Sunflower Sunday (aka Friday Fractal LIV)

If we look at the natural world around us, fractals abound. Sometimes, not. This is the greatest puzzle to me... not that fractals appear in nature, but the fact that not everything is a fractal. Working on this week’s layered set (which took a while, mostly due to unrelated circumstances) I found myself questioning that inconsistency.

As I try to imitate some iterative, aperiodic pattern with my computer, I often find myself layering one fractal on top of another, to match the foreground and background (i.e., tree and sky, or clouds and land.) Sometimes, I’ll use more than a couple. This week, I used five. To list them quickly, there are two Mandelbrot sets, each colored with fBm noise, which create the fuzzy background and the tree-like mass on the left. Another Mandelbrot set, with an epsilon cross applied, creates the branches on the right. A subtle dragoncurve, based on curved lines, creates the stem, while a complex Newton basin creates the flower. Since I was trying to mimic a watery scene, I used a "lake" transformation, applied to the various Mandelbrot set layers:


Then I wished I hadn’t added the ripple-like lake effect. The actual scene was far more still:


A wild sunflower blooms on the banks of Big Dry Creek

There aren’t many still spots along Big Dry Creek in Westminster, CO. A few hundred yards away, this same water spills over a weir, violently splashing with a steady roar. Near that human-made weir, you can find similar structures: beaver dams. We can thank the beavers for the stillness of this scene. This flower bloomed upstream from the series of dams, in a flooded meadow.

Everything seemed perfectly still.... of course, when things "seem perfect" they often are not. When I sat on the bank and looked closely, I noticed the water was indeed flowing, but it was flowing in the wrong direction. This was rather unsettling, at first. Then the ripples absent from my photograph began to appear... again, travelling in the wrong direction:


One could say I got to meet the maker. Unfortunately, the video quality is rather poor. It’s pretty hard to hold a camera steady and wave to a 6-yr-old to not scare the wildlife at the same time. (You can tell when the beaver is startled by my son’s approach.) Next time, I might just skip bringing both camera and son, and try to enjoy the scene all to myself.

Now, I haven’t really addressed the question of why we have fractal things and non-fractal things. Then again, I don’t have a proper answer, either. The last time I talked about sunflowers, I was focusing on the (un)certainties of assigning value. That led me to look at "the garden beyond" the sunflower, and contemplate how that uncertainty related to the shape of our world, specifically dimension. That was followed by discussions on time and chaos theory. In the last one, I ventured forth with the tentative conclusion that because the universe has more than four dimensions, it tends to exhibit fractal tendencies. So, maybe looking at a sunflower in nature, rather than in my planted garden, has inspired me to examine that conclusion more closely.

One way of looking at it is that everything has fractal dimension, but not everything has an "exciting" fractal dimension. Some things appear as ordinary straight lines, or other more concrete forms. They still have a fractal dimension, it just isn’t really worth measuring. Another way to look at it is that each fractal in nature is part of a larger fractal; that scaling repetition is a hallmark of the fractal. Yet, inside each fractal, there are areas which appear chaotic, and others which appear ordinary (ordered). So ordinary forms could be considered the orderly parts of the fractal whole. Either way, my conclusion still seems to be sound.

I still can’t shake the feeling that ordinary forms are beguilingly simple, disguising some complex cause or effect. The earth seems flat until you look at it from the right perspective. The water seems still, but the beaver who made it so might be about to break the stillness.


Ok, so, it isn’t Friday; this one was late. Also, you might note that it wasn’t even Sunday in Colorado yet when this was posted. What can I say? That’s chaos for you.

All images created by the author, fractals using ChaosPro.

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Your observation is at best superficial ... while a number of systems exhibit some characteristics that may appear fractal like they are in fact not ... most of the systems in question exhibit the property for only a few levels and then stop ... this means that they are in fact not fractal based.

Neither fractals in nature nor fractals drawn on a computer are necessarily infinitely iterative. The dragoncurve shown above only iterates as many times as I choose, and does not exhibit repetition at many scales. Yet it is still a fractal; that is, its fractal dimension exceeds its topological dimension. Or, take Mandelbrot's classic example of measuring natural coastlines using fractal geometry. The patterns are similar at each scale (from bay to cove to beach to sandbar to sand grains) until you reach the atomic level, at which point the similarity ends. Fractals do not need to repeat at indefinite scales to be fractals.

But, let me consider your point anyways, and return to the question examined above. If you are correct, then why do we have both things that exhibit fractal properties, and things that do not?

(I'm assuming you read through the posts linked to above--the "Value" series--and understand the context in which this is being discussed.)

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