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  1. Fractals & Chaos In Biology By: Daniel

  2. Introduction Have you ever awed at the fact that the heart beats more than 30 million times a year? Perhaps you have wondered how part of our lungs can have a surface area the size of a tennis court. You’re in luck! Thanks to fractals and chaos we can accurately describe and understand various parts of the human body. Researchers are working around the clock to try to solve biological problems such as cancer and cystic fibrosis. Instead of some new dangerous method, wouldn’t it be wonderful if we could cure diseases by the knowledge we have gained in fractals and chaos theory? Stem cell research, radiation treatment, and any other uncertain cures may all be ruled out if fractals and chaos can take a large role in the field of biology. Come, let’s find out more! Picture from:

  3. What exactly is Fractals and Chaos and why is it so important in Biology? Chaos is traditionally thought of as being confusion, hysteria, and turmoil. Chaos, however, in the sense of chaos theory is the idea that the final outcome of something can be extremely sensitive upon initial conditions. Amazingly, you can actually find brief organized patterns within chaotic systems. Chaotic systems have the three main properties of sensitivity, mixing, and periodicity. As chaos theory is still a fairly new field of research, the properties may be apt to change in the near future. Fractals are the models generated by math equations resulting in chaotic systems. Fractals are very artistic, complex, and intricate. They also have properties which include having a fine structure, being defined by a recursive process, being too irregular to be described by traditional geometry, having self-similarity, and having fractal dimension. In Biology, Chaotic systems can be used to show the rhythms of heartbeats, walking strides, and even the biological changes of aging. Fractals can be used to model the structures of nerve networks, circulatory systems, lungs, and even DNA. (Click to see full view of background)

  4. When was fractals and chaos first researched? When was chaos theory relating to biology first researched? Henri Poincaré, a notable French mathematician, was doing research on celestial mechanics in 1887 when he stumbled upon chaos. A discrete error in one of his solutions later gained fame as the foundation of all chaos theory. Since then numerous notable scientists such as Robert Shaw and Edward Lorenz have researched the subject of chaos. (Lorenz had the butterfly-effect idea) The butterfly effect- A butterfly flapping its wings in China can cause tornadoes in Texas. Picture from: interviste.asp?d=502 Henri Poincaré

  5. When was fractals and chaos first researched? When was chaos theory relating to biology first researched? The father of fractals is often considered to be a man by the name of Gaston Julia. In the early 1900’s, Julia did much research on iterated functions, and even drew some of his famous Julia sets by hand. True, there were some other works out there, such as Sierpinski’s triangle and Koch’s curve, but Julia’s work was a major breakthrough. Until the 1960’s much of the work with fractals was abandoned due to lack of technology. That changed in the 1970’s when Mandelbrot used computers to create what we now know today as the Mandelbrot Set. Gaston Julia, who sadly, lost his nose in WWI Pictures from: The Mandelbrot Set

  6. When was fractals and chaos first researched? When was chaos theory relating to biology first researched? Chaos theory relating to biology was first researched in the early 70’s. Researchers were looking at how chaos theory could be used to model population trends. Several researchers, such as George Oster, Robert May, and Jim Yorke, looked at equations such as this one in their effort to model population: xt+l = l xt(1 –xt). As for human biology, shortly after the Mandelbrot Set was discovered this also took off. Dynamical diseases, a term coined in this era, described diseases that show chaotic systems. Researchers such as Leon Glass and Michael Mackey did research in this field. Now, there are organizations dedicated to research with chaos theory and fractals in the field of biology. Leon Glass Picture from: contributors/ Onward…

  7. Select a topic The human heart Brain, DNA Lungs Click when finished…

  8. The Human Heart The rate of the beating of the human heart can be described as chaotic. The time between heart beats fluctuates due to a variety of factors including exercising, stress, and physical activities. A good way to listen to your heart is with a heart rate monitor or just place your hand on your chest or another place where you can measure your pulse. Then try a variety of exercises that cause your heart rate to change and see how rapidly and slowly your heart goes. Order and Chaos: Healthy hearts vs. Diseased hearts The Sound of the Heart Fractal Geometry of the Heart and Circulatory Structures Click to go to the main menu

  9. Order and Chaos:Healthy hearts vs. Diseased hearts Healthy heart • Has slight variations in the time between one beat to the next. • Has a heart-rate that is a chaotic pattern that is self-similar. Diseased heart • Doesn’t exhibit slight variations in time between each beat • Has a heart-rate that is steady, constant, and predictable or either extremely random Interesting Fact: This concept also applies to human walking. When humans walk with variation in their step they are normal. With the onset of a disease, such as Parkinson’s disease, the human stride is more constant. Evidence for this…

  10. Evidence for chaotic healthy hearts Dr. Ary L. Goldberger is director of electrocardiography at a hospital in Boston, MA. In the years gone past, Mr. Goldberger has done extensive research on heart rates and chaotic patterns. As Dr. Goldberger was in medical school he was learning the traditional taught methods of how hearts should be in equilibrium and should be constant. Yet, after listening to countless heart rates of his patients, he began to notice variations in completely healthy hearts. By researching with his colleagues, Goldberger was able to discover that heart rates show fractal patterns. This is not because of physical reasons, as many might believe, but because of physiological reasons. As for why the fractal patterns break down in diseased hearts is still inconclusive. Goldberger has helped establish a resource center at to help in the discovery process through the share of ideas and data. Dr. Goldberger Picture from: Next page…

  11. Evidence for chaotic healthy hearts As you can see the top graph shows a normal healthy heart. The graph of the healthy heart has more complexity then the bottom graph. Complexity = healthy in many physiological aspects. The bottom time scale graph shows a heart with CHF (congestive heart failure). CHF is just one of the many diseases that causes the heart rate to lose it’s chaotic property. Click on the image for a further view of the normal heart-rate. Picture from:

  12. As you can see the heart rate is very complex and does show signs of self-similarity. What is more surprising is how the heart-rate seems to lose its long range correlation as the heart becomes diseased or break down. (Click to continue…) Picture from:

  13. Both of these images show what happens as the heart goes out of its normal state. The bottom left graph shows a subject with heart failure. This graph has highly periodic values with little variation. The bottom right graph shows a subject with atrial fibrillation. This heart rate is very erratic jumping from the high end of heart rate to the low end, with no particular pattern. Picture from: BACK TO THE HUMAN HEART

  14. Fractal Heart Music • The son of Dr. Goldberger, a man by the name of Zach Davids, actually recorded the melodies of cardiograms on piano. He used much of his father’s research to add the right theme of music to the cardiograms and give them the tempo they needed. This may take a moment to load. It should open up a web page and load. (must have realplayer to view) BACK TO THE HUMAN HEART

  15. Fractal Geometry of the Heart and Circulatory Structures Fractals are very useful in modeling the heart and circulatory structures, and they play an important role in maintaining homeostasis. First, the main areas where fractal geometry can be seen in the circulatory system are: • Arteries and veins -Their cells and organization display the properties of fractals, such as the power-law distribution in the diameter distribution of arteries and veins. • Organization of heart muscle groups - Show properties of self-similarity, fine structure, etc. • Branching of certain muscles inside the heart - resemble the bifurcations seen in fractals such as the Feigenbaum plot • His-Purkinje network - The branches and bifurcation of this electrical system are essential to human biology and resilience. • The tendons that connect the tricuspid valve to the papillary muscles. -These again show bifurcation along with other fractal properties. • The aortic valve leaflets - These are layered providing a huge surface area, while keeping a small volume How does the fractal structure help? -The fractal structure of the veins, arteries, and heart muscles help protect the circulatory system from the strong, violent pumping of the human heart. -The fractal structure, which is usually unnecessary, can come into play when the His-Purkinje network is damaged. This helps the heart be resilient and resistant to damage. - The fractal geometry of the heart could possibly save us everyday.

  16. Fractal Geometry of the Heart and Circulatory Structures Arteries and veins Aortic valve leaflets Pictures from:,

  17. The Lungs What you are hearing is a ventilator, which hopefully you will never have to use. Thanks to the fractal geometry of our lungs, most of us don’t have to use ventilators or other breathing devices. As Benoit Mandelbrot was doing his research on Fractals and Chaos, he proposed that the lung shows signs of fractal geometry. Since then several studies have been done to prove this as being true. Another notable scientist to research fractals in the lungs was Sergey V. Buldyrev. Sergey V. Buldryev Continue… Picture from: ~sergey/home.html

  18. The Lungs The fractal geometry of the lungs helps the lungs resist and overcome problems and physical stress during their growth. By the lungs having fractal geometry they are more efficient. But how? The rate at which the diffusion of air through the alveoli occurs is directly proportional to the surface area of the alveoli and lungs. Also, In the lungs, there are small air sacs, called alveoli, that are responsible for the diffusion of oxygen into the blood. All of the alveoli in a human adult, have a total surface area of about 750 sq ft.! Yet, they have a tiny volume. Thus, the only way to model these alveoli is through fractals. As imagined, the fractal geometry of the alveoli is very high, usually around 2.9 or so. Continue…

  19. So what other parts of the lungs show fractal geometry? As a whole, the lungs show much fractal geometry. Also, in the bronchi and bronchiole tubes, there can be greatly seen the property of bifurcation. If you look back to the Feigenbaum plot, you will find similar such bifurcations. A model of the lungs as a whole. The branching bronchiole tubes (top). The bronchiole tubes and the branching arteries (bottom). A digital representation of lung tissue. Pictures from: Main Menu

  20. Fractals and Chaos in the Brain, DNA Out of the three topics given on the previous page, this one is the most new and waiting to be researched. As research is in its early stages, there have been few discoveries made in how fractals and chaos relates to the brain and DNA. Some discoveries that have been made: • By modeling electrical signals in the brain and nerves, scientists have been able to model the *alpha rhythm using chaos. Due to this discovery, the teaching of what areas in the brain control which neurological functions may be in need of a change. If wave analysis in the brain is indeed modeled by time series graphs it will indeed show that chaotic dynamics is important in neurology. • The Large surface area where neurons are packed in layers in the brain can only be modeled by using fractal geometry. • Brain oscillations and waves is where most of the chaos theory and time series graphs come into play. The Brain itself may very well be organized strictly by the laws of chaos. • DNA sequences can be very similar to fractals in that they display qualities of Brownian self-similarity, which involve small random lines making up lines instead of small, more patterned lines making up lines. • DNA and the Brain both can be modeled by fractal geometry. *Image from: *Alpha rhythm- frequency of brainwaves between 8 -12 Hz.

  21. Fractals and Chaos in the Brain, DNA • By modeling electrical signals in the brain and nerves, scientists have been able to model the *alpha rhythm using chaos. Alpha rhythm, along with other wave frequencies, can often be modeled using time series graphs. Images from: If alpha rhythm is between 8 -12 Hz and has the graph shown try to guess the following. Match the graph to the type of rhythm (frequency band). Delta: .1 – 3 Hz Theta: 4 – 8 Hz Beta: above 12 Hz Alpha rhythm – Does this look familiar to any time plots seen in the study of fractals and chaos? *Alpha rhythm- frequency of brainwaves between 8 -12 Hz. Click to see answer

  22. Fractals and Chaos in the Brain, DNA Beta: above 12 Hz Delta: .1 – 3 Hz Theta: 4 – 8 Hz Images from:

  23. Applications of Fractals and Chaos in the Brain The brain is one of the most intricate parts of the human body. It is very unlikely that science will ever be able to grasp every part of its perplex design, however chaos theory and dynamical equations may very well be a starting ground. In addition to being able to model electrical signals of nerves and the brain, chaos theory may help solve neurological diseases and progress the invention of artificial intelligence. The hit movie, “AI” displayed many types of neurosurgery and inspired feelings about how artificial intelligence would be viewed in the future.

  24. Examples of Fractal Geometry of DNA and of the Brain A Brain Fractal. A fractal of DNA. Images from: fractals/flam3/, www. , . Main Menu

  25. I hope your pursuit of knowledge never comes to an end. I hope you've learned much about Fractals and Chaos in Biology!

  26. See you later! Gaston Julia, displaying one of his many talents. Dr. Goldberger, with his dog, Willy. Pictures from: N23/ilustraciones/...,, illusions/dither.htm Helge Von Koch, with a comical friend. Benoit Mandelbrot.

  27. Credits • Slayden, Cameron. American Association for the Advancement of Science. “Welcome to Modern Biology” Science291, 1177 (2001). (image) • May, Robert. “The chaotic rhythms of life” 10/2/2002 • Microsoft PowerPoint 2002 SP3, Microsoft Clipart Gallery • Rae, Greg.“Chaos Theory: A Brief Introduction” Jan. 31st, 2003 • With sources of: • "Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994, pg. 428. • Gleick, James, Chaos - Making a New Science, Penguin Books Ltd, Harmondsworth, Middlesex, 1987. • Lowrie, Peter, personal interview over the Internet, May 17, 1995. • Rae, Kevin, "Chaos", unpublished paper, submitted to Professor Gould, Modern Physics class, Claremont McKenna College, December 5, 1994. • Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, Penguin Books Ltd, Harmondsworth, Middlesex, 1989. • Browne, Malcolm. “The Fractal Heart.” New York Times, unknown date • Landau, Misia. “Healthy Heart Keeps Polyrhythmic Beat”, March 8th, 2002. Harvard Medical, Dental, and Public Health Schools • Winter, Dan “Tunnelling .. The "Black Hole": In the Human Heart...” 12/26/2000 • Goldberger AL, “Nonlinear Dynamics, Fractals, and Chaos Theory:Implications for Neuroautonomic Heart Rate Controlin Health and Disease”, 1999, PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation101(23):e215-e220 • Berenfeld O, Sadeh D, Abboud S. “Modeling of the heart's ventricular conduction system using fractal geometry: spectral analysis of the QRS complex.” 1993 Mar-Apr;21(2):125- 34, School of Physics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel • Wikipedia contributors, “Main Page”, Wikipedia, The Free Encyclopedia. Nov. 1st, 2005. • Parnell, Lee, “Chaos and Fractals” , 2005, University of Bath, UK, • Helicon Publishing “Lung”, Research Machines plc, 2005, Tiscali Encyclopedia, • Mendelson Jonathan and Blumenthal, Elena, “Chaos Theory and Fractals”, 2000-2003, Mendelson Productions in collaboration with Blumenthal Enterprises, • Kingsley, Danny, “Brain rhythms reveal chaos”, ABC Science Online, August 23rd, 2001, News in Science • Cripe, Curtis T., “Brainwaves and EEG The language of the brain”, Crossroads Institute, November, 2005, Special Thanks to: Mrs. Rainville Rosewood High School Created By: Daniel Ellis