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L-System Fractals

Posted in Recriational Math

An L-system is a set of rules for generating and interpreting an ever-growing sequences of symbols. They are named after Aristid Lindenmayer, a biologist, who evidently used them to study biological growth.

Some L-systems generate fractals, while others don’t. I don’t yet get the difference, but I’ve been exploring various systems in hope of understanding. Basically, what I’ve been doing is modifying the L-systems for the Dragon Curve and C Curve fractals. Here are some of the fractals that I’ve stumbled across:

Flower Dragon

  • Seed: “X”
  • Replacement Rules:
    • “X” becomes “YlFXFlY”
    • “Y” becomes “XrFYFrX”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “r” means, “turn right 90 degrees”
    • “l” means, “turn left 90 degrees”
    • “X” and “Y” are ignored

River Dragon

  • Seed: “X”
  • Replacement Rules:
    • “X” becomes “YFlXFrY”
    • “Y” becomes “YFX”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “r” means, “turn right 90 degrees”
    • “l” means, “turn left 90 degrees”
    • “X” and “Y” are ignored

Hoth Horde C Curve

  • Seed: “X”Replacement Rules:
    • “F” becomes “rFlFr”
    • “X” becomes “YFX”
    • “Y” becomes “XFY”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “r” means, “turn right 45 degrees”
    • “l” means, “turn left 45 degrees”
    • “X” and “Y” are ignored

This curve reminds me of the Battle of Hoth; specifically, it looks to me like the approaching line of AT-ATs (though the curve’s “representation” of an AT-AT is too wide/squat).

Forest C Curve

  • Seed: “X”
  • Replacement Rules:
    • “X” becomes “FrXllXrF”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “r” means, “turn right 45 or 135 degrees”
    • “l” means, “turn left 45 or 135 degrees”
    • “X” is ignored

Architecture C Curve

  • Seed: “X”
  • Replacement Rules:
    • “F” becomes “lFl”
    • “G” becomes “lGl”
    • “X” becomes “XFYFX”
    • “Y” becomes “YGY”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “G” means, “step backward, drawing a line”
    • “r” means, “turn right 45 or 135 degrees”
    • “l” means, “turn left 45 or 135 degrees”
    • “X” and “Y” are ignored

At 45 degrees, the curve appears to consist of classical western columns, while at 135 degrees, the curve reminds me of pagodas, for some reason.

Thornbush Diamond Curve

  • Seed: “FX” (also works with just “X”)
  • Replacement Rules:
    • “F” becomes “Fl”
    • “G” becomes “Gr”
    • “X” becomes “XFrY”
    • “Y” becomes “YFlX”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “G” means, “step backward, drawing a line”
    • “r” means, “turn right 60 degrees”
    • “l” means, “turn left 60 degrees”
    • “X” and “Y” are ignored

This appears to be a version of the Levy Diamond, which you will also find among the number line fractals. One thing that is particularly interesting to me about this version, however, is that it is not symmetrical. Instead of the “left” and “right” halves of the curve “pointing” toward the center, both point left.

Norwegian Coast Curve

  • Seed: “FX”
  • Replacement Rules:
    • “F” becomes “Fl”
    • “G” becomes “Gr”
    • “X” becomes “XFrY”
    • “Y” becomes “XlFY”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “G” means, “step backward, drawing a line”
    • “r” means, “turn right 45 degrees”
    • “l” means, “turn left 45 degrees”
    • “X” and “Y” are ignored

For some reason, this one reminds me of the southwestern coastline of Norway.

Digital Vine Curve

In case it isn’t clear (as it wasn’t to me, at first) that this is a fractal, this is what happens when you zoom in:

  • Seed: “FX”
  • Replacement Rules:
    • “F” becomes “FXY”
    • “G” becomes “GG”
    • “X” becomes “XrF”
    • “Y” becomes “YlG”
  • Interpretation Rules:
    • “F” means, “step forward, drawing a line”
    • “G” means, “step backward, drawing a line”
    • “r” means, “turn right 90 degrees”
    • “l” means, “turn left 90 degrees”
    • “X” and “Y” are ignored

I think this may be my favorite of the lot. It’s so weird.

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