Chaotic Utopia

Friday Fractal VII

The phoenix is one of those enduring symbols from ancient mythology that occurs frequently in modern culture. Phoenix is a name for places and stars (Hollywood and constellations alike.) i-eccdd49f87510177c3528939f6a110be-rfwhole.jpgIt shows up in movies and literature (including the Harry Potter series.) Most importantly (at least for the context used here) this name for a mighty bird, which perished in flame, is a type of fractal. A phoenix set seemed rather fitting for today’s image, as we’ll soon see.

This set (seen whole, at right) is a modification of the original phoenix sets, discovered by
Shigehiro Ushiki. Essentially, I began with this formula:

z(n+1) = z(n)^a + c*z(n)^b + p*z(n-1)

and adjusted a few of the variables until I was satisfied. Here’s the result:


And a similar image, not only to imitate a phoenix, but to honor all the of fathers out there:


An American Robin (
Turdus migratorius
*) sits proudly on a branch of fir.

Happy Father’s Day!

I chose a robin for a Father’s Day image, because they represent so many of the things I love about the dads in my life.

Members of the thrush family, robins have distinctive, melodious calls which delight many birdwatchers. My dad, who inspired my love for science, is always delighting an audience as well. I grew up loving his silly antics, much as my son enjoys them today. i-09f9c5fe95313baea79a2408ab762a31-robinfeeding.jpgLike a robin listening for worms, my dad is patient and observant, traits which have lead to his success in both science and parenting.

My husband has been a great dad, too. Like the robin, he’s always been a hard worker and a loyal dad. As my son put it, “He fixes the computer. He’s the fixer. I like it when he takes me places, too.”

So, to all the dads out there, whether you’re singing to cheer us, fixing our nests, or teaching us to fly, thanks!

A pine note: As a kid, I always thought the little pine cones near the tips of the trees, as seen above, were just baby pine cones on smaller branches. (Of course, I never asked my dad.) As I later learned, the little cones were actually the pollen-bearing male cones. The larger female cones are designed to catch the falling pollen. It makes sense… big ovum, tiny sperm.

A personal observation: The colors of the fledgling cones are so similar to the colors in the robin’s chest, it seems the latter male evolved to camouflage amongst the former males.

And finally, a taxonomy note (*):The Turdus genus. Now you know why they call him “Robin.”

Image of robin feeding chicks was taken by Cal Vornberger. Image of robin perched in fir tree was taken by the author. All fractals were made by the author using ChaosPro


  1. #1 Dan
    June 17, 2006

    A Mandelbrot Set? I’m surprised that it surprises anyone anymore. Recursive activity seems to be an inherent quality in all things, from nature to the future’s market.

  2. #2 DarkSyde
    June 17, 2006

    Mmmmmmm … fractal maps.

  3. #3 Karmen
    June 17, 2006

    Dan, I’ve been delighted by fractals and fractal-like patterns in nature for years… before I even knew what they were. I wouldn’t call it surprised so much as obsessed. Oh, and it’s a Julia set, actually.

    Darksyde, I’m glad you could stop by… and thanks for the link!

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    July 6, 2011

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