This 54-minute NOVA video offers a great introduction to what a fractal is:

Dark matter and dark energy, explored in previous articles (these links are to those articles), continue to be mysteries to physicists. At this point, they are known almost entirely by their effects on space-time. Both theoretically allow for the possibility that general relativity (Einstein's geometric description of spacetime) might not work at the very largest scales in the universe. We already know from another past article dealing with gravity that general relativity doesn't work at the tiniest, or quantum scale, either. This opens up the possibility that general relativity is an incomplete theory.

Most physicists are reluctant to tinker with this brilliant, self-consistent and well-established theoretical framework and it seems wise to exhaust alternative theories for dark matter and dark energy before focusing on general relativity itself. This is what many current experiments are pursuing, as described in the dark energy and dark matter articles.

Still, there is always room to explore even remote possibilities, and a smaller number of (though still many) physicists are concurrently working on alternatives or refinements of gravity theory itself, several of which turn to string theory to explain dark matter and dark energy in terms of gravity. Some theories are even going as far as doing away with the Big Bang origin of the universe itself, an almost universally accepted concept in cosmology. A few theorists are looking to fractal geometry as a possible solution to the general relativity scale problem. It's this possibility that is explored in this series of articles.

Fractals are everywhere, both in popular art and culture as well as in nature. I'm sure you have seen some stunningly beautiful digitally rendered images of fractal geometry and you have seen many objects in nature that also exhibit fractal-like properties.

There are many different kinds of fractals, each one based on a unique underlying mathematical formula. One of most famous examples is the Mandelbrot set, named after the mathematician Benoit Mandelbrot, shown below.

These images are computer renderings created by running numbers through a specific formula over and over again. Imagine that each rectangle is a microscopic view that you zoom in on.

When you are zoomed in 2000X (bottom right), you see something that looks remarkably like the black bug-like image at the top. This magnification means that the original black bug-like image (top) would be as large as 100 meters long if you could see the whole thing. You could continue to zoom in forever and always get new detail, and the complete image would grow infinitely large. As you did so, identical or similar patterns would reappear over and over again. This is called a self-similar pattern. (All four images: Nadimghaznavi;Wikipedia)

Here are two other exquisite fractal animations:

Featherino fractal:

Peacock fractal:

Perhaps a bit less psychedelic, Nature is filled with stunning examples of pattern development, such as frost spreading on a glass surface, below. Many of these patterns resemble the type of self-similarity seen in fractals, except that they seem to reach a limit of scale at some point, perhaps at the molecular level, and of course these patterns are restricted in their overall size.

Schnobby;Wikipedia |

Nature often elevates fractal-like ordering into an art form, as evidenced by the beautiful head of Romanesco broccoli, below left. There are countless examples of animals and plants that have evolved complex fractal-like adaptive structural plans. Some examples are neurons in the brain, the fine branching in the lungs, increasingly fine branching of trees, spiral seashells and the fiddleheads of ferns. Geology too has its share of intricate fractal-like patterning, as in aerial views of branching river systems or of lava flows and ice flows on Earth and on other planets and moons in the solar system.

Jon Sullivan;Wikipedia |

Fractal geometry seems to be part of Nature's toolkit, an observation that begs the question: how fundamental is fractal ordering? Could even spacetime itself have a mysterious underlying fractal nature? For many physicists, this kind of exploration is fringe territory that doesn't merit exploration if it is based only on what might be coincidences of Nature. Do we need to overlay yet another purely hypothetical theoretical structure onto already complicated enough spacetime? And does this line of reasoning even make sense? The universe doesn't look like a giant fractal in telescopes and there is no evidence that there are Horton-Hears-a-Who universes hidden within this universe, which suggest that if you look deep enough into an electron or quark you will see a tiny universe in there.

This kind of hesitation doesn't necessarily kill the possibility that fractals underlie spacetime. Perhaps thinking only in terms of visual patterns is a naive way to proceed. What if the universe incorporates instead something about the process of generating fractal geometry that pervades all of nature? Can the kind of geometry unique to fractals (all are continuous but non-differentiable functions as we will see) be incorporated into the framework of partial differential field equations that make up Einstein's geometry of spacetime? This is the core question in this series. To get to it, we will need to explore fractals in detail before we can ask how they relate, if at all, to the workings of spacetime. And we will need to do a little math.

Physics Meets the Philosophy of Physics

The Scientific Method is the first rule for all scientists, much like the Hippocratic Oath is for all healthcare workers. This means that science is objective, so personal conjecture, bias and opinion have a very limited role, if any. Therefore, philosophy is not often discussed in scientific investigations. However, questions about the universe often have a deeply philosophical, sometimes spiritual, and definitely personal component. A philosophical sideline to our exploration could be "What do fractals mean and how do we fit into a fractal universe?" When I explore fractals online I notice that fractals in particular bring out the philosopher in many of us, perhaps because, thanks to computer technology, they can be rendered into striking images and animations that seem to speak to us at some deep level, much like mandalas, which are complex images full of symbolism that represent the universe in both Hinduism and Buddhism, do. Below is a photo of an exquisite sand mandala created in Britain for the visit of the Dalai Lama in 2008.

Colonel Warden;Wikipedia |

^{2}, which relates matter with energy.

Unfathomable complexity stems from a relatively simple rule or set of rules. How does that complexity arise, and what does it mean for the universe as a system? In Isaac Newton's time, the universe seemed to be finely tuned clockwork, a deterministic set of processes playing themselves out. A deterministic system will always produce the same output from a given starting position or initial state, much like how the gears in a clock work to make the process of keeping time both reliable and predictable.

Quantum uncertainty, developed in the early 1900's, blew this paradigm apart, but even today unpredictability in physics can be difficult to accept.

The mathematics of fractals is especially interesting because it allows fractal processes to exhibit both "clockwork" (predictable processes that run according to classical scientific laws) and degrees of freedom (unpredictability). In fact, chaos theory is closely related to fractal geometry. Computer models of the chaotic or random nature exhibited in weather systems and solar plasma behaviour, for example, are very sensitive to the initial input of parameters. Small changes in input often result in drastic differences in outcome. These kinds of chaotic dynamical systems are described using nonlinear equations. We will explore them soon. When nonlinear equations are solved, and the solutions are plotted in Euclidean space, odd and mesmerizing figures called strange attractors sometimes appear.

Strange attractors sound esoteric but they are incredibly useful in system dynamics. An example is the Lorentz strange attractor. The Lorentz mathematical system is a set of three fairly simple differential equations (we will learn what "differential" means later on) that has chaotic solutions for some, but not all, initial input values. When the chaotic solutions are plotted, you get an image that looks like a butterfly (image bottom right, below).

For example, the set of three differential equations for the Lorentz system give you x, y and z variables which you can plot in three dimensions. These variables allow you three input parameters. When you keep two of them the same and change one, called

*p,*you get solutions that are stable (non-chaotic) and they evolve into something called a fixed point attractor (the tight little red knots, seen right) WHEN

*p*is small (

*p*= 14, 13 and 15 as you go down the top three images). However, when

*p*is greater than 24.74, the solutions are chaotic. In the bottom image, p = 28. This solution is both a strange attractor and a fractal (although its fractal nature is not readily apparent). It is called a Lorentz attractor. When p = 99.96, the solution becomes an even more complex torus knot.

While pretty, the Lorentz attractor is also very useful because it serves as a simplified model for atmospheric convection and it is used in weather prediction modelling. The Lorentz system also arises in simplified models for lasers, dynamos, electric circuits and chemical reactions.

Some chaotic systems are simply chaotic everywhere but sometimes chaotic behaviour is found only in a certain subset of space coordinates within a system. The Lorentz system is an example of this kind of system. In these cases, the chaotic behaviour sometimes takes place on something called an attractor. A large set of initial conditions settles into orbits that converge on the chaotic region. This is strange attractor. Ordinary attractors, where there is no chaotic behaviour, are regions where behaviour is stable and predictable as it converges on some fixed point in space.

With a strange attractor, you never know where on the attractor the system will be at any given point in time and the motion of the system never repeats itself. This is one reason why weather, for example, no matter how good the forecasting models are, can never be predicted more than a week ahead of time with any good accuracy.

It might surprise you to learn that even Lorentz systems, which exhibit some chaotic behaviour, are deterministic, which means they run according to the laws of physics and their future behaviours can be entirely determined by their initial conditions

*in theory*, and yet this does not mean they are

*predictable*. How does this make sense? The reason that chaotic systems, when modeled, have complexity and unpredictability built into them is because the initial state is impossible to measure with prefect precision. You would need to input it with an accuracy of infinite decimal places after the zero, and even the tiniest error could significantly change the outcome.

Even the Shrodinger equation that describes the wave functions of particles is deterministic. There is a continuous evolution of a system's wave function that can be mapped out over time. There are no random choices being made along with way, in other words. However, the relationship between the wave function and observable properties of the system are not deterministic.

Because of this non-deterministic aspect of quantum mechanics, the very building blocks of the universe operate in a system that, by definition, must have unpredictability built into it. It seems impossible to square such a system where the exquisite order seen in galaxies, planets, ferns and even snowflakes arises from particles that are themselves composed of probability clouds, and yet that is where we are. This describes the framework of the universe as one that is simultaneously built of chaos and predictability. Somewhere within the framework, general relativity and quantum mechanics should reside as one complete description, although physicists have not yet been able to make these two fundamental theories compatible with each other. This is an unnerving but much more intriguing picture of the universe than Newton's one.

Another question arises from the complex behaviours of numbers that we've just glimpsed. The physics of the universe can be (not quite perfectly) described as a set of interrelated mathematical formulas. In many cases, processes are far too complex to visualize, so a framework of mathematical formulae does the visualizing for us. In some cases, the mathematical framework describes processes that should not even be possible, at least in what we usually think of as spacetime. I am thinking of the Dirac spinor state of the electron as an example. Does this mathematical backbone of physics imply that the universe itself is built entirely of mathematics, making us, and everything else, artifacts of the play of numbers? I am far from the first person to ask these kinds of philosophical questions. Theoretical physicist Max Tegmark proposes just such a theory - that our physical reality is really a mathematical structure.

I noticed on my reading journey that more than a few physicists strongly disagree with Tegmark. I came up against this idea while writing Holographic Universe. I still find myself backed up against this philosophical wall. The holographic principle for me is not a logical problem but a philosophical one. Theoretical physics seems to consistently push against philosophy. Einstein himself had many wrenching philosophical questions about the universe.

Ultimately these questions ask what is real? Physicists, utilizing the precepts of good science, limit their investigations to isolated artificial systems but yet the universe itself, right down to its most fundamental particles, seems turbulent, chaotic and always just beyond reach. Newton's neat and orderly universe is long gone yet even Einstein could not accept "God playing dice." Theoretical paradigms don't move quickly or easily but fractal theory (and string theories and supersymmetry and many others) indicate some movement in physics beyond Einstein. Fractal theory suggests that we can entertain patterns lurking within chaos, emergent phenomena, and self-organization as just as critical to understanding how the universe works as the physical laws themselves.

If you want to really let the dogs loose and throw religion into this discussion (this is my cue to get a beer), we could say that it seemed that in the age of Newtonian reason, God was absent from the discussion of how the universe works, or at least He was relegated to a ghost left behind in the machine. Now, God is back - in the dynamic and unpredictable nature of the universe and in the wonderful pattern and order created from randomness. Like DNA unfolding into a mysteriously complex living being, the universe is evolving, dynamic and alive. How you feel about the role of chaos in the workings of the universe will likely have some bearing on how you feel about the idea of a fractal universe.

The universe is giant puzzle. I think back to reading Richard Feynman's description of the reflection of light. There's the straight line from A to B (classical reflection) and then there's the famously Feynman-ian quirky (all possible roads taken; quantum) process from A to B. Feynman's work on quantum electrodynamics provides a theoretical base for fractal quantum mechanics as we will see later on. Or, you might also think of Young's famous slit experiment, where electrons somehow sense out their surroundings outside of the boundary of time. Einstein probably endured many sleepless nights pondering quantum entanglement, or "spooky action at a distance" as he called it. Quantum mechanics paints a spooky picture of the universe, one that we can't dismiss. How do we go about investigating it? We go forward on the hope that some single theory ultimately holds it all together and we just have to figure it out. But what can embrace the quantum scale AND what we measure at our everyday (classical) scale AND what cosmologists measure at the largest scales of the universe where dark matter and dark energy don't quite match what general relativity predicts? Will a fractal theory (or any theory!) pull it off?

Next we'll look at how to build a fractal, in Fractal Universe Part 2.

You do a great job. I have always been quietly contemplating the ideas you delve into. I have read QM, Einstein's mirror, special and general relativity, Brian Greene's chaos theory, and others. I guess just existence is fascinating. It is hard for me to wrap my head around the mathematics of it all. For being our supposed universal language, its self inflicted boundaries have unwittingly taken away equivalency. Anyhow, I will keep trying to understand those things I currently don't. It is nice to have someone Out there who I have some ability to relate to. I will continue following your posts. Thankyou. Rod

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