A model of an atom can be likened to a solar system

Um, no. Electrons do not have well-defined trajectories. Quantum particles should not in any way be compared to classical macroscopic objects.

I love fractals and finding examples of them in nature and science, but posts like these annoy me a bit. Saying that there are fractals everywhere is a bit like saying that we don't live in the Platonic plane.

Hierarchical self-similarity does seem to be ubiquitous, and a lot of people have taken this idea and run with it. I think Hofstadter's GEB is one of the crucial modern texts that inspired a lot of people in that direction. Translating these concepts into rigorous theories is difficult; attempts like Wolfram's "A New Kind of Science" haven't succeeded in creating a new discipline.

I think the deepest explanations for the layering of complex structures at multiple scales we see in the universe are being developed at the intersection of the theory of computation and quantum physics. There is a very tight coupling of entropy and information at the quantum scale, as demonstrated by physicists like the amazing W.H. Zurek. The development of quantum computers and work on the P/NP equivalency problem is where I hope to see the long-awaited breakthroughs that may enable us to understand just how math, mind, and physics are intertwined.

Well I was hoping for something more developed. I thought about this stuff too, fractals seem like a good explanation for complex systems.

There seems to be a fractal pattern in how people behave that reflects the pattern of how neurons behave. Neurons silence each other as much as they connect to each other. The loudest neurons, the ones that can silence the most neighboring neurons and connect to more distant loud neurons have the biggest effect on the output. Groups are organized in a similar way. Then groups compete with other groups by trying to silence each other and connect to other successful groups. First a tribe will absorb its nearest neighbors, then reach out to distant tribes.

Also vaguely remember scientists measuring a rain-forest in the Amazon, the thickness of the branches was proportional to thickness of the trunks, which was proportional to the density of the forest. Something useful along those lines.

Wow, what a stupid article.

At least provide examples that are valid. Your atom analogy is just false, it's nothing like the solar system.