Mathematics

Chaos Theory and Consciousness

 

Abstract

Consciousness remains one of the most bizarre phenomena in the universe. Though a well-researched field, science is still to reveal the fundamental nature of consciousness. This is perhaps due to the fact that consciousness is not entirely a biological phenomenon but rather an emergent process, rising out of complex interactions between simpler parts, in a large system.

On the other hand, chaos theory[1] is the branch of mathematics dealing with complex, dynamical systems. Chaos theory has wide-ranging applications – from weather prediction and market research to crowd management and heartbeat inequalities. Fractals[2] form an integral part of chaos theory, and prove that it is possible to generate complex, real-life patterns mathematically.

The question addressed in this article is whether chaos theory can reproduce the complex interactions that give rise to consciousness itself. Consciousness must be treated differently, at a fundamental stage. Once the fundamental nature of consciousness is clear, it is easier to predict its behavior. While it is beyond the scope of present science to achieve this feat, it is reasonable enough to assume that mathematics can, in principle, reproduce the complex patterns and interactions that give rise to consciousness.

Introduction To Chaos Theory And Fractals

Swirling water at the edge of a coastline, craters on the moon, turbulent phase transitions, population growth, and weather forecasting. What links and explains so many phenomena is an entirely-new science – the science of unpredictability, the science of order within disorder, the science of pattern, the science of chaos.

Mathematics is, undoubtedly, the most fundamental subject. It can be referred to as “applied logic” or, in other words, the purest form of human thought. Chaos theory is one of the newest branches of mathematics. It was difficult to set up chaos as a mainstream science; however, today it is clear that chaos theory is a highly important, practical and complex science. It may seem that everyday objects, like fluids, are well-understood but on closer inspection, it is clear that even such simple objects show complex, chaotic behavior. As James Gleick says in his book titled “Chaos”: “Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does – one water molecule, one cell of heart tissue, one neuron – and what millions of them do.” [3] Chaos theory, as is evident from the name, deals with complex dynamical systems, which show chaotic behavior, that is, cannot be predicted easily. Chaos theory also helps in dealing with nonlinear perturbed systems, which can be dealt with by approximating linear perturbation techniques. In linear dynamics, we work on an idealized situation to get an approximate result, and try to incorporate small perturbations or disturbances in the system to account for the phenomena that were ignored initially. However, the complex world around us can hardly be understood this way. Prediction is a messy business, and nothing can be predicted with a 100% certainty. But still, prediction is important in modern science. While predicting complex behavior using a set of equations is not possible, we can run computer simulations to be able to make a good prediction.

Above all, chaos theory is a new way of looking at nature. At one time, it was believed that it was, in principle, possible to predict the state of the universe at any time in the past or the future, provided that the positions and momenta of all the particles in the universe was known[4]. However, modern physics has proved that it is actually impossible to know both the position and momentum of even a single particle with absolute certainty[5].

Animal Population

One of the best ways to understand chaos theory is to look at animal population. Let us assume that the equation represents the growth of a population. Here, represents the population for the next year, while is the population for that existing year. represents a rate of growth, which may change. The term keeps the growth within bounds; as increases, falls. [It should be noted that, in this model, for convenience, population is expressed as a fraction between zero and one, where zero represents extinction, and one the maximum possible population.] If the population falls below a certain level one year, it is liable to increase in the next. But if it rises too high, competition for space and resources will tend to bring it within bounds.

Any population, as it has been determined, will reach equilibrium after many initial fluctuations. The population gradually goes extinct for small values of . For bigger values of , the population may converge to a single value. For greater values still, it may fluctuate between two values, and then four, and so on. But everything becomes unpredictable for greater values. The line representing the population function, gradually, though initially single, breaks into two, four…… and then goes chaotic. The population-versus- graph for the situation produces a curious result.

When is between 0 and 1, the population ultimately goes extinct. Between = 1 to = 3, the population converges to a single value. At about = 3.2, the graph bifurcates (breaks into two), since at this value of , the population does not converge to a single value, but fluctuates between two values. For greater values of , the bifurcation speeds up; and after a quick succession of period doublings soon the graph becomes chaotic. This means that, for those corresponding values of , the population fluctuates unpredictably between random values, and never exhibits a periodic behavior. However, on closer inspection, it is evident that the graph becomes predictable at certain points, between the chaotic portion. These can be referred to as “windows of order amidst the chaos”. After the initial chaotic region, suddenly the chaos vanishes, leaving in its wake a stable period of three. This, then continues to double – 6, 12, 24 and goes chaotic again…

Figure 1: Bifurcation Diagram. This is what the graph looks like. Along the y-axis, the population (x) has been plotted; while along the x-axis, different values of r (or the rate of growth) have been plotted.

The chaotic portion of the graph is actually a fractal. A fractal is a complex pattern that repeats endlessly on closing in. On zooming in, it is evident that the chaotic part, in the above graph, repeats the same pattern endlessly. To quote Lewis F. Richardson, “Big whorls have little whorls / which feed on their velocity / And little whorls have lesser whorls / and so on to viscosity.”[6]

However, a fractal might not always be self-similar, i.e., reveal similar patterns on zooming in. Even the coastline of Great Britain is a fractal. A fractal, roughly speaking, is a complex pattern which provides a way of measuring the roughness.

Figure 2: Fractal. The shape of the graph of the bifurcation diagram becomes unpredictable and chaotic for greater values of r. The pattern is a fractal.

On further investigations, mathematician Mitchell Feigenbaum found that, on dividing the width of each bifurcation section by that of the next one, the ratio always converges to a constant value, now known as the Feigenbaum constant, 4.6692016090.. What was curious was that, for all bifurcation diagrams, no matter what function has been used, this number remained the same. In chaos theory, the concept of scaling plays a vital role. Feigenbaum believed that scaling (across different ranges) was the key to understanding perplexing phenomena like turbulence. It was also proven that the rules of complexity are universal, and applies to all dynamical systems, regardless of their constituents.

This behavior can be observed with dripping water. Initially, water will fall drop-by-drop. Then, on speeding up the flow of the water, it will drip in pairs and so on, and then it follows a chaotic behavior. Thus, the characteristic chaotic behavior, as shown in the bifurcation diagram, applies to uncountable real-life systems – from dripping water to the amazingly-complex Mandelbrot set.

Mandelbrot Set

On a two-dimensional complex number plane, iterating a function produces many interesting patterns. (For instance, if the roots (both real and complex) of the equation are plotted on the complex plane, and a program is run to determine to which solution the numbers on the complex plane approaches, then a complex shape is generated, which is cut into three identical slices. The boundary of the shape, on closer inspection, reveals complex fractal patterns. The most notable of such patterns is, perhaps, the Mandelbrot set. The Mandelbrot set (named after mathematician Benoit Mandelbrot) is constructed from a two-dimensional complex number plane. It follows the equation: , where is a complex number. One starts by setting , and in this case, the equation becomes . Then, using as the input, one iterates the equation: , and so on. If, for a given value of , the corresponding results get bigger and bigger; the point (on the complex number plane) does not lie in the Mandelbrot set. Otherwise, it lies within the set. [For example, at = – 1, the results always fluctuate between two numbers: 0 and – 1. Thus, – 1 lies within the Mandelbrot set. At = 1, however, the results blow up to infinity.] Doing this for all the points on the complex plane produces the amazingly-complex Mandelbrot set.

The bifurcation diagram, interestingly, is exactly what the Mandelbrot set resembles from the side, in three-dimensions.[7]

Figure 3: The bifurcation diagram is an integral part of the Mandelbrot set. Rotating the above picture sideways, and viewing it in two-dimensions reveals the Mandelbrot set. The pointy “needle” portion of the Mandelbrot set corresponds to the chaotic part in the bifurcation diagram; the smaller circle in the Mandelbrot set corresponds to the first bifurcation lines and the bigger heart-shaped portion corresponds to the single, non-bifurcated curve in the bifurcation diagram.

Many similar patterns (like the original set) can be found on closing in on the Mandelbrot set. As one keeps zooming in at different parts of the set, infinitely-many beautiful, repeating patterns (which may be similar to the set itself, but never an exact copy) are revealed. (In fact, in principle, any shape can be generated from the Mandelbrot set, provided we could zoom in at the right places, and for long enough. The Mandelbrot set can be regarded as nothing less than the ‘face of God’. According to mathematician Roger Penrose, the Mandelbrot set is evidence for mathematical realism. It is so complex that it could not, possibly, be invented, but only discovered.

Figure 4: Mandelbrot Set: This is one of the most famous and beautiful fractals. It is really wonderful that this pattern can be generated mathematically. Interestingly, the ratio of the radii of successive circles on the real line in the Mandelbrot set, is the Feigenbaum constant.

Figure 5: Close-up view of the Mandelbrot Set reveals the endless, intricate patterns, especially near the boundaries of the set.

One of the most important predictions of chaos theory is that systems with slightly-different initial conditions give rise to fundamentally-different results. The most popular example is the butterfly effect. A butterfly flapping its wings can give rise to a chain of events which might end up creating a thunderstorm in some distant place. This is only an example, and this idea applies to everything in our universe. Tiny changes in the initial conditions produce results that are very different from each other and are, thus, unpredictable. Even the Mandelbrot set reflects this. It is evident on zooming in that tiny changes in the positions of the numbers chosen (on the complex plane), ends up in entirely different areas. [The black portion of the set represents numbers that are predictable, while the function goes chaotic (that is, diverges) if the number lies in the blue region. The color gradients represent how close the numbers of that region are to the set. The use of different colors also reveals the detailed, intricate patterns.]

Another way to visualize this is using topology. If a sheet of space is transformed by stretching and squeezing, then points that were initially close might end up far away in the transformed space. Also, points that were initially far might end up close to each other.

Weather Prediction

The applications of chaos theory in weather prediction are widely known. Clouds are, undoubtedly, one of the most interesting fractals in nature. They are formed by the condensation of tiny droplets of water, which occur on a random basis under suitable conditions. However, once clouds are formed, they tend to attract more tiny water droplets at certain points around them. Clouds are one of the most uniform fractal objects present in the earth, and it is impossible to determine how far away a cloud might be by looking at it. They look the same at all scales. Mathematician and meteorologist Edward Lorenz wanted to predict weather conditions. He used three differential equations[8]:

Here, represents the ratio of fluid viscosity to thermal conductivity, represents the difference in temperature between the top and bottom of the system and is the ratio of the box width to the box height (the entire system is assumed to be taking place in a 3-dimensional box). In addition, there are three time-evolving variables: (which equals the convective flow); (which equals the horizontal temperature distribution) and (which equals the vertical temperature distribution). For a set of values of and , the computer, on predicting how the variables would change with time, drew out a strange pattern (now referred to as the Lorenz attractor). Basically, the computer plotted how the three variables would change with time, in a three-dimensional space.

Figure 6: Lorenz Attractor. The lines curved out seem to be attracted to two points.

In the above fractal, no paths cross each other. This is because, if a loop is formed, the path of the particles would continue forever in that loop and become periodic and predictable. Thus, each path is an infinite curve in a finite space. Though this idea seems strange, this can actually be demonstrated by a fractal. Essentially, a fractal continues infinitely; though it can be represented in a finite space.

Attractors function in a phase space. A phase space pictorially represents dynamical systems. Each point on a phase space represents the state of the dynamical system at that time. Plotting such points for successive time intervals gives rise to an attractor. An attractor can be a simple one. For example, if we plotted a two-dimensional velocity-versus-position graph (the phase space) for a simple pendulum, we would see that the curve traced out on the phase space, as the pendulum swings, will be a curve that spirals inward to the origin. This is because, due to friction, the swinging pendulum will gradually come to a stop at its mean position (i.e., at this point, both the velocity and the position are zero.)

Figure 7: Phase space for a pendulum with friction. It seems that the curve (on the phase space) is attracted to a fixed point. Also, no matter whatever disturbances this system is exposed to, it will always come to a rest, sooner or later, due to friction. Thus, such a system is predictable and is not sensitive to initial conditions.

For more complex systems (like a double pendulum or a three-body gravitational system), the curve on the phase space becomes complex and chaotic. It should be infinite in length, i.e., no path should intersect and form a loop at any point. However, this infinite pattern must be capable of representation in a finite phase space. This is possible only if the curve is a fractal.

Also, contrary to common misconception, a dynamical system does not always end up in a chaotic and unpredictable state. A system might have more than one equilibrium state, both acting like attractors. The intermediate stages might be chaotic, but a dynamical system might end up in a stable state, too, in which case the final state of the system always remains predictable.

Turbulence, or the unpredictable behavior of fluids under certain circumstances, remained a problem in fluid dynamics. Turbulence, as was verified experimentally, was not taking place simply due to accumulation of complexity. The sudden change from predictable to turbulent behavior in fluids was most difficult to exactly explain. The concept of strange attractors, as it turns out, can explain such phenomena. A strange attractor is a complex attractor that is fractal in nature. The concept of strange attractor can explain numerous random phenomena in nature – and beyond. The Lorenz attractor is also a strange attractor. Even the orbits of stars in galaxies have been studied to show chaotic behavior. For complex and chaotic three-dimensional phase spaces, scientists use techniques like studying two-dimensional cross-sectional slices of the curve.

Lorenz also found that even slight changes in the inputs can create drastically dissimilar outputs. (This is referred to as the butterfly effect, and is technically known as “sensitivity to initial conditions.”) He modelled a mini-weather in a computer, which functioned based on twelve nonlinear equations. Then, he considered the three nonlinear equations above. He also examined the phenomenon of convection, which is the fluid motion associated with the rising of hot gas or liquid. Complex, chaotic behavior was observed even in hot gases and liquids. When it gets hot enough to set the fluid in motion, chaotic behavior is observed. But what is interesting about chaos is that a system can, simultaneously, be chaotic, yet stable. As in the case of Lorenz attractor, no matter what perturbations the system is exposed to, one always gets back the infinite, complex fractal, which is, in itself, chaotic. A great example is the Red Spot in Jupiter. This swirling spot always remains, perfectly self-organized, among the surrounding chaotic atmosphere. A complex system can give rise to turbulence and coherence at the same time.

The Sierpiński Triangle

Fractals are, in fact, an integral part of nature. Fractals can be observed almost everywhere and can also be generated in many curious ways. An interesting way is the chaos game[9]. Three non-collinear points (say, A, B and C) are chosen on a plane, such that they form an equilateral triangle. A random starting point (say, P) is chosen anywhere on the plane. The game proceeds by following certain conditions. A die is rolled. If the outcome is 1 or 2, the point halfway between the points P and A, is marked. Similarly, if the outcome is 3 or 4, the midpoint of the line segment joining the points P and B is marked. For outcomes 5 or 6, the midpoint of the line segment joining the points P and C is marked. For similar outcomes more than once, the midpoint of the line segment joining the last obtained point, with A, B or C (depending on the outcome), is marked. If this is continued for long enough, the collection of all the points resemble a beautiful fractal called the Sierpiński triangle. The Sierpiński triangle has an infinite length, because the fractal continues infinitely. However, the area tends to zero, since the black regions (figure 8) are not included in the fractal, and have zero area.

This fractal, with an infinite length, is more than a one-dimensional pattern, but less than a two-dimensional figure, since its area is zero. The dimension of the Sierpiński triangle lies between 1 and 2, therefore it has a fractional dimension. Though this seems absurd, the dimension of a fractal is basically a measure of its roughness.

[It should be noted that a fractal does not always have fractional dimensions. For instance, the Hilbert curve, on more and more iterations, covers up the area of a square, and is two-dimensional.]

If a one-dimensional line is broken into two equal halves, i.e., if it is scaled by one-half; its mass is also scaled down by one-half, since two such halves will reproduce the original line. Similarly, if we scale the side of a square by one-half, its mass is scaled by one-fourths, since it takes four squares (each of a length one-half the length of the original square) to reconstruct the original square. One-fourth is just one-half raised to the power of two, and this number is the dimension of the square, which is two. Similarly, just as a line is one-dimensional and a square two-dimensional, a cube is three-dimensional because on being scaled by one-half, the mass is scaled down by one-eighth (or one-half raised to the third power), and it takes eight copies of the smaller cube to generate the original cube. For a Sierpiskiński triangle, on scaling it by one-half, we get a similar, but smaller pattern, three of which, when arranged in the right pattern, gives back the original triangle. Thus, the mass has been scaled by one-thirds. Following the above line of reasoning, this means that one-half raised to the power of (say) , should equal one-third. And is the fractal dimension of the Sierpiskiński triangle.

Thus, , which gives . By taking the logarithm of both the sides, to the base 2, we get , which is the fractal dimension of a Sierpinski triangle.

Figure 8: Sierpiński Triangle. No points lie on the black regions, except the initial random point, which may be chosen to be anywhere on the plane. Interestingly, the Sierpiński triangle behaves like an attractor. All points on the plane seem to be attracted in a certain pattern, away from the black regions. Such a system, though, is not sensitive to initial conditions. This is because, no matter wherever we choose the starting point to be, they will always form the same pattern, provided we plot points as per the rules of the “chaos game”.

This activity may be done with more than three points, to generate more complex fractals.

The Koch Snowflake

Something as beautiful and as complex as a snowflake can be constructed mathematically. The Koch snowflake can be constructed from an equilateral triangle. The sides of the triangle are trisected and the middle part is removed from each side. With the removed portion as a side of it, another equilateral triangle is constructed, which is again treated similarly. On continuing this process infinitely, a fractal is formed, which resembles a snowflake. What is interesting about the Koch snowflake is that, though it has a finite area, it actually has an infinite length. It is lesser than a two-dimensional but greater than a one-dimensional figure. Such things are described by a fractional dimension, which, though not possible in Euclidean geometry, is a characteristic of fractal geometry. The fractal dimension of the Koch snowflake lies between 1 and 2. (It is 1.2618.)

Figure 9: Koch Snowflake: Fractal generated from a simple equilateral triangle. No matter how much we zoom in, the fractal will endlessly repeat its original pattern.

Interestingly, if one of the angles of the original triangle is set to be very small (i.e., approaches zero), then the part of the fractal generated by that end of the triangle, will be a space-filling curve. This means that the pattern generated will fill out all of space (at that region). Indeed, even a one-dimensional line can be iterated repeatedly in a certain manner, to generate the Hilbert curve.

Figure 10: Hilbert Curve. As the number of iterations tends to infinity, the one-dimensional line actually moves through all the points on the two-dimensional space, though the idea seems counter-intuitive.

Space-filling curves might also have great applications. A high-resolution picture can be filled using a space-filling curve, and a computer program could be written to define a sound frequency to each point on the picture. This might be used to develop a device that would let blind people visualize pictures by ‘listening’[10].

Barnsley’s Fern

Even something as complex, as beautiful and as realistic as a fern can be mathematically generated. This fractal is called the Barnsley’s fern.

Figure 11: Barnsley’s Fern: A beautiful and realistic fern-shaped fractal generated mathematically.

Sierpiński Carpet

Another interesting fractal is the Sierpiński carpet. It can be generated by dividing a square into a 3X3 matrix, i.e., nine squares. Then, the middle square is removed. This operation is then repeated on the eight remaining squares, and so on infinitely. When this same activity is carried out on a three-dimensional cube, the Menger sponge is formed. Interestingly, it has an infinite surface area but zero volume.

Figure 12: Menger Sponge.

The Study of Chaos

The study of chaos has revealed that, underlying the chaotic randomness in nature, there lies order, which becomes perceptible only on the average. When, for instance, Lorenz studied random weather patterns, he discovered the Lorenz attractor. To quote Sir Arthur Conan Doyle[11], “while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant.”

The study of chaos also revealed that chaos and randomness, in some sense, creates information. This is because, in the modern view, information depends on randomness. More the randomness, more the complexity, more the variation, more the disorder; more is the amount of information contained. This is obvious, since if everything were to collapse into an orderly, indistinguishable form, no information would, then, have survived; and one would have no way to perceive differences between things. Just as turbulence transmits energy from large scales to small scales, creating vortices; similarly information is transmitted back from the small scales to the larger ones. And this is done by strange attractors, which magnify the initial randomness into large ones, analogous to how the butterfly effect magnifies initial conditions and gives dramatically different outputs.

Consciousness: What Is So Special About It?

At the most fundamental stage, it is reasonable to assume that all possible phenomena must take place, probably in different systems. (The other alternative is that absolutely nothing should have taken place, which cannot account for the existence of the universe.) The universe, thus, is the result of the evolution of one such possible phenomenon.

Consciousness is a fundamental property and at a fundamental stage, anything that can perceive the passage of time can be assumed to be conscious, where consciousness does not imply emotions and things characteristic of human consciousness. At the most elementary stage, consciousness must have been the inevitable process that must have formed in the process of evolution of certain phenomena among all the possible phenomena. It is evident that over time, the nature of consciousness has changed. The more a particular system (among all the possible systems) has evolved, the less fundamental consciousness has become.

Any variation that causes an overall change in a system, as a whole, can be safely assumed to be a variation that has been accounted for, by the system. This means that this variation has been ‘observed’. By the above definition of ‘consciousness’, such a system can be assumed to be conscious. If there was no consciousness at that stage, the very existence of humankind would be questionable. It is probable, then, that there would only exist a superposition of all possible states[12].

It is also a strong possibility that with the emergence of a greater number of, more complex, and thus less fundamental consciousnesses, different ‘illusions of reality’ are produced, with some trace of the ‘actual reality’. The argument can be that a consciousness can only perceive one state, yet the other states, as per our fundamental working hypothesis, are equally likely to occur. By this line of reasoning, it may be the case that all humans create their individual ‘illusions of reality’.

In fact, rather than assuming consciousness to be a part of a greater system (like the universe), it might be more appropriate to assume that the consciousness exists as a whole, while the memories of the past, dreams of the future (etc.) are etched into the consciousness. Each human consciousness is a single system, with its own ‘version of reality’; and no two such versions can ever be the same. The actual reality might be an average representation of the different versions of reality.

Schrödinger emphasized that consciousness can be experienced only in singular, and never in plural. As he put it, “Quantum physics thus reveals a basic oneness of the universe”[13]. To a consciousness, it is evident on deeper thought, that only the present matters, and neither the past nor the future has any physical existence to it. To quote Schrödinger again, “Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.” Thus, we must not try to find certain conditions a system should fulfill to be conscious. It is important to realize that consciousness is fundamental and can be generalized, assuming the human consciousness to be a complex, special and less fundamental manifestation of the general consciousness.

Thus, we may conclude that particles formed at that fundamental stage can be assumed to be ‘conscious’; rather their properties must be referred to as ‘fundamental consciousness.’ All particles exhibit chaotic behavior at such stages. Following this line of reasoning, it is obvious that chaotic behavior is characteristic of consciousness. Over time, as the system evolves into more complex systems, consciousness becomes less fundamental.

How Are Consciousness And Chaos Related?

The link between consciousness and chaos might not be apparent. The great physicist and philosopher Erwin Schrödinger, in his book “What is Life?”[13], quoted regarding consciousness that: “The reason for this was not that the subject [consciousness] was simple enough to be explained without mathematics, but rather that it was much too involved to be fully accessible to mathematics.” Schrödinger, so many years back, had already formed the then-unusual idea that life was both orderly and complex. He saw aperiodicity as the source of life’s special qualities. His ideas on life and biology also inspired Watson and Crick’s work on DNA.

Chaos theory has numerous applications in physiology, especially cardiology. The idea is that mathematical tools could help biologists and physiologists to understand the complex systems of the human body, without a thorough knowledge of local detail. Chaos theory successfully explained the sudden, aperiodic and chaotic behavior of the heart, called ventricular fibrillation. According to chaos theory, the fibrillation is the result of disorder of a complex system, like the human heart. Though all individual parts of the heart seem to work perfectly, the whole system becomes chaotic, and eventually fatal. (This intuitively shows that the reductionist approach does not always work in science. Often, it is the entire system as a whole that is to be considered, instead of breaking it down to smaller and smaller parts.) Ventricular fibrillation is not a behavior that returns to stable conditions on its own; rather this fibrillating state is itself stable chaos. Research on the mosquitoes’ body cycle reveals that a burst of light at a special, certain time would cause the biological clock to completely break down and go random. However, bursts of light at any other random time, does not cause any long-time unpredictability in the mosquitoes’ biological clock. Fractal geometry also allows the formation of bounded curves of great lengths, and that is how the lungs manage to accommodate so large a surface area inside a small volume, which in turn, increases the efficiency of the respiratory system. Fractal geometry has also been used to model the dynamics of the HIV virus, which is responsible for AIDS. Bone fractures are fractal and even the surface structures of cancer cells display fractal properties, and this property can be manipulated to detect cancerous cells at an early stage. Another interesting application of chaos theory in the medical sciences is the phenomenon of mode locking. In this phenomenon, one regular cycle locks into another. This accounts for the ability of biological oscillators (like heart cells and neurons) to work in synchronization. The principles of chaos theory must also apply to the most complex, nonlinear and dynamic organ in the human body – the brain. The brain is not in an equilibrium state; it is dynamic and chaotic.

Fractals might also be intimately connected to psychology. The world around us is complex and fractal. Thus, as Nigel Lesmoir-Gordon writes in his book “Introducing Fractal Geometry” [14], “It is entirely conceivable that the low level of fractal complexity in modern inner cities is a strong contributing factor to the high incidence of depression reported in these kinds of environment.” This may be why we still are fascinated by the complex architecture of the ancient times. This might also, scientifically, explain why poets have commented on the hopelessly materialistic world. The human mind is not satisfied with high buildings with a simple, rectangular design. It may also be the case that humans are fascinated by ancient buildings simply because they do not spend most of their lives in such places. But fractal geometry undoubtedly is intimately connected to nature, and possibly to our minds too.

Consciousness is an emergent property. As neuroscientist David Eagleman explains in his book “The Brain: The Story of You”[14], the most appropriate way to look at consciousness is not to focus on the parts, but on the interaction between the parts. A single neuron among the millions of neurons in a human brain, is, by itself simple enough. It carries out its functions in a perfect, predictable manner, that is, sends signals via neurotransmitters across synapses. It is unlikely that consciousness can ever be understood by looking at a single neuron. What matters is the complex interaction between the neurons. Each neuron performs its own simple functions; but this large scale interaction, among the millions of neurons gives rise to something for which an individual neuron cannot account for: consciousness.

There are, as with fractals, “windows of order” amidst the chaos in the brain. For instance, the overall pattern in the brain becomes similar and more predictable every night, when the person is asleep. But if all the interactions in the brain are examined at any given instance of time, the behavior is complex, unpredictable and chaotic. To quote Erwin Schrödinger, “…we find complete irregularity, co-operating to produce regularity only on the average”[13]. It has been discovered that fractal patterns exist throughout the body – from the heart to the way blood vessels branch. As it has already been demonstrated in fractal geometry, a simple function, on repeated iterations, can produce an infinitely complex pattern. Thus, complex outcomes do not always require complex inputs.

Large-scale, complex interactions between simple parts is not uncommon. Such interactions can be seen in anthills, cities etc.. It may be assumed that these interactions must be in a precise range to support consciousness and that anthills and cities are not conscious of their own because the interaction is not in the right range. But it may also be the case that anthills and cities are conscious of their own, in a very different sense. Once it is understood that consciousness is a fundamental property and may not require emotions, movement, or any physical support to function, then the link between consciousness and chaos becomes obvious.

Most complex fractals are actually generated from simple movements. For instance, in the chaos game, the interaction of some simple points, in a particular way, gave rise to the Sierpiński triangle fractal. Even the amazingly-complex Mandelbrot set arises out of the interaction of simple points (representing certain numbers) in the complex number plane. The basic idea is the same: large scale interactions of simple units to give rise to something greater.

Even in a complex system, like a city, chaotic behavior can be observed. During the morning, all students and teachers seem attracted to schools. In an anthill, all the worker ants seem attracted to food, and so on. It is probable that these movements might form a fractal. When the universe is considered as a whole, the movements of all the particles in it, and the presence of ‘attractors’ in the form of black holes, might give rise to a fractal.

Finally, what is most notable about life and consciousness is the unpredictability. Life may take unpredictable turns at any moment. And the exact perception at any given moment, of a consciousness, is always different than the perception of that consciousness at any other moment. Due to life’s unpredictability, a loop is never formed; and this property is common among certain fractals, like the Lorenz attractor. No matter how close, no path in the Lorenz attractor intersects, and the behavior remains as unpredictable as ever.

Further Discussion

Immortality has been humankind’s long-lasting desire. In order to achieve immortality, modern neuroscientists propose to map the entire human brain (that is, cut it into ultra-thin slices and construct a map of each and every neuron and how they are arranged across the synapses, in high detail). This, in itself an insurmountable task, is only half the process. After this, a three-dimensional model of the brain (called connectome), with all the neural circuitry, has to be constructed and electrical impulses generated in the precise manner. Even after all this, it is questionable whether true human consciousness would emerge. But the idea is acceptable. In this view, consciousness does not need a physical support to survive; and in presence of the right interactions, can be run anywhere and forever [15].

The impossibility of this idea can be accounted for by the amount of information stored in the brain. According to modern physics, entropy[16] is analogous to information. Information can be assumed to be responsible for the existence of different objects in the universe. For instance, it is the information of the arrangement of atoms/molecules that differentiates a wooden plank from an iron rod, or water. More fundamentally, it is the difference in the arrangement of electrons and number of protons in an atom that defines what type of atom it is. Thus, it is evident that a huge amount of information is required to account for the complexity and uniqueness of each human brain.

The above process, though not impossible, is definitely improbable considering current technology. We may, instead, conduct more research on fractals and try to understand how and what fractal patterns resemble the patterns formed in the human brain. Generating fractals using computers is a lot easier. Another important point is that self-similar fractals do not depend on the scale. No matter to what extent a self-similar fractal is magnified, it reveals the same pattern over and over again. However, the brain, with all its neural circuitry and complex interactions, cannot be assumed to be analogous to a complex fractal. This is because the neural circuitry does not go on infinitely. Also, the size of the brain is crucial to its functioning. The brain depends on the size to function properly, and if we happen to design an organism with a differently-sized brain, then the entire physiological structure of that organism needs to be altered, to ensure its survival. Thus, from this argument, it is reasonable to assume that consciousness does not depend on the physical (or biological) part. In principle, consciousness depends on complex patterns. These patterns might be fractals, or resemble fractal patterns to a certain finite extent. It might be the case that a certain, fixed amount of threshold complexity and chaotic behavior is a prerequisite for consciousness of the order of the human consciousness. Though the brain is a support for the human consciousness, it is not the only way consciousness can be generated. Consciousness is a chaotic system, and indeed, the human consciousness shows unpredictable behavior, with occasional regularities.

Conclusion

The world is made of intricate details and patterns. Fractals are omnipresent; some of which are imaginable, while others are not. Even the universe might be a self-generating fractal, endlessly creating other universes out of itself. Thus, it is very probable that any natural object can be formed mathematically.

Mathematics, thus, is not an abstract subject with minimal practical applications. Mathematics not only has great practical applications, but also may be able to recreate the entire universe. Mathematics must be the key to understanding the fundamental nature of reality. The world surrounding us is really built up of these endlessly-repeating, intricate patterns called fractals, which, though seem to be works of art, are actually the amazing works of mathematics.

Mathematics, in particular, the study of nonlinear dynamics, has successfully addressed questions that biology had failed to do. Even the theories of natural selection and evolution can be studied in more detail, using the principles of chaos theory. As Joseph Ford said, evolution is chaos with feedback. The universe is randomness and dissipation; and randomness with direction can give rise to surprising complexity, while dissipation is essentially an agent of order.

Chaos theory has successfully proven the inherent ideas about complexity and unpredictability to be incorrect. Indeed, neither do simple systems always behave in a simple way, nor does complex behavior always imply complex causes. Also, most importantly, the laws of complexity hold universally, regardless of the constituents of the system. Chaos theory addresses the very questions scientists have been afraid to ask, due to the fear of being called a lunatic. It is reasonable to assume that chaos theory, combined flexibly with our present knowledge, might hold the key to understanding the fundamental nature of reality itself. To quote Doyne Farmer, “Here was a coin with two sides. Here was order, with randomness emerging, and then one step further away was randomness with its own underlying order.” Also, the idea of pattern is a fundamental property of nature. As mathematician Manjul Bhargava shows[17], the number of petals in a daisy must always be a number from the Fibonacci sequence, starting with 1: 1, 2, 3, 5, 8, 13, 21, 34.. Such patterns are observed everywhere in nature. As my teacher says, “nothing exists randomly just for the sake of existing, patterns have a very good reason for existing.”

Finally, if mathematics can generate something as complex as the Mandelbrot set, then it seems reasonable that, in theory, human consciousness can also be generated mathematically. To quote Sir Arthur Conan Doyle[18], “For strange effects and extraordinary combinations we must go to life itself, which is always far more daring than any effort of the imagination.” Though this feat is beyond the reach of science at present, the evidence is clear that it is surely possible. The question is not ‘if’; it is ‘when’.

References

[1] Matt DeCross, Satyabrata Dash, Christopher Williams et al. “Chaos Theory”. 2020. https://brilliant.org/wiki/chaos-theory/

[2] The Editors of Encyclopaedia Britannica. “Fractal”. 1998. https://www.britannica.com/science/fractal

[3] James Gleick. “Chaos: Making a New Science”. Viking Books. 1987.

[4] Stanford Encyclopedia of Philosophy. “Causal Determinism”. 2003. https://plato.stanford.edu/entries/determinism-causal/#:~:text=Causal%20determinism%20is%2C%20roughly%20speaking,analysis%20in%20the%20eighteenth%20century.

[5] The Editors of Encyclopaedia Britannica. “Uncertainty principle”. 1998. https://www.britannica.com/science/uncertainty-principle

[6] Lewis Fry Richardson. “Weather Prediction by Numerical Process.” Cambridge University Press. 1922

[7] Derek Muller (Veritasium). “This equation will change how you see the world (the logistic map)”. 2020. www.youtube.com/watch?v=ovJcsL7vyrk

[8] William Harris. “How Chaos Theory Works”. 2014. https://science.howstuffworks.com/math-concepts/chaos-theory4.htm

[9] Ben Sparks (Numberphile). “Chaos Game – Numberphile”. 2017. www.youtube.com/watch?v=kbKtFN71Lfs

[10] Grant Sanderson. “Hilbert’s Curve: Is infinite math useful?”. 2017. https://youtu.be/3s7h2MHQtxc

[11] Sir Arthur Conan Doyle. “Sherlock Holmes: The Sign of Four”. Spencer Blackett. 1890.

[12] Bernardo Kastrup, Henry P. Stapp, Menas C. Kafatos. “Coming to Grips with the Implications of Quantum Mechanics”. 2018. https://blogs.scientificamerican.com/observations/coming-to-grips-with-the-implications-of-quantum-mechanics/

[13] Erwin Schrödinger. “What is Life?”. Cambridge University Press. 1944.

[14] Nigel Lesmoir-Gordon, Will Rood, Ralph Edney. “Introducing Fractal Geometry”. Totem Books. 2020.

[15] David Eagleman. “The Brain: The Story of You”. Canongate. 2015.

[16] Jason Brownlee. “A Gentle Introduction to Information Entropy”. 2019. https://machinelearningmastery.com/what-is-information-entropy/

[17] Manjul Bhargava (NDTV). “Poetry, Daisies and Cobras: Math class with Manjul Bhargava”. 2015. https://youtu.be/siFBqH-LaQQ

[18] Sir Arthur Conan Doyle. “Sherlock Holmes: The Red-Headed League”. George Newnes. 1892.

Figure References

Fig. 1: “Bifurcation Diagram”. https://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fig2.9.GIF

Fig. 2: “The logistic map – Fractal Forums”. https://nocache-nocookies.digitalgott.com/gallery/17/4917_25_05_15_12_10_12.png

Fig. 3: “Mandelbrot 3D”. https://live.staticflickr.com/3053/2987125185_bbf85927d0.jpg

Fig. 4: “Mandelbrot set – Wikipedia”. https://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_mandelbrot_set.jpg

Fig. 5: “Mandelbrot set No. 82”. https://farm4.staticflickr.com/3773/33177249451_60805780fd_b.jpg

Fig. 6: “The Lorenz Attractor, a thing of beauty”. https://lh3.googleusercontent.com/proxy/y9k4p1MTviBDhTPojoYN7FGMHzBvh4OEZKh7NJTeS2S3ic2Ap2W9OtFW1P78hV-u0PDKVFEenmryiwHA6JPJqJBddt53kQ

Fig. 7: “Pendulum with friction”. https://slideplayer.com/slide/3845649/13/images/2/Pendulum+with+friction.jpg

Fig. 8: “Fractals”. https://www.sfu.ca/~rpyke/335/sierpinski_bk.jpg

Fig. 9: “Koch Snowflake Zoom”. https://sites.google.com/a/maret.org/advanced-math-7-final-project-2014/_/rsrc/1468872090410/architecture-and-arts/fractals-koch-snowflake/koch_snowflake.jpg?height=300&width=400

Fig. 10:Why does the Hilbert curve fill the whole square?”. https://i.stack.imgur.com/LSlil.png

Fig. 11: “Barnsley Fern – Album on Imgur”. i.imgur.com/DAxe2QL.jpg

Fig . 12: “Menger sponge – Wikipedia”. https://upload.wikimedia.org/wikipedia/commons/d/de/Menger_sponge_%28Level_0-3%29.jpg

About the Author

Arpan Dey, aged 15 years, is a student of Delhi Public School, Burdwan, West Bengal, India. He is interested in physical sciences and mathematics. He wishes to pursue quantum mechanics in the future. He is also an aviation enthusiast. His ideals in life, apart from his parents and teachers, include Srinivasa Ramanujan, S. N. Bose, Bohr, Einstein, Schrodinger, Dirac, Feynman etc.

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