1: } b \to a$ $\text{Rule 2: } a \to ab$

We can summarize these transformations in the following table:

This recursive replacement process is known as an L-system (specifically, the Fibonacci L-system). If we start with a single baby pair (b), the sequence evolves as follows:

Month 1: b
Month 2: a
Month 3: ab
Month 4: aba
Month 5: abaab
Month 6: abaababa
Month 7: abaababaabaab
Month 8: abaababaabaababaababa
Month 9: abaababaabaababaababaabaababaabaab
Month 10: abaababaabaababaababaabaababaabaababaababaabaababaababa

Interestingly, the Fibonacci sequence emerges in three ways here:

1. The total length of the string.
2. The total count of a symbols.
3. The total count of b symbols.

Visualizing the L-System Flow

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Unconventional Applications

When I encounter these patterns, I immediately think of how the Fibonacci L-system can be applied to linguistic strings. For instance, I recently mentioned Zyzzyva—a genus of weevils likely named by a biologist with a desire to be the very last entry in any dictionary.

I've used this logic to generate "growing" versions of various words:

• Weevil Names:

  • Yva
  • Zva
  • Zyva
  • Zyzva
  • Zyzzyva
  • Zyzzyzyzva
  • Zyzzyzyzzyzzyva
  • Zyzzyzyzzyzzyzyzzyzyzva
  • Zyzzyzyzzyzzyzyzzyzyzzyzzyzyzzyzzyva
  • Zyzzyzyzzyzzyzyzzyzyzzyzzyzyzzyzzyzyzzyzyzzyzyzzyzyzva

• Pizza Toppings:

  • eroni
  • proni
  • peroni
  • peproni
  • pepperoni
  • peppepeproni
  • peppepeppepperoni
  • peppepeppeppepeppepeproni
  • peppepeppeppepeppepeppeppepeppepperoni
  • peppepeppeppepeppepeppepeppeppepeppepeppeppeppepeproni

• Matriarchal Lineages:

  • Grandma
  • Grandmam
  • Grandmamma
  • Grandmammamam
  • Grandmammamammamma
  • Grandmammamammammamammamam
  • Grandmammamammammamammamammammamammamma
  • Grandmammamammammamammamammammamammammamammammamammamam

Note: I cheated slightly with the "Grandma" sequence by ignoring the middle 'a' during substitution. If I hadn't, the result would be the unpronounceable Grmammandmammamammamma.

A brief search for other variations revealed results involving "mammaries" or "titties", but to avoid turning this into a display of juvenile sniggering, I will stop here.

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Final Thoughts

It turns out there was already a previous article on a very similar subject.

As a side note, I attempted to enlist Claude's help in writing this piece, but the AI refused, claiming the entire premise was "too stupid." Consequently, I had to write it myself.

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