Thursday, January 23, 2014

Fractal Universe Part 3

Fractal Spacetime - A Whole New Paradigm

You can't just fall in love with fractals and decide, what the Hell, space-time itself (the well from which fractals are drawn) must be fractal too. As you have no doubt discovered in your reading, all phenomena are described mathematically in physics. In order to put fractal spacetime in a context that can be tested and explored, physicists must figure out a mathematically consistent theory for fractal spacetime. Much of this article is devoted to that process. It is unavoidably complex. I know many of us are uncomfortable with differentiation (calculus) let alone variations on it. I don't think it is necessary to understand every process described here (I wouldn't want to be tested on some of them!). Instead, I think it's much more interesting to see how physicists go through the process. These kinds of mathematical journeys are at the heart of physics. They tell as much about what and how the people are thinking, as they do about the phenomenon at hand. A subtext of this series of articles is how the thinking of scientists evolves, how the process of science moves.

Fractal spacetime is a possibility in quantum mechanics. In 1948, Richard Feynman described the paths of photons as they struck a mirror and reflected off of it in a shocking and brilliant way: each photon of light actually has not just one trajectory but all possible paths to the mirror and back. What we see as a single reflected path of light is the probability amplitude for all the possible paths, even crazy, long, curved paths. Ultimately, these possibilities can be mathematically described and given a geometry. And in this geometry, each photon pathway is a continuous non-differentiable trajectory, making the geometry itself fractal, if we recall from the previous article that fractal math is continuous non-differential math. These photon trajectories can be described by a fractal dimension that jumps from nonfractal behaviour (whole integer dimensions; regular spacetime) at large everyday scales to fractal behaviour (a dimension that may exist in between whole integers) at the quantum scale of physics. These in-between fractal dimensions are strange and impossible to visualize; they remind me of Platform 93/4 at King's Cross Station in Harry Potter. This classical-to-quantum/fractal transition is thought to occur at the de Broglie scale (in the nanometer (nm) or  billionth of a meter range). Above this scale, we see light reflected in ordinary straight lines as we expect it to. Below this scale, things get weird. If we could see it, reflected light would be individual photons going every which possible way, but with the vast majority of them following (fractal) trajectories that are very close to the classical trajectory.

A Classical to Quantum/Fractal Transition at the Very Small Scale

Why a transition at the de Broglie scale? Every subatomic particle has a wavelength thanks to its wave nature, which is inversely proportional to its momentum. A (1 eV) photon has a de Broglie wavelength of around 1240 nm. An electron, on the other hand, has a de Broglie wavelength about a thousand times shorter than that. Unlike the photon, it has rest mass-energy that gives it much more momentum, and much shorter wavelength. This shorter wavelength is why electron microscopes, using electrons rather than photons, have much higher resolution. What is a de Broglie wavelength?

De Broglie figured out how electron orbits inside atoms remain stable rather than crashing into the nucleus, which is exactly what would happen according to Newton's laws for gravity. De Broglie realized that (a) electrons must act like waves and (b) these waves must fit around the nucleus as whole integer waves, called standing waves. Electrons can increase their energy in an atom by moving to a new shorter wavelength standing wave. This means that electron energy comes in discrete packets and it gives electrons their observed quantum (packet-like) energy levels. It also gives a resonant scale structure to atoms. Feynman suggested that at scales smaller than these standing waves, particle behaviour transitions into continuous non-differentiable behaviour, which is characteristic of fractal behviour.

De Broglie wavelength is thought to be the large-end cut-off point for fractal quantum behaviour. There is also a smallest possible cut-off point in fractal geometry in general, and it might come as a surprise that it is not zero. Planck length is the lowest possible universal scale in physics, about 1.6 x 10-35 m. Most physicists working in this area take Planck length to be fractal geometry's lowest limit and the reasons for it are quite technical. I think it's worth noting, and quite interesting, that while the mathematics of fractals allows a fractal structure such as the Koch snowflake to have infinite magnification and therefore it can theoretically approach infinitely small segment lengths that go smaller than Planck length, spacetime fractal behaviour that is measurable in any meaningful way has  a Planck length limit imposed on it. To be described by our current physical laws, fractal geometry has to follow the universal limit of meaningful scale in physics. This doesn't necessarily mean that reality has a Planck length limit; it's seems to be more of an uncomfortable compromise between old physics and new physics, at least to me.

Quantum Fractal Behaviour: From a Differential Equation to a Partial Differential Equation

This leads us to the mathematics behind fractals and the mathematics behind quantum mechanics. Can they be reconciled? In order to describe how a quantum state changes over time, physicists turn to the Schrodinger equation, formulated in 1925. This incredibly important equation is the quantum equivalent of Newton's second law, a classical law that describes the motion of a system. Newton's second law is often written as a differential equation where motion can be described as a smoothly changing system. Schrodinger's equation is written as a partial differential equation. A partial differential equation is a differential equation that contains unknown multivariable functions and their derivatives. A differential equation, in contrast, deals only with functions of a single variable and their derivatives. Because of this,  a partial differential equation can offer a more complete description of a complex real system. These kinds of equations are used in many areas of physics such as electrostatics, electrodynamics, fluid flow and elasticity in solid state physics - in addition to quantum mechanics. They have a unique ability to treat distinctly different phenomena as variations on a single theme, bringing their dynamics together into a single language other words. Because of their multivariable functions, partial differential equations can also describe a system changing in multiple dimensions rather than just one, another big advantage. The cost is that they are usually far more difficult to solve than ordinary differential equations are.

Partial differential equations can be generalized as stochastic partial differential equations, a word we will explore shortly. It is this generalization that paves the way toward bringing the math of quantum mechanics and fractal geometry together.

From Brownian Motion To Fractal Quantum Mechanics

The Schrodinger equation describes both the wave function of the quantum system and the evolution of the quantum state of the system over time. Interestingly, although we think of the Schrodinger equation as being useful only for describing quantum scale behaviours, it is a partial differential equation that can be generalized to describe any physical system including macroscopic systems. The trick will be going from this kind of equation to a non-differential equation, one that truly describes the close relationships between quantum mechanics, fractal geometry and chaos.

Luckily there is a phenomenon that offers a potential bridge between fractals and quantum mechanics, and that is Brownian motion, a phenomenon that has humble origins. The name, Brownian motion, was coined in 1827 to describe the random motion of particles suspended in a liquid or a gas. When you see dust particles dancing in a beam of sunlight, you are observing this kind of motion. Below is a simulation of five yellow particles that collide with a set of 800 particles, each one leaving behind a blue trail of perfectly random motion. One red velocity vector for one yellow particle is also shown. This is a computer model of Brownian motion.

Brownian motion, discovered in classical physics, can be shown to underlie modern particle theory. This motion is a very simple example of something called a continuous stochastic (or probabilistic) process. A continuous stochastic process is a collection of random variables that can be used to represent the evolution of some random value or system over time. This sounds pretty close to a partial differential equation, doesn't it? It is the opposite of a deterministic process, which can evolve in only one way and can be described using an ordinary differential equation. An example of a simple continuous deterministic process is the curved trajectory of a projectile. You can reverse time and trace back the motion to its origin. Brownian motion, on the other hand, is an example of a chaotic system. It's impossible to trace back this kind of motion because it is random. In fact, you can think of most stochastic processes as idealizations of a more primary chaotic process, one which also underlies fractal geometry and is described in Part 1 of this series.

Nelson's stochastic quantum mechanics formally puts this connection between fractal mathematics and quantum mechanics together. It describes a quantum particle, such as an electron, as being subject to an underlying Brownian motion of unknown origin, which, in turn, is described by two processes called  Markov-Weiner processes, one backward and one forward. Combining these processes gives the electron its wave function and it transforms Newton's dynamics into the Schrodinger equation, which describes quantum behaviour, as we saw earlier. The Schrodinger equation is a partial differential equation that describes how the quantum state of the electron, for example, changes over time. It describes its wave function in other words. If you remember at the beginning of this article, I mentioned that, according to Feynman, the (classical) reflection of light breaks down into the continuous non-differential trajectories of individual photons at the de Broglie scale. This fractal dimension at the quantum scale is also the fractal dimension of Brownian motion, described mathematically as a Markov-Weiner process.

In other words, by introducing quantum Brownian motion (the Markov-Weiner process) to the Schrodinger equation (a partial differential system), quantum mechanics can be described in terms of fractal geometry, a continuous non-differential system.

One thing that makes this Brownian motion idea so interesting is that it changes how we think of spacetime at the quantum scale. Spacetime is a bit of a mystery at this scale. General relativity doesn't seem as relevant here because gravity's influence is minuscule and the strong force, which holds the nuclei inside atoms together, becomes accessible. Spacetime, viewed at the quantum scale, appears to have a mysterious non-zero energy that is not associated with any of the fundamental forces, called vacuum energy. Nelson's stochastic quantum mechanics turns the concept of vacuum energy on its head. Vacuum energy is the energy at any point in spacetime that allows the creation of virtual particles and their antimatter twins to pop into existence. Once they form, they immediately annihilate each other and this quantum froth of activity results in changing or fluctuating vacuum energy on a very tiny, quantum, scale. Brownian motion makes the vacuum fluctuations of spacetime the driver of quantum mechanics, if we consider that the underlying Brownian motion is describing quantum fluctuation. This makes vacuum fluctuations the "Brownian motion of unknown origin" mentioned earlier. Usually, physicists think of it the other way around: quantum mechanics (the uncertainty principle) is the driver of quantum fluctuations.

These Brownian processes suggest that spacetime at the quantum level is fractal rather than flat and Minkowskian (Euclidean-like three dimensions of space plus one dimension of time), if we assume that the trajectories of Feynman's virtual photons are part of a fractal curve. Because fractal spacetime is nondifferentiable, it implies there must be an infinite number of geodesics (virtual trajectories) that the photons choose from.

A fractal interpretation of quantum mechanics means that the physical properties defining not only the virtual photon, but any particle, (properties such as mass, momentum, spin, velocity, etc.) can be defined as geometric structures of its fractal trajectory. A photon, or any particle, is no longer a point with momentum that follows a trajectory (more specifically one of its virtual trajectories recalling Feynman's work). Now a particle IS the fractal structure of its trajectory.

Quantum spin is now a purely geometrical property of the virtual trajectories of the particle. The whole infinite family of possible geodesics can be extended to describe the wave-particle nature of a particle. The probability cloud of an electron, for example, owes itself to the non-differentiability of fractal spacetime and the infinite family of geodesics that results from it.

From Fractals to Quantum Jewels

A very recent article (2013) by Natalie Wolchover, called A Jewel At The Heart Of Physics, describes an intriguingly similar geometric description of particle-particle interactions that, like fractals, challenges our current understanding of spacetime. A jewel-like geometric object, calculated by physicist Nima Arkani-Hamed and his doctoral student, Jaroslav Trnka, encodes the probabilities of outcomes for particle interactions in its volume, drastically simplifying calculations of particle interactions in the process. Click on the link above to see an approximate image of this beautiful and mesmerizing jewel. Like fractal quantum theory, this object suggests that quantum interactions are the consequence of geometry. This object, called an amplituhedron, forms the basis of a new quantum field theory that might be help researchers find a quantum theory for gravity that will seamlessly connect two mutually exclusive theories together - quantum mechanics and general relativity. This would amount to nothing less than discovering the theory of everything. Like fractal quantum theory, the probabilities associated with quantum phenomena are natural outcomes of the object's geometry.

The new object really shines in the field of high-energy physics, and that is where it was born. Calculating all the possible outcomes of even a very simple gluon-gluon collision, for example, requires the calculation of millions of different possible scattering amplitudes. This is done by running Feynman diagrams through a powerful computer program and, even with the best technology available, complex collision probabilities can be practically unsolvable. A construction that took decades to come together, the amplituhedron effectively takes all these calculations and turns them into one function. Instead of tediously plotting out millions of position-time variables, the amplituhedron couches the scattering process in terms of variables called twistors. A handful of twistor diagrams can describe very complex particle interactions. These twistor diagrams correspond to the volumes of pieces that fit together to construct the amplituhedron. The Feynman diagrams can piece the amplituhedron together too, but they are far, far less efficient.

They also found a "master" amplituhedron with an infinite number of facets. Its volume represents the total amplitude of all physical processes. Lower dimensional amplituhedra, representing the interactions of limited numbers of particles, live on the faces of this master structure. I am very eager to see how this very new theory plays out and how fractal quantum theory relates to it. Are these two different aspects of a single description?

Fractal Spacetime in General Relativity?

Fractal geometry seems to meld much more successfully into quantum mechanics than it melds into the much larger scale where relativity becomes the basic description of spacetime, and at the largest super-galactic scales, fractal geometry runs into possibly fatal trouble.

The geometry of spacetime is currently described by a set of field equations for general relativity. These equations are based on the idea that spacetime is flat (the topology) and curved to a manifold by a metric that is described by continuous (smoothly changing) differential geometry. This is how spacetime is curved by momentum and we experience this curve as gravity.

There is the hope, as I mentioned earlier on in this series, that by coming up with this kind of fractal theory of spacetime, what physicists observe as the effects of dark energy could be distortion-type effects of the underlying fractal geometry of spacetime becoming significant over great distances across space. In view of the large-scale structure of the universe (the scale of galaxy clusters and larger), some physicists such as Luciano Pietronero (1987) have attempted to model the distribution of galaxies on a fractal pattern. He claimed that a fractal dimension could be detected over a wide range of scale in the universe, one which seemed to hint that there is both randomness and hierarchal structuring at work at these very large scales in the universe. Other researchers have also examined the large-scale structure of the universe for signs of fractal geometry. An interpretation of recent cosmology data (David Hogg et al., 2005) suggests that, while the universe seems to be fairly clearly homogenous at the level of galaxy distribution (mass is smoothly spread out as evidenced by early Sloan survey results), it may exhibit a fractal dimension at distances of about 60 light years.

While fractal geometry offers an incredibly enticing new paradigm in physics, recent galaxy survey data cannot be ignored. In fact, it tears a big and problematic hole in fractal cosmology. The very same galaxy surveys (for example, the WiggleZ Survey concluded in 2012) that recently proved the existence of dark energy (the increasing expansion rate of the universe) tells us that the universe is very homogenous at very large scales. Despite some of the earlier claims described above, this homogenous galaxy distribution offers no sign of any fractal-like patterning at this scale. This means that, while you would expect some kind of galaxy clustering (a fractal pattern) at ever-larger scales, none are seen. There is no deviation from pure randomness of mass distribution, what you would expect from a homogenous universe. These observations do not support fractal theory being the answer to explain dark energy, which shows its effects especially at larger and larger scales, and they pose a problem for any universal fractal spacetime theory. The Sloan Digital Sky Survey, scanned over a decade and completed in 2013, created the largest ever three-dimensional map of galaxies in the universe, a map that would require half a million HD TV's to view its full image, which contains over a trillion pixels of information. These results together with the WMAP data definitively show that both the large-scale structure of the universe (at scales larger than 250 million light years) and the cosmic background radiation are evenly distributed, or homogenous, meaning that there is no other ordering, fractal or otherwise, of the universe at the galactic scale and larger. They do not lend support to fractal theory being the answer to explain the general relativity anomalies of dark energy or dark matter, which begin to show their effects at very large scales of spacetime.

So, is the universe fractal or not? One possible way around this is to think of the universe as snow: it is made up of fractal flakes but it transitions to a smooth and uniform sea of white as you step back. In the same way, the fractal nature of spacetime can only be observed at the quantum scale.

To describe space-time in terms of fractal geometry, the theory needs some kind of scale relativity and a continuous non-differential geometry that is completely dependent on the scale of the observation, so that at scales larger than de Broglie scale, the scaling part is dominated by the differential general relativity part. The geometry is still there underlying the theory but it's hidden at this scale. At scales smaller than de Broglie scale, the scaling part is dominated by fractal geometry instead, as the differential general relativity part becomes less relevant. The theoretical scale change from quantum to everyday to galactic is a bit like the change that happens during symmetry-breaking in gauge particle theory, or during a phase transition, except that here, the underlying scale symmetry is always intact.

I should make a distinction here between this kind of overall scale relativity and the scale invariance of fractals themselves. Fractal images reappear no matter what scale you are looking it them from. Some people call this an example of scale invariance but it is better described as closely related self-similarity.

The Fractal Advantage

The universe at its largest scale does not appear to have any fractal geometric ordering. And yet, hints at an underlying fractal nature can be seen everywhere around us. As described above, it is possible to describe a transition from the (classic) macroscopic scale of physics to quantum mechanics, where underlying fractal geometry seems to show much promise (we will see more examples of this promise in a moment). It may still be possible that fractal geometry underlies spacetime at all scales, but at macroscopic and larger scales it must be hidden from observation. Even though spacetime at these scales does not exhibit any fractal nature, fractal-like ordering seems to be at work in various biological processes, structures and in geology. What makes this kind of ordering favourable in living systems in particular? Is it a cost-saving or simplification advantage? Does it impart structures (like shells and trees for example) with greater strength or durability? A 2012 paper called Fractal Structures Do More With Less investigates possible advantages of fractal-like design and hierarchal structuring in construction. Researchers found that, for example, when more hierarchal levels are added to structures less material is needed to support a given load. Trabecular (spongy) bone is a similar architectural example taken from biology, in which evolution favours hierarchal fractal-like ordering in order to maximize strength, all while minimizing the amount of input material required as well as minimizing weight. Similarly, tree branching reveals fractal-like ordering, a quality that Leonardo da Vinci noticed and called his "rule of trees" more than 500 years ago. This kind of ordering is especially obvious after deciduous trees lose their leaves in the fall. Tree branching may offer two advantages - hydrological (this arrangement most efficiently transports sap) and structural (it increases the tree's resistance to various stresses like snow load and wind). A 2011 study, based on computer-modeled trees, suggests that fractal ordering in trees protects them against wind damage in particular. There is a great deal of current research on the appearance and roles of fractal-like ordering in nature, but perhaps the most convincing evidence for fractal ordering in matter comes from solid-state physics.

Evidence of Fractals from Solid State Physics

There is increasing evidence that fractal geometry underlies processes at the molecular, atomic and quantum scale.

For example, during phase transitions from regular conductors into superconductors, when electrons in a material organize themselves in ways that resemble bosonic (force particle) behaviour rather than the fermionic behaviour that particles of matter normally display, fractal geometry pops up. Even very good conductors experience at least some electrical resistance. However, during this phase change the wave functions of electrons spread out over the whole material in a special way that allows it to conduct electricity without any resistance at all. High-temperature superconductors are especially mysterious, because the molecular jiggling that goes on at temperatures well above absolute zero should destroy the kind of ordering that is necessary for a spread-out wave function. Physicists have recently found a clue about how this phenomenon is made possible. In 2010, physicist Antonio Bianconi discovered that oxygen atoms inside ceramic compound high-temperature superconductors appear to be in random positions and to take on complex geometries that display self-symmetry, a fractal behaviour. Larger fractals correspond with higher superconductivity temperatures. No one yet knows exactly how fractal ordering seems to stabilize wave functions and make high-temperature superconductors possible.

Ali Yazdani and his colleagues at Princeton University in the US observed a fractal pattern created when electrons interfere with one another. They observed the material gallium arsenide undergoing a phase transition under a scanning tunneling microscope (it gives you atomic-scale resolution) and found that a fractal pattern is observed as it changes from a metal into an insulator. When this happens, the wave functions of the electrons change from being shared across the whole material (metallic state) to being localized at individual atomic lattice sites (insulator state). During transition, the electron wave functions get squashed together and begin to affect each other in a complicated pattern of constructive and destructive interference and this is when a fractal pattern develops. Their results were published in 2010.

Last year (2013), physicists found the first proof of a decades-old theoretical fractal pattern called the Hofstadter Butterfly. Hofstadter, a graduate student in the 1970's, discovered that electrons confined inside a crystalline atomic lattice would race around in circles when placed in a powerful magnetic field. The motion of the circling would soon become complicated (chaotic). When plotted on a graph, the motion revealed a fractal pattern that looked like a butterfly, shown below as a computer rendering, although fractals were not known at the time.

In the diagram above, the horizontal axis is energy and the vertical axis is magnetic flux through the material. Warm and cold colours represent positive and negative values for Hall conductance (a voltage difference), respectively. Like all fractal images it shows self-similarity. Small fragments of the structure contain a (distorted) copy of the entire structure.

Pablo Jarillo-Herrero at MIT in Cambridge found that by stacking a sheet of graphene with a sheet of boron nitride and applying a magnetic field, he observed discrete changes in conductivity, stepwise jumps that corresponded to the same splitting of electron energy levels that Hofstadter observed. Wolfgang Ketterle, also at MIT, is currently trying to go a bit further by making supercooled rubidium atoms act like electrons by trapping the atoms in regularly spaced pockets and guiding them with lasers and gravity to mimic the circular motions of electrons in a magnetic field. If he succeeds, he may be able to show fractal ordering at the atomic level.

Some physicists are wondering if these kinds of fractal organization observed with electrons and atoms might offer quantum clues about why living systems tend to show a preference for fractal-type structures. It's possible that, like many natural and geologic structures, the natural world at the quantum level favours fractal structures.

Other researchers are making a possible link between string theory and fractal geometry. General relativity treats spacetime as 4-dimensional with three spatial dimensions and one time dimension. String theory predicts the existence of extra dimensions in spacetime. M-theory, for example, predicts 11 dimensions. A new possibility is that the dimensions in spacetime change with fractal scale, allowing small scales to exhibit fractal properties. For example, such a theoretical framework could describe quantum relativity, or quantum gravity in other words, where gravity at quantum scales appears fractal. This idea expands upon the idea that fractal spacetime is composed of non-integer dimensions, rather than the whole integer dimensions (at all scales) of spacetime described by Euclidean space, Minkowski space and the curved spacetime of general relativity. By giving fractal spacetime non-integer dimensions, the properties of spacetime depend on the scale of observation. In this case, the very dimensions of spacetime are scale-dependent.

Fractals Push Against the Differential Heart of Physics

Fractal geometry at the quantum level seems to be gaining momentum because there is so much promise, as well as some enticing experimental evidence coming together, as physicists try to sew together a consistent fractal theory for quantum particle behaviour. Meanwhile, research into fractal biology, geology and several other fields is taking off. But the possibility of a fractal cosmos seems far less promising. At best, a possible underlying fractal nature seems to be hidden from view.

Physicist Tom Palmer thinks that fractal geometry might be alive and well in the cosmos after all - if we look for it in the right place. He argues that each physical system around us has an invariant set, a mathematical ground state, in which it is unable to lose any more information. If you take a large star, for example, you will find that it has an enormous amount of data held within all the atoms that make it up (information like quantum spins, mass, momentum, energy state, etc.). When it starts to collapse in on itself at the end of its life, some of that data is lost. When it collapses all the way into a black hole, much more data is lost. The black hole is a minimal information ground state where no more data can be lost, and this is the invariant set which underlies that star's information. This kind of logic can be extended to the universe as a whole, and the invariant set of the universe might be fractal in nature. This approach could lead to an explanation for some of the most puzzling paradoxes in spacetime such as nonlocality - the ability of two entwined particles to communicate with each other across vast distances of space, or the ability of a single particle to exist in more than one location in space at the same time. It seems reminiscent of the Holographic approach to spacetime.

Perhaps the greatest promise offered by fractals is the possibility of creating more accurate mathematical models of how nature works. It seems, as we saw earlier, that reforming quantum mechanics into a non-differential equation opens up a whole new way of investigating quantum processes. A 1993 paper by Laurent Nottale discusses the movement away from differential math toward non-differntial math in physics. Since the time of Isaac Newton, differentiatial calculus has been used to describe most physical phenomena. There are countless examples of physical and biological processes (any phenomenon that changes over time) that are described in terms of one or more differential equations. This Wikipedia link lists many examples of differentiatial calculus at use in physics, engineering, biology and economics to describe processes such as radioactive decay, diffusion, animal population dynamics and evolutionary changes, just to name a few.

Calculus (differentiation and integration), developed in the mid 17th century, was a great breakthrough in physics because it offered a way to model continuous change in systems. Yet, there is no underlying principle in place that says the fundamental laws of physics must be differentiable. What if the basic reality of the universe is more accurately modelled using non-differentiable mathematics, and that fractal geometry underlies all physical processes even though it may be hidden at larger scales? If the universe really does have a fundamentally fractal backbone, it would mean reconstructing physical laws in terms of continuous but non-differentiable equations. Quantum mechanics, for example, becomes mechanics in non-differentiable spacetime.

Fresh new possibilities like this remind us that even established paradigm-making theories are not sacred. There may be other overlooked assumptions waiting to be questioned.

A Few Final Questions:

Fractal geometry seems to impart some kind of efficient process to particle interactions, an efficiency that nature at larger everyday scales seems to draw from. Is it the geometry itself or is there something deeper from which it draws? Is fractal geometry universal, and if so how is it hidden from view at the cosmic scale? Will fractals be the link between the macro universe and the micro universe that allows physicists to find a theory for quantum gravity?

Thursday, January 16, 2014

Fractal Universe Part 2

From Math Comes Fractals

What exactly is a fractal? An excellent online resource for fractals is the online educators guide at

A fractal is usually defined as any object or mathematical set that displays self-similarity on all scales. The object doesn't have to have exactly the same structure at all scales but the same "type" of structure must appear. In the Mandelbrot set mentioned in the previous article for example, the black figure reappears over and over but in some cases it appears distorted.

The mathematical set, or formula, is often very simple. The formula for the Mandelbrot set is shown left. You start by plugging in a constant value for C for each test of this equation you want to perform. Z starts out as zero. The equation gives you a new Z. You plug this back into the equation at old Z and run it again, and so on. Each time you run it through is called an iteration. It's called a set as well as a formula because you are generating a collection of numbers.

We can rewrite the formula as Zn+1 = Zn2 + C

Let's start with a value of C = 1.

Z1 = Z02 + C = 0 + 1 = 1
Z2 = Z12 + C = 1 + 1 = 2
Z3 = Z22 + C = 4 + 1 = 5
Z4 = Z32 + C = 25 + 1 = 26
Z5 = Z42 + C = 676 + 1 = 677

If you graphed these results against n, you would get an upward parabolic curve because the numbers increase exponentially (to infinity). Most starting values of C will go to infinity like this, but not all.

If you start with C = -  0.5, you get entirely different results.

Z1 = Z02 + C = 0 + -0.5 = -0.5
Z2 = Z12 + C = 0.25 + -0.5 = -0.25
Z3 = Z22 + C = 0.0625 + -0.5 = -0.4375
Z4 = Z32 + C = 0.1914 + -0.5 = -0.3086
Z5 = Z42 + C = 0.0952 + -0.5 = -0.4084
Z6 = Z52 + C = 0.1638 + -0.5 = -0.3362
Z7 = Z62 + C = 0.1130 + -0.5 = -0.3870
Z8 = Z72 + C = 0.1498 + -0.5 = -0.3502

When you graph these results you get an oscillation of values which gets smaller and smaller. Eventually the results converge on a value of - 0.365.

If you start with C = -1, you get a value of Z that oscillates between two fixed points:

Z1 = Z02 + C = 0 + -1 = -1
Z2 = Z12 + C = 1 + -1 = 0
Z3 = Z22 + C = 0 + -1 = -1
Z4 = Z32 + C = 1 + -1 = 0
Z5 = Z42 + C = 0 + -1 = -1

This oscillation will continue forever.

The Mandelbrot set is made up of all the values for Z that stay finite, so most solutions such as those for when C = 1 are thrown out because Z in those cases goes to infinity.

So far we've looked at simple starting values for C: 1, -0.5 and -1. To graph the Mandelbrot set and see the beautiful image, we need to plot results along two axes on a plane. The plane we use is not your basic X-Y Cartesian plane. Instead we use something called a complex plane where the real axis, X, contains all the numbers we've already dealt with that don't go to infinity. The Y axis, however, is made up of imaginary numbers. The most basic imaginary number is i, and it is equal to the square root of -1 (something that can't exist).

Instead of plotting integers, in other words, like we do on an ordinary Cartesian plane, we plot complex numbers, each of which has two components - a real number and an imaginary number. A real number is an ordinary number. It can be positive or negative, a fraction or a whole number. An imaginary number is a real number times the special number, i. A complex number, therefore, has the form a + bi. An example of a complex number would written as -0.75 + 0.1i. For the Mandelbrot set, both Z and C are complex numbers.

Values for Z centred around the complex number, -0.75 + 0.1i, for example, give you a portion of the Mandelbrot set called the Sea Horse Valley (where the black blob head and body connect) when plotted. The squiggly "hairs" look like sea horses if you use your imagination. To see several more zooms of this valley, Wikipedia offers a great set of images in striking detail here.

(Wolfgang Beyer;Wikipedia)
This is how you go from a simple formula to the beautiful image, but you need to run the equation many millions of times to get a detailed two-dimensional image of a fractal.

The mathematics behind the Mandelbrot set is actually very old but it was not until the computer age that its true beauty as a fractal was discovered. Benoit Mandelbrot, a mathematician, coined the word fractal when he discovered the Mandelbrot set image in 1979. Luckily he had access to IBM computers and he could create his own fractal images on them, letting the computer do the tedious job of computing all the values.

The math of the Mandelbrot set is fairly straight forward but the solutions show some  unpredictability built into the system. You are squaring old Z every time you run through the equation, so you expect new Z to get bigger and bigger and approach infinity. For most starting values of C that's what happens, and there's no surprise. However, for some values of C, the new Z converges on a single value or it alternates between a fixed set of values, something you don't expect. These points correspond to the black shapes in the Mandelbrot fractal. Around the edges of each black shape, all the values of C make the equation tend toward infinity. The colour is proportional to the speed at which equation expands toward infinity.

Watch this 5-minute video as it zooms in on a Mandelbrot set, one of the deepest zooms ever performed, 2.1 x 10275 times. It demands incredible computing power. The background music is called "Research Lab" by Dark Flow, a perfect choice. A magnification this deep would mean that the original figure in the animation would be far larger than the diameter of the universe!

If you have some programming skill, you can set up your own algorithm and program your PC or Mac to create your own Mandelbrot image and then zoom in on it. Andrew Williams at offers an online tutorial to get you going.

Non-differentiability of Fractals

The Mandelbrot set is an example of a fractal curve. A fractal curve is any geometric pattern that is repeated at smaller and smaller scales to produce shapes and surfaces that cannot be represented by classical Euclidean geometry. It is a curve that bends and curls at every level of magnification.

The Koch snowflake is one of the earliest mathematical fractal curves to be discovered. It appeared in a 1904 research paper. The animation below right goes through seven iterations of the curve, from a triangle to a complex snowflake outline:

The Koch curve, like all fractal curves, has some interesting properties. For example, it has infinite length. Each iteration creates four times as many line segments as the one before, and the length of each segment is one-third the segment length in the previous iteration, so the total length increases by one third with each iteration. The length of the curve after n iterations is (4/3) n x the original triangle's length, where n goes to infinity.

If we look closely at the Koch snowflake we can get a clue about how this kind of mathematical curve is non-differentiable.

First, to find out what a differential curve is, take any kind of nonfractal curve, like the one drawn in black in the graph below, for example.

You can take many slope measurements along this black curved line by drawing many lines (in red) tangent to the curve at every point along the curve, and this way you will get a set of numbers that describe its slope. You are basically treating the slope as a head-to-tail series of very tiny straight lines. To do this, you turn each line into the hypotenuse of a right triangle (orange). This way you can get two values - rise over run. The more measurements you take, the better your approximation is to the real slope. If you could take an infinite number of measurements like this, you would get a prefect reproduction of the slope. To get this kind of perfect measurement without infinite work, you can take a function of these measurements, which in mathematics are called derivatives. The slope of the tangent line is equal to the derivative of the function at each point on the curve (three points on the curve are marked green as examples). The process of finding each derivative is called differentiation. Instead of calculating slope at every single point by hand, you let the function do the heavy lifting for you. A graph of the function at each point gives you a perfectly accurate slope or curve. Differentiation offers a way to describe not just curves but any smoothly changing phenomenon, like a chemical reaction, for example. This is the basis of calculus, which is widely used in physics to describe how systems change over time. Calculus is fundamental to physics because with it you can study how matter and energy interact, and how they change as they do so. Calculus offers a model, an approximation, of the real system, one that is accessible for you to analyze.

If we take our Koch snowflake and try to take the same kinds of measurements for slope everywhere on it, we'll find that it's impossible to do because you can never get a segment of curve short enough to take your slope measurement. Each curve breaks down into infinitely more segments. Remember, the pattern reappears over and over again at smaller and smaller iterations to infinity. This makes the Koch snowflake, and all fractal curves, non-differentiable at any point.

Fractal Dimensions

If you plot the Mandelbrot set or any fractal set on a log-log versus scale graph, you will get a straight line. The slope of the line gives you something called the fractal dimension. A fractal dimension is very unique, and much different from what we ordinarily think of in three-dimensional space.

Any ordinary point has a dimension of zero. A line has a dimension of one (length). A square is 2-dimensional (D; length + height) and a cube is 3-D (length + height + depth). This way of describing dimension is called topological dimension. It works for Euclidean geometry and we are all fairly familiar with it, but it doesn't work for fractals.

An example that illustrates this problem is the fractal Peano curve. We call it a curve even though there is no waviness to it. A curve, in fact, is defined in math any continuous function which is built from unit intervals.

Three iterations are shown below.

Antonio Miguel de Campos;Wikipedia
The shape at the far right becomes a perfectly solid 2-dimensional square after many iterations, but its topological dimension remains 1 because it is made up of 1-dimensional lines. The formal definition of a fractal is any figure where the fractal dimension (2 for the Peano curve) is higher than the topological dimension (1 for the Peano curve).

Fractal dimensions don't have to be whole integers, but they can be, as in the case of the Peano curve. The Koch snowflake has a fractal dimension that isn't a whole integer. Remember the length of its curve after n iterations is (4/3)n x the original triangle's length, where n goes to infinity. Unlike the Peano curve, the fractal dimension of the Koch snowflake is very difficult to visualize, so we can use another technique instead. We can calculate its fractal dimension by taking log 4/log 3, which is 1.26186. This is greater than the dimension of a line but less than the dimension of a two-dimensional surface. This is something you never see in Euclidean geometry OR in the geometric description of space-time. The metric tensor that describes spacetime has three whole integer (Euclidean) dimensions for space and 1 whole integer dimension for time. In situations describing space-time close to or at the speed of light or where gravity bends space-time, the geometry becomes non-Euclidean but dimensions in these cases are still treated as whole integers.

Fractal dimension gives you an idea of the complexity of the fractal curve. A fractal curve with a dimension very close to 1, such as 1.1, behaves much like an ordinary line does. A fractal curve with a dimension close to 2 fills space much like an ordinary 2-dimensional surface does. A fractal curve with a dimension close to 3 fills space almost like a volume does.

So far we have seen fractals that are built by using a type of iteration called generator iteration. With this kind of iteration, you simply substitute certain geographic shapes with other geometric shapes.

To build a Koch snowflake you substitute this:

for this:

The swapping out can actually be done using any kind of function. For
example, you can repeatedly apply geometric transformations such as rotation and/or reflection, or you can take a mathematical formula and substitute one or more different mathematical formulae for the initial formula. Formula iteration (called IFS) produces some of the most complex fractals. Examples of this type of iteration are the Mandlebrot set and the strange attractor set we saw in the previous article. Meteorologists use this kind of iteration to construct weather models.

The Barnsley fern is an exceptionally pretty example of a fractal created through four transformations called affine transformations. The formula for one transformation in two dimensions is shown below.

Transformations in  mathematics are often shown as matrices (the square brackets above). These kinds of transformations preserve points, straight lines and planes but not angles between lines or distances between points. Each of the leaves of the fern frond shown right is related to one another by the affine transformation. For example the red leaf can be transformed into the dark blue leaf by a combination of reflection, rotation, scaling and translation.

By playing around with coefficients (a, b, c, d, e and f above) in the transformation formula, you can make mutant fern varieties, such as the three shown below. The more iterations you run through, the more complex each fern diagram gets. As you can imagine, IFS models are especially useful for creating computer-generated imagery (CGI).

 (images to the left:DSP-user;Wikipedia)

From Simple To Complex

So far, we have looked at how fractals are built. We can see how the geometry of a fractal doesn't lend itself to differentiation or to ordinary Euclidean geometry, and we can see how a simple starting point can become very complex through many iterations, but we  haven't really looked into how fractals are related to chaos. Where does the chaotic nature of weather, for example, come into the formula iteration process just mentioned?

To answer this, we need to take a closer look at functions and numbers. Let's take a simple formula:

xnew = bx(1-x)

For every X value, you map it to bx(1-x). You run it through bx(1-x) to get your next new x value. You take that value and run it through bx(1-x) again, and so on. This is mathematically what you are doing when you do a formula iteration.

I'm using the example in Fractals Unleashed Tutorial Chapter 13. Let's take this formula now and make b = 1.5 and we'll start with x = 0.234. We get:

.234, .269, .295, .312, .322, .327, .330, .332, .333, .333, .333 . . .

After a while, the iteration gets stuck on 0.333.

Let's make b = 3.20 and we'll start with the same x, 0.234: We get

.234, .574, .783, .544, .794, .524, .798, .516, .799, .513, .799, .513.   . .

Now the numbers start jumping back and forth between 0.799 and 0.513. These two different results are similar to what we found when we played with the Mandelbrot set. If we use b = 3.45, the results will settle into jumping between 4 numbers and if we use b = 3.54, we will find them jumping around 8 different numbers! As we increase b, the size of the number cycles goes up. What's going on?

This is the process called bifurcation  in mathematics. These iterations can be plotted on a bifurcation map. An example of a bifurcation map is shown below for the functional equation xn+1 = rxn(1-xn). This formula is often used to approximate the evolution of animal populations over time.

The value, r, plays a similar role to b, above. Like the earlier function, as you increase the value of r, the results begin to jump between 2 numbers (A), then 4 (B), then 8 and so on.

What is especially fascinating here is that you can take a section of the map above and magnify it, and you will see that it's a fractal, as demonstrated below.

These bifurcation fractals are called Feigenbaum fractals, after Mitchell Feigenbaum, a pioneer in chaos theory. In fact, this fractal shows the close connection between chaos and fractals. As the value of r (or b in our earlier example) increases, the system moves toward chaos (all those jumping results). This is actually a very simple example of chaos. There doesn't seem to be any obvious randomness you might expect but there is unpredictability. Changing the input, r, unexpectedly gives rise to bifurcations at certain values. In any chaotic system, a tiny difference in input results in a huge range of diverging possible outcomes. If you take a vertical slice of a bifurcation in the map above, you will get a strange attractor (described in the previous article) for that specific value of r. This mathematical system is an example of a nonlinear system. This means that the output is not directly proportional to the input. All chaotic systems are nonlinear systems. Most systems in nature are nonlinear. Scientists often approximate natural behaviours by using much easier linear equations to describe them, but there is a price for doing so. When this approximation is done, there is a risk of having chaotic elements or even singularities (input values for which the results are meaningless) hiding somewhere in the linearization. This is sometimes where catastrophic building failures have their origin. After an earthquake, a few seemingly unimportant structural components in a building fail, for example, and this leads to an unexpected cascading catastrophic failure of the whole structure.

Bifurcation fractals are examples of discrete chaotic dynamical systems. Most systems in nature are continuous dynamical systems. Continuous dynamical systems with Euclidean geometry can never be chaotic. However, most continuous dynamical systems in nature have non-Euclidean geometry (fractal geometry being just one example), and these systems can often be chaotic. Below, Euclidean geometry is compared with elliptic and hyperbolic geometry, two common non-Euclidean geometries.


The non-Euclidean non-differential mathematics behind fractals and their close relationship to chaos and other nonlinear systems seem to be characteristics shared in common with continuously changing natural systems. Yet, most of the mathematics of physics is historically grounded in Euclidean geometry and differentiable functions. This math offers relatively easy and workable models for nature, but perhaps it should come as no surprise that they don't quite work when we try to describe certain phenomena. If nature is fundamentally math as Max Tegmark suggests, then there is increasing evidence that it is made of fractal math. This implies that the mathematics we need to describe nature should be non-linear, non-differentiable and non-Euclidean. Most theory in physics, however, is written as differentiable functions mapped onto Euclidean space - an approximation that might be too rough to use for some models. We've seen the naturalness of fractal geometry all around us in nature but what about the fundamental structure of spacetime itself?  Maybe the traditional framework of mathematics is partly to blame for the scale problem of spacetime as well. In recent articles (Dark Matter and Dark Energy) we wondered what might lie beyond Einstein's geometry of spacetime. In the next article we are ready to ask what lies beyond our customary mathematics used in most physics. Is the math fundamentally wrong? We will explore the possibility of fractal space-time next, in Fractal Universe Part 3.

Tuesday, January 14, 2014

Fractal Universe Part 1

Why a Fractal Universe?

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.

One of my favourite examples is the Lichtenberg Figure, shown below right. I used this image in my series on lightning. High voltage applied to one point in a block of Plexiglass ® creates highly branched discharges throughout, which are thought to extend all the way to the molecular level. They are etched white because the electric potential breaks down the structure of the material.

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
As we ponder the fractal, another question might also ask itself. A fractal image is an utterly complex, beautiful and infinite image that evolves from simplicity as numbers are simply re-inserted over and over again into a mathematical formula. Complexity is generated through a strange play of numbers with each other (we will explore how this happens later on). In a roughly analogous way, we can think of many complex phenomena in physics that rely on just a handful of basic formulae, each one relating one function or value to another function or value. A good example is Einstein's deceptively simple but incredibly far-reaching E= mc2, 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.