Chaos Tamed

By Shelley Penner
January 4, 2013


Peacock's Tail and Apophysis 2 fractal 1 by Shelly Penner ©

Peacock’s Tail                                               Apophysis 2.0 – fractal #1

Two definitions for the word ‘fractal’:

  1. Patterns in chaos – the tendency for similar or related elements to grow and act in similar ways, which creates repetitive patterns within the apparent randomness of nature.

  2. Artwork created by applying color gradients to mathematical equations that create detailed and virtually infinite repeating patterns.


Frost Grass and Ultra Fractal #2 by Shelly Penner  ©

Frosty Grass                                                            Ultra Fractal – fractal #2


Gerber Daisy Center and Ultra Fractal #3 by Shelly Penner ©


Gerber Daisy Center                                                              Ultra Fractal – fractal # 3



The average human brain is only designed to recognize and deal with a very limited amount of data at one time. Any excess just confuses us and we perceive it as an indecipherable, unmanageable mess, which we call chaos. But in actuality, there is no such thing as chaos.


Sea Foam and Ultra Fractal #4 by Shelly Penner ©

Sea Foam                                                               Ultra Fractal – fractal # 4



Nature only appears chaotic when viewed at close range, because our minds are overwhelmed by all the countless and minute details of which nature is composed. Step back and take a larger view – the details disappear and chaos assumes a flow and rhythm of repeating patterns. For example, waves pounding a shoreline -- each wave is individual and unique; each wave is generated and directed by a multitude of varying factors and influences too numerous to comprehend. Yet seen from a hundred feet in the air, they all follow a set series of patterns that in turn erode the shoreline into a repeating pattern of curving beaches and headlands. Each wave is different, each shoreline is uniquely detailed, but the overall pattern is repeated again and again with infinite small variations on coastlines around the world.


Tidal Pool with Sea Anemones and Ultra Fractal #5 by Shelly Penner ©

Tidal Pool with Sea Anemones                                 Ultra Fractal – fractal # 5


So what is it that creates these patterns in nature? What gives order to the chaos of details?


Potential, or rather the limits of potential, provide the guidelines that marshal the details of nature into ordered patterns. Each tiny element of creation has its part to play and a finite amount of energy with which to play it, and when similar elements have similar limits to their potential, the same outcome gets repeated over and over with just enough variation to spread the results out into a pattern.



Tent caterpillars and Ultra Fractal #6 by Shelly Penner ©

Tent Caterpillars                                                 Ultra Fractal – fractal # 6


Artificial fractals created by mathematics work in a similar way. By dividing the results back into the equation many, many times, you get a repeating pattern of increasingly small numbers. Apply a range of colors to the numbers and you can create images of incredible beauty that, interestingly enough, sometimes reflect similar forms and patterns to those observed in nature.


Captive Barred Owl and Apophysis fractal #7 by Shelly Penner ©

Captive Barred Owl                                                   Apophysis 2.0 – fractal # 7


Frost on window and Ultra Fractal #8 by Shelly Penner ©

Frost on a Window                                                   Ultra Fractal – fractal # 8


Viewed from close and personal range, nature is often messy, painful and cruel. But viewed over a lifetime of seasons, the flux and flow of nature assumes the dynamic power of waves on a shoreline, and viewed from the standpoint of planetary biological history, it takes on the grace and elegance of a repeating pattern of rise and fall, in and out, Creation breathing and pulsing with the vigorous, dynamic energy that permeates all matter.



Pale Swallowtail and Apophysis fractal #9 by Shelly Penner ©

Pale Swallowtail                                                         Apophysis 2.0 – fractal # 9

Apophysis is a freeware program created by Mark Townsend, who says he gives it away for free because he “wants to promote the creation of beauty”. Every time you open the program, it generates new equations, which you can then color and manipulate in a multitude of ways to create new images. Apophysis fractals are called ‘flames’.
You can download Apophysis software here:

Nootka Rose and Ultra Fractal #10 by Shelly Penner ©

Nootka Rose                                                             Ultra Fractal – fractal # 10

Ultra Fractal is a dynamic fractal explorer that starts with equations created by talented mathematicians. You can zoom deeper and deeper into the details, apply color gradients and algorithm and stack layers to change the patterns. Cost ranges from $39 to $129 US, depending on which edition you buy. I really recommend the Creative Edition ($69 US), which can do layers. Layers add a whole new dimension and are so fun to play with!
You can purchase Ultra Fractal here:


Shelly Penner portrait.

Shelley Penner is a professional artist and amateur photographer, living and working on Vancouver Island, British Columbia. The website below includes many galleries of her photos as well her paintings, fractals and other art forms




Links to additional Resources on Fractals & Chaos


Apophysis software interface the program is available for free.



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