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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 19 – MONDAY 29TH NOVEMBER Henri Poincaré At the turn of the 20th century, Henri Poincaré (1854‐1912) was considered to be the finest living mathematician. One of the last to specialize in virtually every known branch of the subject, both pure and applied, he became a professor at the Sorbonne (Paris) in 1881, and wrote over 500 papers and 30 books. He received numerous awards for this work, and was elected president of the Académie des sciences in 1906. The Three‐Body Problem Newton’s laws of motion imply that to fully describe the motion of two bodies that interact solely with each other, for example a planet and its moon, we need to know the position (x, y, z) and the velocity (vx, vy, vz) based on their initial values, where x, y, and z represent coordinates in 3‐dimensional space. There are hence six equations that need to be solved. A natural extension of Newton’s work was to try and determine what happens when there are three bodies that act upon each other due to gravity such as the sun, the earth, and the moon. However in this case we need to know the position and velocity of each body, which requires six three‐dimensional vectors, and hence 18 equations. The three‐body problem was considered by both Euler and Lagrange, who made substantial progress, but by 1887 a solution had still not been found. During that year, King Oscar II of Sweden and Norway announced a competition to celebrate his 60th birthday. It comprised four questions, one of which was the general solution to the n‐body problem, submitted by Weierstrass, who had looked at the problem, and was one of the judges for the competition. The prestige, and associated cash prize, spurred Poincaré to try and find a solution. He showed that the general n‐body problem is difficult to make progress on since it leads to chaotic behavior with no repetition of path (i.e. the bodies fall out of orbit, which has serious implications for our own solar system). However Poincaré was able to solve a restricted 3‐body problem, and his solution (which turned into a 270 paper once some minor flaws were corrected) was considered worthy of the prize, with Weierstrass writing “I have no difficulty in declaring that the [work] in question deserves the prize. Tell the King that this work cannot, in truth, be considered as supplying a complete solution to the question we originally proposed, but that it is nevertheless of such importance that its publication will open a new era in the history of celestial mechanics.” Chaos Theory While some work was done trying to advance the ideas of Poincaré in the first half of the 20th century, the subject of chaos theory did not surface until the 1960’s, with the advent of high speed computers. Edward Lorenz (1917‐2008) became the person whose development of the theory allowed it to enter mainstream mathematics, even though his work was mainly concerned with predicting weather patterns. While running a simulation he noticed that by changing the initial input value by a small fraction, very different results were obtained. This is the characteristic of a chaotic dynamical system. Lorenz rose to prominence when he gave a talk in 1973 describing what he called the butterfly effect, which posed the question “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”, which indicates that the flapping wing can cause a small change in the initial condition of the system, which produces a chain of events leading to large‐scale alterations of the outcome. While the theory can become very complicated, by considering a simple function of the form f(x) = x2 + c, we can see how chaos can arise. Ex. Consider the function f(x) = x2 − 0.75. If we let x = 1.75, then f(x) = 2.31. Substituting the value x = 2.31 into the function we get 4.59. Continuing this produces the values of x, f(x), f2(x), f3(x), … We call this an iterative process, and the initial value of x the seed. By choosing a value of 1.75 for our seed we get an increasing large value for the function, and after only 6 iterations we get a value of approximately 30 billion, indicating that the limiting value is infinite. However if we change the seed to 1.5, then the successive values of the iterative process will all be 1.5 (we call x = 1.5 a fixed point), and as we can see from the table below, ‐0.5 is also a fixed point. We say that 1.5 is a repelling fixed point or source, since small changes in the seed produce a very different limiting value, whereas ‐0.5 is called an attracting fixed point, or sink, since all seed values near ‐0.5 will result in fn(x) converging to ‐0.5. ±1.75 2.31 4.59 20.38 414.93 172173.29 1010 1021 1042 1084 10167 Bifurcation Diagrams ±1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 ±1 0.25 ‐0.6875 ‐0.2773 ‐0.6731 ‐0.2969 ‐0.6618 ‐0.3120 ‐0.6526 ‐0.3240 ‐0.6445 ±0.75 ‐0.1875 ‐0.7148 ‐0.2390 ‐0.6929 ‐0.2699 ‐0.6771 ‐0.2948 ‐0.6650 ‐0.3077 ‐0.6553 ±0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ‐0.5 ±0.25 ‐0.6875 ‐0.2773 ‐0.6731 ‐0.2970 ‐0.6618 ‐0.3120 ‐0.6526 ‐0.3240 ‐0.6445 ‐0.3340 ‐0.5 One way to visualize what is taking place in a chaotic dynamical system is (to use a computer) to draw a bifurcation diagram, which shows the fixed points of the system as a function of the bifurcation parameter, which in the example above is c. If we zoom in to the area where we go from two to four fixed points (between ‐1.5 and ‐1.3) then we see that the region exhibits self‐similarity, i.e. that a small part of the graph has the same shape as the whole graph. This is a typical property of fractals. Fractals Although the term was not created until 100 years later, the Weierstrass example of a function that is continuous but nowhere differentiable (see lecture 17) was the first fractal. This was followed in 1904 and 1915 respectively by the now famous fractals of Helge von Koch (1870‐1924) and Wacław Sierpiński (1882‐1969) In the Sierpinski triangle, we start with an equilateral triangle and split it into three as shown below by halving the sides, removing the triangle in the center. We then repeat this process indefinitely. To calculate the dimension of the Sierpinski triangle we use the formula derived by Felix Hausdorff (1868‐1942), which allows for objects, in particular fractals, to have a fractional dimension. While the formal definition is very technical, for our purposes we will use the formula below. Dimension log number of self similar pieces
log magnification factor So in the case of the Sierpinski triangle, where 1 triangle becomes 3, and then 9 etc., with the length of the sides being (successively) 1, ½, ¼, etc, we get Dimension = 1.585 In the Koch snowflake, we start with an equilateral triangle, remove the inner third of each side, and create two sides of an equilateral triangle outside the original. This is then repeated on the new twelve sided figure, and continued indefinitely. Dimension log number of self similar pieces
log magnification factor = 1.262 Benoît Mandelbrot Benoît Mandelbrot (1924‐2010) was born in Poland, but moved to France as a child to escape the Nazis, before emigrating to the United States in 1957. He worked for IBM for the next 32 years, spending his last years before retirement in 2005 as a professor at Yale. Mandelbrot is most famous for his work in fractal geometry, using the term “fractal” for the first time in a 1975 book. However he came to prominence in 1967 for a paper titled How Long Is the Coast of Britain? Statistical Self‐Similarity and Fractional Dimension. It examined the coastline paradox, which argues that the length of a coastline depends on what you use to measure it. Unit = 100 km, length = 2800 km Unit = 50 km, length = 3400 km Mandelbrot argued that, due to self‐similarity, coastlines behave like fractals. If we continue to reduce the unit of measurement then the length will increase without bound. This is known as the Richardson effect, after Lewis Fry Richardson (1881‐1953), who measured the coastline of Britain to have dimension 1.25, that of South Africa to be 1.02, and Norway’s to be 1.52. Note: Since then, the dimension of many irregular objects has been estimated, including that of broccoli (2.66) and cauliflower (2.88). Julia Sets Earlier we looked at how chaos can arise from functions of the form f(x) = x2 + c. An extension of this is the work done by Gaston Julia (1893‐1978), who in 1918 published (and won the grand prize of the Acedémie des sciences) for a 199 page paper Mémoire sur l'iteration des fonctions rationelles, which included a precise description of the set of complex numbers z for which the nth iteration, fn(z), stays bounded as n tends to infinity. However it was Mandelbrot who harnessed the power of modern computers so that the geometric beauty of the Julia sets could be visualized, with some examples being given below. c=1‐ c=0.285+0.01i c=0.64i c=‐0.4+0.6i ...
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