Fractals in Biological Systems
Angel Chang<angelx@alum.mit.edu>
February, 1993

Science had always focused on nature. Its curiosity knows no bounds as it seeks to know all nature has to tell. By reductionism, scientists were able to understand much of the orderly world but their ears and eyes remain closed to the one of chaos. For the traditional scientist, explanation for the random and unexpected behavior of nature escapes them while for others the study of the irregular and erratic is opening a whole new universe to them. This magical world of chaos would reveal nature's secrets in the stunning forms and images of fractals. Called the geometry of nature, this new mathematics is changing the face of science. Bringing together mathematicians, physicists, and biologists, it describes with surprisingly simple rules the complex and beautiful images of nature. Fractals is everywhere you look; it is in the shape of a cloud, the unevenness of ground, the clusters of stars, the shift of the current, the ever branching of the trees, and most of all it is within us. From the aperiodic beating of our hearts, to the way the circulatory system and our lungs are constructed, to the very function of our brain, the rules and characteristics of fractals can be found. It is evident fractals will represent the future of many areas of sciences, especially in biology.

Among the important contributions chaos had made to science was that it gave biologists the ability to explore the complex systems of the human body, changes in population and even the creation of life. Organs such as the lung are semi-fractal in nature with such properties as intricate detail and infinite outlines (Cherbit 165-171). James Gleick had said, "In the mind's eye, a fractal is a way of seeing infinity" (Gleick 98), and this is precisely what it is, for fractals are characteristic of having infinite detail and length. The classic example of this is this question posed by Benoit Mandlebrot, "How long is the Coast of Great Britain?" Though the immediate reaction to this question would be that the answer would be a very precise and definite answer. That, however, is not the case and surprisingly the answer differs from one source to another. According to Mandlebrot, the true length of the coastline would be infinite. To estimate the length of a coastline, a surveyor uses a standard measuring device, one that would surely miss any detail smaller that it. Thus with smaller and smaller measuring devices, less detail would be missed and the length of the coastline would emerge to be greater and greater (See Appendix A). One's common sense would suggest that inevitably the coastline would still converge to a constant value. But since this process continues to the atomic level, and perhaps even beyond, the true length of the coastline could be said to be infinite (Gleick 95-96). An example of a mathematical object that contains a length without limit within a finite area would be the Koch snowflake or the Koch Island (See Appendix B). The lung can also serve to illustrate this idea of infinity originating from the realization that the length or even the area of an object is dependent upon the resolution at which it is measured (Cherbit 65). Though it seems impossible to compare the length of any fractal object since they all would be immeasurable, this could be accomplished through the idea of fractional dimensions.

In the world of fractals, with all its peculiar array of monstrosities, there is a group of curves that have confounded and been discarded by mathematicians of the nineteenth century as oddities. These so called space-filling curves are lines that twists and turns to fill all the points in a given plane but never crosses over its self. The wondering puzzle of these curves was of what dimension were they? By the Euclidean definition, a line is of one dimension and a plane of two, but what of a line that covers every point in a plane? The answer lay in the concept of fractal dimensions which had escaped from the confines of integral dimensions into fractional ones. Instead of standard Euclidean dimensions of zero for a point, one for a line , two for a plane, and three for a solid, fractals often possess fractional dimensions. Just as the classic example of "How long is the Coast of Great Britain?", "Of what dimension is a ball of twine?" is another question posed by Mandlebrot to help explain fractal properties. The obvious answer to this would be it is a three-dimensional object. But upon closer examination, it seems the answer would be dependent upon your point of view. From a great distance away, the ball of twine would be no more than a point with a dimension of zero. From closer up, however, it is spherical, occupying a three-dimensional space. But as you examine the ball more closely, you see it consists of one dimensional strings twisted together. Upon still closer examination, the string is made up of three dimensional columns which in turn are made up of one dimensional fibers. And so on goes this process from three dimensional to one and back again, yet there is never a specific point in time at which this occurs. Still, there is a constant degree of irregularity across the different scale able to be measured (Gleick 96-98). The measurement of discontinuity or the extent to which a set of points occupies space is defined to be fractal dimensions (Pickover 402). In space-filling curves such as the Hilbert and the Peano curves (See Appendix C & D), their dimension is decided upon to be of two but they are still considered fractals. The Cantor Dust is another fascinating set of points which has an infinite number of elements yet a total length of zero (See Appendix E). The use of fractal dimensions in the study of living organisms is apparent for an organism could be described as no more than a highly convoluted surface hidden beneath a solid body. For if one looks beneath the covering skin, he would have found numerous systems with fractal surfaces and vascular branching and organs confined inside a finite volume within infinite surface area indicative of fractal dimensions (Peitgen, Jurgens, Saupe 210-211).

Though not all fractals have the characteristic of fractal dimension, each and every one of them are self-similar and iterative. It is Hans Lauerier's view that, the essential property of a fractal is continuing self-similarity. Fractal dimension is just a by-product' (Gill-Hoffstadt 83) demonstrates the importance of self-similarity compared to the other features of fractals. Still, it would be a mistake to believe an structure a fractal simply because it is self-similar (Peitgen, Jurgens, Saupe 203). Technically, the word fractals is defined as an object which displays increasing detail with increasing magnitude which corresponds to the idea of self-similarity, the repetition of details at different scales. Originally Mandlebrot have described them as "shapes that are equally complex in details as in overall form" (Pickover 402). This complexity is often formed by iterations or the repetition of an operation. The Sierpinski gasket and carpet (See Appendix F), the Mandlebrot Set and the Julia Sets all illustrates the self-similar and iterative nature of fractals (See Appendix G,H,I,J). Iterations in the human body often wears the mask of recycling cells as the body returns almost every cell in it every seven years. In the pancreas, cells are changes every twenty-four hours and the stomach lining is changed every three days. Even in the brain, this iterative process takes place as about 98% of the protein is recycled every month. On a larger scale, even the process of aging can be thought of as an iteration of cells. With death the ultimate attractor, the body is drawn closer and closer toward it with each fold and divergence of the iteration of cells (Briggs & Peat 68).

An example of the limit cycle and of its ability to resist changes through feedback is the predator-prey cycle. Take for instance, there was a lake with a wealth of trouts but few pikes. The pike population would grow rapidly as it consumes the apparently unlimited supply of trout. But as the pike's source of food reduces, the number of pikes dwindle. With fewer pikes, the trout population would then multiply and the cycle would continue (See Appendix K). Astonishingly and unexplained, this system would always settle into its original limit cycle whether more trout is dumped into the lake or most of the trout population is killed by disease. With more organisms interacting in the predator-prey systems, the limit cycle transcends from the two dimensional plane into the third dimension and beyond. The limit cycle would change to a torus attractor, a doughnut like limit cycle, as several systems of the predator-prey cycle interacts. And in the realistic world of predator and prey, the more complex torus attractor dominates (Briggs & Peat 37-40).

In the study of population biology can be found some of the most important concepts of chaos. Using the simple equation xnext = rx ( 1 - x ) to represent the fall and rise of the population of a group from one year to the next, ecologists discovered an amazingly complex system (See Appendix L). The expected results of this equation were that no matter the starting point, the system would always converge to an equilibrium but instead there was chaos. Robert May, who studied the system globally, discovered that with a low input, the system does converge to one steady state. With a higher input, however, it oscillates between two values and with a even higher value, the system becomes completely unpredictable. From bifurcation of one to two to four to eight, each after coming faster and faster until beyond the 'point of accumulation' where there is only chaos (Gleick 59-80). Surprisingly, there are islands of stability within this sea of chaos, where bifurcation repeats again and again until there is once more only chaos (See Appendix M). Feigenbaum discovered that within such bifurcating system, there is a universality. It seems that there is a geometric convergence with the scaling of 4.669 showing as a bit of regularity underneath the unexpected (Gleick 172). With a better understanding of such oscillating cycles for epidemics of measles, polio, and rubella, a new impetus is given to theoretical biology. Ecologists are going back to data that had before been complicated to handle as they change their views. It also opens a better understanding of the human body as physiologists begin to realize that organs are not static but arises from the complexity of oscillation (Gleick 59-80).

In the human body, biological systems with the properties of fractals are everywhere. With a limited amount of space but often needing to extend its surface area to the maximum, dimensional magic is often performed by biological organs. The lung is such an amazing structure, effortlessly packing in a surface area of seemingly unbelievable size into a confined amount of space. Since the ability of an animal to absorb oxygen is roughly proportional to the surface area of the lung, the lung has to have the greatest surface area possible. By practicing dimensional magic, the human lungs are able to fit in a surface area larger than a tennis court into its limited space (Gleick 109,110). Other than the fractal nature of contorting a two-dimensional object into one of three dimensions, the bronchial tubes of the lung also follow an interesting fractal sequence. The Fibonacci sequence: 1,1,2,3,5,8,13,21,..., obtained by adding together the two previous elements to form the next, can be used to described the ratios of the lengths of the bronchial tubes. However, this sequence shifts scale after ten generations and according to West and Goldberger, this shifting of fractal scales allows for greater efficiency (Briggs & Peat 107).

Taking up from three to five percent of the entire volume of the human body, the blood of the circulatory system is nevertheless able to reach every part of the body. By branching and dividing repeatedly, no cell is more that three or four cells from this complex arrangement of veins. The fractal nature of this system as it iterates endlessly can be described by the word bifurcations. Such intricate structures as the circulatory system can be found throughout the body. Undulations within undulations is revealed in the tissues of the digestive tract (Gleick 107,108). The vascular system, the nervous system, the exhaust system along with the circulatory system, all comes from the heart and must be coordinated to be never far from any part of the body. Beyond the ability of the body to form the most efficient arrangement of these structure, it also has to coordinate and merge them perfectly (Briggs & Peat 107).

The ability of these oscillators to work in synchronization is accounted for by mode-locking or entrainment. Defined as when one regular cycle locks into another, entrainment can diminish an organism's ability to adapt to the ever changing environment. But since it is still essential for the survival of an organism to respond to changes, a problem arises as to how well can such a biological system adapt to changes. According to Ary Goldberger, the fractal nature in the physical structures of these organs allows these systems to adjust to a variety of rhythms. Recognition of fractal processes as broad spectrums of information and the periodic as narrow and monotonous gives scientists an insight to the inner workings of the body. Understanding that it is the irregular and fractal that provides one with health and that chaotic rhythms are normal will provide a new view to solve many disorders of the human body (Gleick 293).

Beside the complexity of those organs, the human body is also governed by fractal rhythms and aperiodic cycles. The rhythms of the heart, oscillating at the borderline between chaos and order, has such a fractal nature and is illustrative of this point made by Arnold Mandell," When you reach an equilibrium in biology, you're dead" (Gleick 298). If the beat is too periodic, heart failure might be the result, but a heart attack might occur if it is too aperiodic. Instead, the heart follows the Feigenbaum phenomenon and period doubling. Regular under certain circumstances and chaotic under others, the normal beating of the heart is indeed both orderly and irregular (Briggs & Peat 107-108). Another characteristic of fractals is encountered with the fibrillating heart. For the normal heart, an electrical signal is send in a regulated wave through the entire three-dimensional structure, causing each cell to contract and then relax. This wave is somehow broken up in the fibrillating heart leaving the organ never at once entirely relaxed or in contraction. This uncoordinated wave can cause the blockage of arteries and can lead eventually to the death of the contracting organ. In the fibrillating heart, each part might be contracting and relaxing when the signal is received, passing it onward to the next cell. But despite the perfect workings of each individual cell, somehow the workings of the whole goes awry. Regarded also as the science of wholeness, focusing not on the parts that make up the whole but on the whole, chaos is clearly important in the study of this problem. Among the hopes of cardiologists for the future is that with the new insight fractals have given them, they will be able to predict those at risk of a fibrillating heart and to create a fibrillating device with the true rhythms of the heart (Gleick 281-284, 288-292).

Biological clocks or circadian rhythms also seem to follow a fractal pattern. Experimenting with mosquitoes, Winfree had found that with a constant temperature and light, the insect had a inner cycle of twenty-three hours. In the outside world with day and night, a jolt of light resets the mosquitoes' biological clock. By shining carefully regulated doses of artificial light on the mosquitoes, Winfree found it either advances or delays the inner cycle. He also found a singularity, a point in the cycle at which a precisely timed burst of light could cause the complete breakdown of the biological clock. This point could be compared to the 'point of accumulation' in the bifurcating population, where order gives way to chaos, demonstrating again universality, the idea that different systems can behave identically. In humans, Winfree discovered two nonlinear oscillators, the sleep-wake cycle and the body-temperature cycle. Both cycles were able to restore itself after slight perturbations in the real world but Winfree found a different case in isolation. Having a cycle of twenty-five hours with the low occurring during sleep, the body-temperature cycle would detach itself from the sleep-wake cycle after several weeks and become erratic and unpredictable. Knowing more about this irregular biological clock could help biologists find solution to many connected problems as of now unsolved, such as human jet lag and isomnia (Gleick 285-288).

The laws of fractals governs not only the numerous structures found within ourselves but also plays an important role in the evolution of life. From the first stirring of life from the chaotic interface between solid, liquid, and gases, the principles of fractals and chaos were involved. The very foundations of life, the DNA and RNA, is in itself fractals. Schrodinger had described the building block of life as an 'aperiodic crystal' based on irregularity which has the "astonishing gift to concentrate a stream of order" onto itself, keeping from decaying in this world of chaos (Gleick 299). Linked in a feedback loop, the hypercycle, the structures replicate iteratively. Displaying self-similarity, but never exactly identical with each iteration (Briggs & Peat 154). With fractals, the encoding of the amazing complexity of the body in such a small structure is also explained. When a structure is fractal in nature, it can be described with simple and iterative rules, no matter how complex it may seem by Euclidean standards. Another changing view of the DNA is that it is not the strictly deterministic and static factor it was once thought to be. Instead, through feedback, the DNA also seems to be changing day to day in relation to its environment as discovered by McClintock. He also described it as the feedback center which balances the negative with the positive (Briggs & Peat 160-166). Fascinating scientists, the possibility of controlling the DNA will make fractals of a major importance in the future of this area of biology.

From this new science of chaos is also born a new view of evolution, one that is the opposite of the traditional beliefs of Darwin. Rather than the belief that the fittest is those who compete, the feedback theory of evolution stated that those who survive are those who cooperates first proposed by Lynn Margulis. There are several major developments in evolution through feedback and symbiosis (See Appendix N). When the cyanobacteria first entered a host cell, the host protected its DNA with a membrane, creating the first cells with a true nucleus. Feedback and symbiosis were at work again as a rod-shaped oxygen-breathing bacteria entered a host and formed the mitochondria. Through the invasion of the cyanobacteria, which become the chloroplasts of plant cells, plants were formed. By symbiosis and feedback, axons, and dendrites evolved, leading to the development to the brain and neurons (Briggs & Peat 154).

Described by Ilya Prigogine as a creature of chaos, the nonlinear brain is another amazing structure born of feedback in the creation of information. Through feedback, the brain is able to function as a whole and to coordinate its relationship with the environment. Matti Bergstrom explain the brain as consisting of two parts interacting together to form thought and behavior, the 'information' end and the 'random' end. All input are processed by the 'random generator' in a structure to complex to decode with the information generator sorting through and making a coherent output. Sensations and emotions are nuances amplified by the numerous feedback loops in the brain is yet another theory presented by William Gray and Paul LaViolette. Pertaining to the storage and retrieval of the memory, there seems to be no specific part for even after large portions of the brain is cut out, information is still retained. The idea of wholeness and fractals is encountered again as scientists propose that each local region of the brain contains the whole. According to Gerald Edelman, the information embedded in feedback between neuron groups and the environment may change from day to day, as the background pattern of chaos changes. Since a memory is pattern of relationships which is continually changing, this could explain why some memories are easier to retain that others and why memories of events changes with the passing of time. Even the location of each neuron is dependent upon the fractal growth of neurons, not predetermined by genes. Forming under the guidance of 'adhesion' molecules and feedback, the growth of neural networks from nerve fibers is random and irregular (Briggs & Peat 166-174). The significance of these new fractal concepts of the brain lay in the future ability to understand and cure mental diseases and in the possibility of artificial intelligence.

From these theories of the nonlinear brain emerged the possibilities of fractals and artificial intelligence. NetTalk, a computer network modeled on the neural network of the brain, uses this concept to learn how to pronounce English words. Learning through trial and error, the computer developed its own rules. Scientists could choose a random 'seed' from any ten 'neurons' and would be able to reproduce the entire code scheme of the computer. They could also cut out several 'neurons' and still expect NetTalk to be able to pronounce word, albeit a bit fuzzily. This evidence seems to support the fractal theories of the working human brain. It indicates a wholeness, that it is not the parts that make up the whole that is the most important but the whole itself. In other computers modeled on the neural network, associative memory is displayed. With each 'neuron' responding to input from the others, scientists theorize that if the conditions are right, this nonlinear feedback would cause the computer to become intelligent through self-organization. The idea that intelligence comes from chaos is exemplified in Freeman's statement "Chaos is what makes the difference in survival between a creature with a brain in the real world and a robot in an artificial environment" (Briggs & Peat 172,174).

Ilya Prigogine had believed that "Chaos is what makes life and intelligence possible" (Briggs & Peat 166), and this view of chaos is held by many. The future role of chaos in the science of biology is inevitable as modern scientists seek to manipulate life and intelligence. They play with the strands of DNA hoping to learn all its secrets and they experiment with the brain attempting to create a self-thinking structure. To do these, they have learned to accept and embrace the importance and existence of the science of wholeness, chaos. As biologists attempt to understand the unique and magical workings of the human body, they encounter again and again the phantoms of fractals. Like ghosts, the images of fractals are hidden everywhere, waiting for mankind to discover and explorer. In the past, chaos was not but a mystical creature with no scientific base, but in the future it promises to change the face of science. Ary Goldberger had felt that "In 1986 you won't find the word fractals in a physiology book ... in 1996 you won't be able to find a physiology book without it" (Gleick 282) and this accurately describe the impact that chaos will have not only in biology but all areas of science.