What exactly is chaos? The name "chaos theory" comes from the fact that the systems that the theory describes are apparently disordered, but chaos theory is really about finding the underlying order in apparently random data.
When was chaos first discovered? The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn't predict the weather itself. However this computer program did theoretically predict what the weather might be.
One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run.
When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. (See figure 1.) Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.
By all conventional ideas of the time, it should have worked. He should have gotten a sequence very close to the original sequence. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Surely the fourth and fifth, impossible to measure using reasonable methods, can't have a huge effect on the outcome of the experiment. Lorenz proved this idea wrong.
This effect came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings.
This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment. Such things are impossible to avoid in even the most isolated lab. With a starting number of 2, the final result can be entirely different from the same system with a starting value of 2.000001. It is simply impossible to achieve this level of accuracy - just try and measure something to the nearest millionth of an inch!
From this idea, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory.
Lorenz started to look for a simpler system that had sensitive dependence on initial conditions. His first discovery had twelve equations, and he wanted a much more simple version that still had this attribute. He took the equations for convection, and stripped them down, making them unrealistically simple. The system no longer had anything to do with convection, but it did have sensitive dependence on its initial conditions, and there were only three equations this time. Later, it was discovered that his equations precisely described a water wheel.
The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)
This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment. Such things are impossible to avoid in even the most isolated lab. With a starting number of 2, the final result can be entirely different from the same system with a starting value of 2.000001. It is simply impossible to achieve this level of accuracy - just try and measure something to the nearest millionth of an inch!
From this idea, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory.
Lorenz started to look for a simpler system that had sensitive dependence on initial conditions. His first discovery had twelve equations, and he wanted a much more simple version that still had this attribute. He took the equations for convection, and stripped them down, making them unrealistically simple. The system no longer had anything to do with convection, but it did have sensitive dependence on its initial conditions, and there were only three equations this time. Later, it was discovered that his equations precisely described a water wheel.
At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other. (James Gleick, Chaos - Making a New Science, pg. 29 )
The equations for this system also seemed to give rise to entirely random behavior. However, when he graphed it, a surprising thing happened. The output always stayed on a curve, a double spiral. There were only two kinds of order previously known: a steady state, in which the variables never change, and periodic behavior, in which the system goes into a loop, repeating itself indefinitely. Lorenz's equations were definitely ordered - they always followed a spiral. They never settled down to a single point, but since they never repeated the same thing, they weren't periodic either. He called the image he got when he graphed the equations the Lorenz attractor.
In 1963, Lorenz published a paper describing what he had discovered. He included the unpredictability of the weather, and discussed the types of equations that caused this type of behavior. Unfortunately, the only journal he was able to publish in was a meteorological journal, because he was a meteorologist, not a mathematician or a physicist. As a result, Lorenz's discoveries weren't acknowledged until years later, when they were rediscovered by others. Lorenz had discovered something revolutionary; now he had to wait for someone to discover him.
Another system in which sensitive dependence on initial conditions is evident is the flip of a coin. There are two variables in a flipping coin: how soon it hits the ground, and how fast it is flipping. Theoretically, it should be possible to control these variables entirely and control how the coin will end up. In practice, it is impossible to control exactly how fast the coin flips and how high it flips. It is possible to put the variables into a certain range, but it is impossible to control it enough to know the final results of the coin toss.
A similar problem occurs in ecology, and the prediction of biological populations. The equation would be simple if population just rises indefinitely, but the effect of predators and a limited food supply make this equation incorrect. The simplest equation that takes this into account is the following:
next year's population = r * this year's population * (1 - this year's population)
In this equation, the population is a number between 0 and 1, where 1 represents the maximum possible population and 0 represents extinction. R is the growth rate. The question was, how does this parameter affect the equation? The obvious answer is that a high growth rate means that the population will settle down at a high population, while a low growth rate means that the population will settle down to a low number. This trend is true for some growth rates, but not for every one.
One biologist, Robert May, decided to see what would happen to the equation as the growth rate value changes. At low values of the growth rate, the population would settle down to a single number. For instance, if the growth rate value is 2.7, the population will settle down to .6292. As the growth rate increased, the final population would increase as well. Then, something weird happened.
As soon as the growth rate passed 3, the line broke in two. Instead of settling down to a single population, it would jump between two different populations. It would be one value for one year, go to another value the next year, then repeat the cycle forever. Raising the growth rate a little more caused it to jump between four different values. As the parameter rose further, the line bifurcated (doubled) again. The bifurcations came faster and faster until suddenly, chaos appeared. Past a certain growth rate, it becomes impossible to predict the behavior of the equation. However, upon closer inspection, it is possible to see white strips. Looking closer at these strips reveals little windows of order, where the equation goes through the bifurcations again before returning to chaos. This self-similarity, the fact that the graph has an exact copy of itself hidden deep inside, came to be an important aspect of chaos.
An employee of IBM, Benoit Mandelbrot was a mathematician studying this self-similarity. One of the areas he was studying was cotton price fluctuations. No matter how the data on cotton prices was analyzed, the results did not fit the normal distribution. Mandelbrot eventually obtained all of the available data on cotton prices, dating back to 1900. When he analyzed the data with IBM's computers, he noticed an astonishing fact:
The numbers that produced aberrations from the point of view of normal distribution produced symmetry from the point of view of scaling. Each particular price change was random and unpredictable. But the sequence of changes was independent on scale: curves for daily price changes and monthly price changes matched perfectly. Incredibly, analyzed Mandelbrot's way, the degree of variation had remained constant over a tumultuous sixty-year period that saw two World Wars and a depression. (James Gleick, Chaos - Making a New Science, pg. 86)
Mandelbrot analyzed not only cotton prices, but many other phenomena as well. At one point, he was wondering about the length of a coastline. A map of a coastline will show many bays. However, measuring the length of a coastline off a map will miss minor bays that were too small to show on the map. Likewise, walking along the coastline misses microscopic bays in between grains of sand. No matter how much a coastline is magnified, there will be more bays visible if it is magnified more.
One mathematician, Helge von Koch, captured this idea in a mathematical construction called the Koch curve. To create a Koch curve, imagine an equilateral triangle. To the middle third of each side, add another equilateral triangle.
Keep on adding new triangles to the middle part of each side, and the result is a Koch curve. (See figure 4) A magnification of the Koch curve looks exactly the same as the original. It is another self-similar figure.
The Koch curve brings up an interesting paradox. Each time new triangles are added to the figure, the length of the line gets longer. However, the inner area of the Koch curve remains less than the area of a circle drawn around the original triangle. Essentially, it is a line of infinite length surrounding a finite area.
To get around this difficulty, mathematicians invented fractal dimensions. Fractal comes from the word fractional. The fractal dimension of the Koch curve is somewhere around 1.26. A fractional dimension is impossible to conceive, but it does make sense. The Koch curve is rougher than a smooth curve or line, which has one dimension. Since it is rougher and more crinkly, it is better at taking up space. However, it's not as good at filling up space as a square with two dimensions is, since it doesn't really have any area. So it makes sense that the dimension of the Koch curve is somewhere in between the two.
Fractal has come to mean any image that displays the attribute of self-similarity. The bifurcation diagram of the population equation is fractal. The Lorenz Attractor is fractal. The Koch curve is fractal.
During this time, scientists found it very difficult to get work published about chaos. Since they had not yet shown the relevance to real-world situations, most scientists did not think the results of experiments in chaos were important. As a result, even though chaos is a mathematical phenomenon, most of the research into chaos was done by people in other areas, such as meteorology and ecology. The field of chaos sprouted up as a hobby for scientists working on problems that maybe had something to do with it.
Later, a scientist by the name of Feigenbaum was looking at the bifurcation diagram again. He was looking at how fast the bifurcations come. He discovered that they come at a constant rate. He calculated it as 4.669. In other words, he discovered the exact scale at which it was self-similar. Make the diagram 4.669 times smaller, and it looks like the next region of bifurcations. He decided to look at other equations to see if it was possible to determine a scaling factor for them as well. Much to his surprise, the scaling factor was exactly the same. Not only was this complicated equation displaying regularity, the regularity was exactly the same as a much simpler equation. He tried many other functions, and they all produced the same scaling factor, 4.669.
This was a revolutionary discovery. He had found that a whole class of mathematical functions behaved in the same, predictable way. This universality would help other scientists easily analyze chaotic equations. Universality gave scientists the first tools to analyze a chaotic system. Now they could use a simple equation to predict the outcome of a more complex equation.
Many scientists were exploring equations that created fractal equations. The most famous fractal image is also one of the most simple. It is known as the Mandelbrot set (pictures of the mandelbrot set). The equation is simple: z=z2+c. To see if a point is part of the Mandelbrot set, just take a complex number z. Square it, then add the original number. Square the result, then add the original number. Repeat that ad infinitum, and if the number keeps on going up to infinity, it is not part of the Mandelbrot set. If it stays down below a certain level, it is part of the Mandelbrot set. The Mandelbrot set is the innermost section of the picture, and each different shade of gray represents how far out that particular point is. One interesting feature of the Mandelbrot set is that the circular humps match up to the bifurcation graph. The Mandelbrot fractal has the same self-similarity seen in the other equations. In fact, zooming in deep enough on a Mandelbrot fractal will eventually reveal an exact replica of the Mandelbrot set, perfect in every detail.
Fractal structures have been noticed in many real-world areas, as well as in mathematician's minds. Blood vessels branching out further and further, the branches of a tree, the internal structure of the lungs, graphs of stock market data, and many other real-world systems all have something in common: they are all self-similar.
Scientists at UC Santa Cruz found chaos in a dripping water faucet. By recording a dripping faucet and recording the periods of time, they discovered that at a certain flow velocity, the dripping no longer occurred at even times. When they graphed the data, they found that the dripping did indeed follow a pattern.
The human heart also has a chaotic pattern. The time between beats does not remain constant; it depends on how much activity a person is doing, among other things. Under certain conditions, the heartbeat can speed up. Under different conditions, the heart beats erratically. It might even be called a chaotic heartbeat. The analysis of a heartbeat can help medical researchers find ways to put an abnormal heartbeat back into a steady state, instead of uncontrolled chaos.
Researchers discovered a simple set of three equations that graphed a fern. This started a new idea - perhaps DNA encodes not exactly where the leaves grow, but a formula that controls their distribution. DNA, even though it holds an amazing amount of data, could not hold all of the data necessary to determine where every cell of the human body goes. However, by using fractal formulas to control how the blood vessels branch out and the nerve fibers get created, DNA has more than enough information. It has even been speculated that the brain itself might be organized somehow according to the laws of chaos.
Chaos even has applications outside of science. Computer art has become more realistic through the use of chaos and fractals. Now, with a simple formula, a computer can create a beautiful, and realistic tree. Instead of following a regular pattern, the bark of a tree can be created according to a formula that almost, but not quite, repeats itself.
Music can be created using fractals as well. Using the Lorenz attractor, Diana S. Dabby, a graduate student in electrical engineering at the Massachusetts Institute of Technology, has created variations of musical themes. ("Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994) By associating the musical notes of a piece of music like Bach's Prelude in C with the x coordinates of the Lorenz attractor, and running a computer program, she has created variations of the theme of the song. Most musicians who hear the new sounds believe that the variations are very musical and creative.
Chaos has already had a lasting effect on science, yet there is much still left to be discovered. Many scientists believe that twentieth century science will be known for only three theories: relativity, quantum mechanics, and chaos. Aspects of chaos show up everywhere around the world, from the currents of the ocean and the flow of blood through fractal blood vessels to the branches of trees and the effects of turbulence. Chaos has inescapably become part of modern science. As chaos changed from a little-known theory to a full science of its own, it has received widespread publicity. Chaos theory has changed the direction of science: in the eyes of the general public, physics is no longer simply the study of subatomic particles in a billion-dollar particle accelerator, but the study of chaotic systems and how they work.
Siddhartha Gautama, known as the Buddha, was born in the sixth century B.C. in what is now modern Nepal. His father, Suddhodana, was the ruler of the Sakya people and Siddhartha grew up living the extravagant life of a young prince. According to custom, he married at the young age of sixteen to a girl named Yasodhara. His father had ordered that he live a life of total seclusion, but one day Siddhartha ventured out into the world and was confronted with the reality of the inevitable suffering of life. The next day, at the age of twenty-nine, he left his kingdom and newborn son to lead an ascetic life and determine a way to relieve universal suffering.
For six years, Siddhartha submitted himself to rigorous ascetic practices, studying and following different methods of meditation with various religious teachers. But he was never fully satisfied. One day, however, he was offered a bowl of rice from a young girl and he accepted it. In that moment, he realised that physical austerities were not the means to achieve liberation. From then on, he encouraged people to follow a path of balance rather than extremism. He called this The Middle Way.
That night Siddhartha sat under the Bodhi tree, and meditated until dawn. He purified his mind of all defilements and attained enlightenment at the age of thirty-five, thus earning the title Buddha, or "Enlightened One". For the remainder of his eighty years, the Buddha preached the Dharma in an effort to help other sentient beings reach enlightenment.
Quote:
"To enjoy good health, to bring true happiness to one's family, to bring peace to all, one must first discipline and control one's own mind. If a man can control his mind he can find the way to Enlightenment, and all wisdom and virtue will naturally come to him."
--Siddhartha Gautama
Sigmund Freud was born May 6, 1856, in a small town -- Freiberg -- in Moravia. His father was a wool merchant with a keen mind and a good sense of humor. His mother was a lively woman, her husband's second wife and 20 years younger. She was 21 years old when she gave birth to her first son, her darling, Sigmund. Sigmund had two older half-brothers and six younger siblings. When he was four or five -- he wasn't sure -- the family moved to Vienna, where he lived most of his life.
A brilliant child, always at the head of his class, he went to medical school, one of the few viable options for a bright Jewish boy in Vienna those days. There, he became involved in research under the direction of a physiology professor named Ernst Brcke. Brcke believed in what was then a popular, if radical, notion, which we now call reductionism: "No other forces than the common physical-chemical ones are active within the organism." Freud would spend many years trying to "reduce" personality to neurology, a cause he later gave up on.
Freud was very good at his research, concentrating on neurophysiology, even inventing a special cell-staining technique. But only a limited number of positions were available, and there were others ahead of him. Brcke helped him to get a grant to study, first with the great psychiatrist Charcot in Paris, then with his rival Bernheim in Nancy. Both these gentlemen were investigating the use of hypnosis with hysterics.
After spending a short time as a resident in neurology and director of a children's ward in Berlin, he came back to Vienna, married his fiance of many years Martha Bernays, and set up a practice in neuropsychiatry, with the help of Joseph Breuer.
Freud's books and lectures brought him both fame and ostracism from the mainstream of the medical community. He drew around him a number of very bright sympathizers who became the core of the psychoanalytic movement. Unfortunately, Freud had a penchant for rejecting people who did not totally agree with him. Some separated from him on friendly terms; others did not, and went on to found competing schools of thought.
Freud emigrated to England just before World War II when Vienna became an increasing dangerous place for Jews, especially ones as famous as Freud. Not long afterward, he died of the cancer of the mouth and jaw that he had suffered from for the last 20 years of his life.
Quote:
"When you think of this dividing up of the personality into ego, super-ego and id, you must not imagine sharp dividing lines such as are artificially drawn in the field of political geography. We cannot do justice to the characteristics of the mind by means of linear contours, such as occur in a drawing or in a primitive painting, but we need rather the areas of colour shading off into one another that are to be found in modern pictures. After we have made our separations, we must allow what we have separated to merge again. Do not judge too harshly of a first attempt at picturing a thing so elusive as the human mind."
--Sigmund Freud
Albert Einstein was born at Ulm, in Wrttemberg, Germany, on March 14, 1879. Six weeks later the family moved to Munich and he began his schooling there at the Luitpold Gymnasium. Later, they moved to Italy and Albert continued his education at Aarau, Switzerland and in 1896 he entered the Swiss Federal Polytechnic School in Zurich to be trained as a teacher in physics and mathematics. In 1901, the year he gained his diploma, he acquired Swiss citizenship and, as he was unable to find a teaching post, he accepted a position as technical assistant in the Swiss Patent Office. In 1905 he obtained his doctor's degree.
During his stay at the Patent Office, and in his spare time, he produced much of his remarkable work and in 1908 he was appointed Privatdozent in Berne. In 1909 he became Professor Extraordinary at Zurich, in 1911 Professor of Theoretical Physics at Prague, returning to Zurich in the following year to fill a similar post. In 1914 he was appointed Director of the Kaiser Wilhelm Physical Institute and Professor in the University of Berlin. He became a German citizen in 1914 and remained in Berlin until 1933 when he renounced his citizenship for political reasons and emigrated to America to take the position of Professor of Theoretical Physics at Princeton. He became a United States citizen in 1940 and retired from his post in 1945.
Einstein always appeared to have a clear view of the problems of physics and the determination to solve them. He had a strategy of his own and was able to visualize the main stages on the way to his goal. He regarded his major achievements as mere stepping-stones for the next advance.
At the start of his scientific work, Einstein realized the inadequacies of Newtonian mechanics and his special theory of relativity stemmed from an attempt to reconcile the laws of mechanics with the laws of the electromagnetic field. He dealt with classical problems of statistical mechanics and problems in which they were merged with quantum theory: this led to an explanation of the Brownian movement of molecules. He investigated the thermal properties of light with a low radiation density and his observations laid the foundation of the photon theory of light.
Albert Einstein received honorary doctorate degrees in science, medicine and philosophy from many European and American universities. During the 1920's he lectured in Europe, America and the Far East and he was awarded Fellowships or Memberships of all the leading scientific academies throughout the world. He gained numerous awards in recognition of his work, including the Copley Medal of the Royal Society of London in 1925, and the Franklin Medal of the Franklin Institute in 1935.
Einstein's gifts inevitably resulted in his dwelling much in intellectual solitude and, for relaxation, music played an important part in his life. He married Mileva Maric in 1903 and they had a daughter and two sons; their marriage was dissolved in 1919 and in the same year he married his cousin, Elsa Lwenthal, who died in 1936. He died on April 18, 1955 at Princeton, New Jersey.
Quote:
"It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing -- a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned above. This was demonstrated by Cockcroft and Walton in 1932, experimentally."
--Albert Einstein
Charles Darwin (1809-1882) was a British naturalist, who revolutionized the science of biology by his demonstration of evolution by natural selection. Darwin's On The Origin Of Species By Means Of Natural Selection, Or The Preservation Of Favoured Races In The Struggle Of Life, was published on November 24, 1859, and sold out immediately.
Darwin was born in Shrewsbury on 12 February 1809. In 1827 he started theology studies at Christ's College, Cambridge. His love to collect plants, insects, and geological specimens was noted by his botany professor John Stevens Henslow. He arranged for his talented student a place a on the surveying expedition of HMS Beagle to Patagonia. Despite the objections of his father, Darwin decided to leave his familiar surroundings.
The voyage took five years from 1831 to 1836. Darwin returned with observations he had made in Teneriffe, the Cape Verde Islands, Brazil, the Galapagos Islands, and elsewhere. During the voyage he had contracted a tropical illness, which made him a semi-invalid for the rest of his life. By 1846 Darwin had published several works based on the discoveries of the voyage and he became secretary of the Geological Society (1838-41).
From 1842 Darwin lived at Down House, Downe. In 1839 he had married his cousin Emma Wedgwood, and when not devoting himself to scientific studies, he led the life of a country gentleman. In the 1840s Darwin worked on his observations of the origin of species for his own use. He began to conclude, although he was deeply anxious about the direction his mind was taking, that species might share a common ancestor.
Darwin's great work, The Origin of Species by Means of Natural Selection was heavily attacked because it did not support the depiction of creation given in the book of Genesis. Darwin's argument that natural selection - the mechanism of evolution - worked automatically, leaving little or no room for divine guidance or design. All species, he reasoned, produce far too many offspring for them all to survive, and therefore those with favorable variations - owing to chance - are selected.
At Darwin's hands evolution matured into a well-developed scientific theory, which have been a constant target of religious or pseudo-scientific attacks. However, Darwin himself did not at first explicitly apply the evolutionary theory to human beings. He knew that his challenge to the Biblical doctrine would cause stress to his friends and family, among them his religious wife.
However, T.H. Huxley published in his Man's Place in Nature (1863) an application of the theory and Darwin followed him in The Descent Of Man, And Selection In Relation To Sex (1871) and Expression Of Emotions In Man And Animals (1872), which showed the similarities between animals and man in the expression of emotions and was the start of the science of ethnology. Darwin's voyage with the Royal Navy's H.M.S. Beagle is recorded in the Journal Of Researches (1836), a blend of scientific reporting and travel writing.
Darwin died in Down, Kent, on April 19, 1882 and is buried in Westminster Abbey.
Quote:
"Man with all his noble qualities, with sympathy which feels for the most debased, with benevolence which extends not only to other men but to the humblest living creature, with his god-like intellect which has penetrated into the movements and constitution of the solar system- with all these exalted powers- Man still bears in his bodily frame the indelible stamp of his lowly origin."
--Charles Darwin
Edward Lorenz is Professor Emeritus of Meteorology at the Massachusetts Institute of Technology in Cambridge, Massachusetts. He received his B.A. from Dartmouth College in 1938 and his M.A. from Harvard in 1940, both in Mathematics. He received his M.S. in 1943 and Sc.D. in 1948, both from M.I.T. in Meteorology, becoming professor at M.I.T. in 1955. He has received honorary degrees from seven universities. He is a fellow of the American Academy of Arts and Sciences, the American Meteorological Society (AMS), the National Academy of Sciences, Indian Academy of Sciences, Norwegian Academy of Science and Letters, and the Royal Meteorological Society (RMS). He has won the Clarence Leroy Meisinger Award from the AMS, the Carl-Gustaf Rossby Research Medal from the AMS, the Symons Memorial Gold Medal from the RMS, the Holger and Anna-Greta Crafoord Prize from the Royal Swedish Academy of Science, the Elliott Creson Medal from the Franklin Institute, the Kyoto Prize from the Inamori Foundation, the Roger Revelle Medal from the American Geophysical Union, the Louis J. Battan Author's Award from the AMS, I.M.O. Prize from the World Meteorological Organization, and the Buys Ballot Medal from the Royal Netherlands Academy of Arts and Sciences.
His research in the 1950s initially focused on the general circulation of the atmosphere, but transitioned to simplified governing equations for the atmosphere. It was working on a particular set of simplified equations that he discovered imperceptable changes in the initial conditions could produce drastically divergent solutions, a concept we now call chaos. This sensitive dependence to the initial conditions was described in his foundational 1963 paper, "Deterministic nonperiodic flow." According to the ISI Web of Science, Lorenz (1963) has been cited in the scientific literature over 3700 times. This equation set and the attractors described by this equation set have been called the Lorenz equations and Lorenz attractors, respectively. His later work has defined the predictability limits of the atmosphere through observations and state-of-the-art numerical weather prediction models. His 1972 talk at the American Association for the Advancement of Science entitled "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?" coined the term "butterfly effect," which is now the popular description for chaos. In 1993, he published a book based on his Jessie and John Danz lectures at the University of Washington entitled "The Essence of Chaos." He has published over 70 scientific papers, including two in 2005 entitled "Designing Chaotic Models" (Journal of Atmospheric Sciences) and "A Look at Some Details of the Growth of Initial Uncertainties" (Tellus) at the age of 88 years.
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"It implies that two states differing by imperceptible amounts may eventually evolve into two considerably different states. If, then, there is any error whatever in observing the present state - and in any real system such errors seem inevitable - an acceptable prediction of the instantaneous state in the distant future may well be impossible."
--Edward Lorenz
Read more about the Chaos Theory in my general interests section!
The Philosophy Of Time Travel
Forward
I would like to thank the sisters of the Saint John Chapter in Alexandria, Virginia for their support in my decision.
By the grace of God, they are:
Sister Eleanor Lewis
Sister Francesca Godard
Sister Helen Davis
Sister Catherine Arnold
Sister Mary Lee Pond
Sister Virginia Wessex
This intent of this book is for it to be used as a simple and direct guide in a time of great danger.
I pray that this is merely a work of fiction.
If it is not, then I pray for you, the reader of this book.
If I am still alive when the events foretold in these pages occur, then I hope that you will find me before it is too late.
Roberta Ann Sparrow
October, 1944
Chapter One: The Tangent Universe
The primary universe is fraught with great peril. War, plague, famine and natural disaster are common. Death comes to us all.
The Fourth Dimension of Time is a stable construct, though it is not impenetrable.
Incidents when the fabric of the fourth dime(n)sion becomes corrupted are incredibly rare.
If a Tangent Universe occurs, it will be highly unstable, sustaining itself for no longer than several weeks.
Eventually it will collapse upon itself, forming a black hole within the Primary Universe capable of destroying all existence.
Chapter Two: Water and Metal
Water and Metal are the key elements of Time Travel.
Water is the barrier element for the construction of Time Portals used as gateways between Universes at the Tangent Vortex.
Metal is the transitional element for the construction of Artifact Vessels.
Chapter Four: The Artifact And The Living
When a Tangent Universe occurs, those living nearest to the Vortex will find themselves at the epicenter of a dangerous new world.
Artifacts provide first sign that a Tangent Universe has occured.
If an Artifact occurs, the Living will retrieve it with great interest and curiosity. Artifacts are formed from metal, such as an Arrowhead from an ancient Mayan civilisation, or a Metal Sword from Medieval Europe.
Artifacts returned to the Primary Universe are often linked to religious Iconography, as their appearance on Earth seems to defy logical explanation.
Divine intervention is deemed the only logical conclusion for the appearance for the Artifact.
Chapter Six: The Living Receiver
The Living Receiver is chosen to guide the Artifact into position for its journey back to the Primary Universe.
No one knows how or why a Receiver will be chosen.
The Living Receiver is often blessed with a Fourth Dimensional Powers. These include increased strength, telekinesis, mind control, and the ability to conjure fire and water.
The Living Receiver is often tormented by terrifying dreams, visions and auditory hallucinations during his time within the Tangent Universe.
Those surrounding the Living Receiver, known as the Manipulated, will fear him and try to destroy him.
Chapter Seven: The Manipulated Living
The Manipulated Living are often the close friends and neighbours of the Living Receiver.
They are prone to irrational, bizarre, and often violent behaviour. This is the unfortunate result of their task, which is to assist the Living Receiver in returning the Artifact to the Primary Universe.
The Manipulated Living will do anything to save themselves from Oblivion.
The Manipulated Dead
The Manipulated Dead are more powerful than the Living Receiver. If a person dies within the Tangent Dimension, they are able to contact the Living Receiver through the Fourth Dimensional Construct.
The Fourth Dimensional Construct is made of Water.
The Manipulated Dead will manipulate the Living Receiver using the Fourth Dimensional Construct (see Appendix A and B).
The Manipulated Dead will often set an Ensurance Trap for the Living Receiver to ensure that the Artifact is returned safely to the Primary Universe.
If the Ensurance Trap is succesful, the Living Receiver is left with no choice but to use his Fourth Dimensional Power to send the Artifact back in time into the Primary Universe before the Black Hole collapses upon itself.
Chapter Twelve: Dreams
When the Manipulated awaken from their Journey into the Tangent Universe, they are often haunted by the experience in their dreams.
Many of them will not remember.
Those who do remember the Journey are often overcome with profound remorse for the regretful actions buried within their Dreams, the only physical evidence buried within the Artifact itself; all that remains from the lost world.
Ancient myth tells us of the Mayan Warrior killed by an Arrowhead that had fallen from a cliff, where there was no Army, no enemy to be found.
We are told of the Medievel Knight mysteriously impaled by the sword he had not yet built.We are told that these things occur for a reason.
Time has been studied by philosophers and scientists for 2,500 years, and thanks to this attention it is much better understood today. Nevertheless, many issues remain to be resolved: what time actually is; whether time exists when nothing is changing; what sort of time travel is possible, why the time dimension has an arrow but a space dimension does not; whether the future is real; how to analyze the metaphor of time's "flow"; whether there was a time before the Big Bang; whether tensed or tenseless concepts are semantically basic; what is the proper formalism or logic that captures the special role that time plays in reasoning; and what are the neural mechanisms that account for our experience of time.
Of these issues, the general public is interested mostly in time travel. Einstein's special theory of relativity implies that you can travel into someone else's future by high speed travel in space. Travel into the past is more controversial, but doesn't appear to be inconsistent with relativity theory. Several physicists have produced novel suggestions on how to create time machines that exploit regions of space-time that curve back onto themselves, either naturally or by human intervention.
Philosophers of time are deeply divided on the question on what sort of ontological differences there are among the present, past and future. Presentists argue that necessarily only present objects and present experiences are real; and we conscious beings recognize this in the special "vividness" of our present experience. The growing-universe theory is that the past and present are both real, but the future is not yet real. The most popular view is that there are no significant ontological differences among present, past and future. This view is called "eternalism" or "the block universe theory."
This raises the issue of tenseless versus tensed theories of time. The block universe theory implies a tenseless theory. In the earliest version of this theory, it is declared that tensed terminology (such as "will win" within the sentence "The Lakers will win the basketball game") is not semantically basic, but instead is analyzable into tenseless terms (such as "does win at time t" and "happens before" and "is simultaneous with"). Once all tenseless facts are fixed, all tensed facts are thereby fixed. In later versions of the tenseless theory, the claim is not made that tensed terminology is removable or reducible, but only that the truth conditions of tensed remarks can be handled with tenseless facts. On the other hand, advocates of a tensed theory of time say that tenseless terminology is not semantically basic but can be analyzed in tensed terms, and tensed facts are needed. For example, a tensed theory might imply that the world involves irreducible tensed properties such as presentness or now-ness or being-in-the-present, and no adequate account of the present tensed fact that it's now midnight can be given without these tensed properties. So, the philosophical debate is over whether tensed concepts have semantical priority over untensed concepts, and whether tensed facts have ontological priority over untensed facts.