Chapter 1 - Geometry and the Perfect Universe [title pic] Ever since the dawn of modern science in the ancient civilisations of ŠGreece and the Middle East, people have tried to explain the universe Šin simple terms. [Pythagoras pic] The discovery of geometry in Ancient Greece led philosophers such as ŠPythagoras to suggest that there was some natural mathematical order Šthat described the universe. [perfect solids pic] Pythagoras discovered the five perfect solids, each of them described Šsimply in geometrical terms. He found natural counterparts to his Šperfect solids in the mineral world. [perfect solids / minerals pic] However, while these crystals replicated the simple beauty of geometry Šthese perfect crystals are rare. Pythagoras must have at some time Špicked up a pebble and wondered how its irregular, rough surface could Šfit into his geometric universe. [Dark ages pic] The fall of the classical civilisations and the rise of religious Šbigotry during the dark ages prevented the advancement of science for Šover fifteen hundred years. [Kepler pic] Johanes Kepler was born in 1571. From his youth he was fascinated with Šthe stars. He devoted his life to the search for a natural geometrical Šorder that described the positions of the six known planets in the Šsolar system. [Kepler circle-triangle] Whilst teaching mathematics he discovered that by inscribing an Šequilateral triangle inside a circle representing the orbit of Saturn, Šanother circle drawn inside the triangle represented the orbit of ŠJupiter. [Kepler thingy] Kepler worked for the rest of his life trying to fit the orbits of the Šplanets in between the five Pythagorean perfect solids. He drew Šdiagrams and built models of his vision of the universe as he saw it. [Kepler thingy 2] Classical thought in Kepler's time talked of the planets as points of Šlight inscribed on massive crystal spheres. Two thousand years Špreviously Pythagoras had talked of the 'harmony of the spheres', Šthinking that the orbits of the planets were in some way related to Šthe geometric progression of the notes on a musical scale. Kepler himself said "Geometry provided God with a model for the ŠCreation, Geometry is God Himself". [Picture needed] As time passed Kepler became increasing frustrated with the fact that Šhowever hard he tried he could not get the planets to match his model Šexactly. He thought that his limited astronomical data, collected from Ša time before the telescope, must be incorrect. [Tycho Brahe] Eventually he obtained the observations of the Dutch Astronomer Tycho ŠBrahe. From Kepler's point of view this made things worse. Not only Šdid the new data not fit his geometrical model of the universe, it Šalso proved that the orbits of the planets were not, as previously Šassumed circular, but were elliptical. [elliptical orbit diagram] Kepler was devastated. The universe could not be explained purely in Šterms of geometry. The fact that he had discovered one of the most Šimportant discoveries in astronomy in two thousand years was little Šconsolation. Geometry had failed him. [fade out] Chapter 2 - The early fractal pioneers [newton] Isaac Newton, in his investigations into Gravity carried on with ŠKepler's work on the motion of the planets. Whilst studying this he Šcame across a problem. [two planets] With two planets of the same mass, working the gravitational Šattraction between them was easy. [two planets with gravitational attraction shown] At any point the gravitational field would attract an object to one of Šthe two planets, the area of gravitational attraction for each planet Šis easily defined. [three planets] With three or more planets it suddenly becomes more complex. [three planets with simple attraction] Using approximations Newton was able to prove that the in the areas Šimmediately around each planet a point would be attracted to that Šplanet, but in the space between the three planets there appeared to Šbe a chaotic region where the final outcome could not be predicted. [newtbasin picture] Now, with modern computers we can calculate the appearance of these Šchaotic zones, and we find infinitely complex patterns in the Šgravitational fields. [fade] [picture] Helge von Koch was a Swedish mathematician. In 1904 he described a Šmathematical structure that he called his 'coastline'. [kock coastline] Koch realised when he discovered this structure that it had some Šunusual properties. Firstly, the line is infinitely long. Despite this Šthe line fits into a finite space. A triangle can be drawn around the Šshape, a finite geometrical shape with an infinite complex line inside Šit. Secondly, at no time does one portion of the coastline meet another. [magnify anim] Thirdly, by magnifying a small portion of the shape we can see that it Šis identical to the whole shape. It is self similar. [other shapes - Sierpinski, et al] Helge von Koch, and other mathematicians of the 19th and early 20th Šcenturies, described several similar shapes, fantastic mathematical Šcurios with which to amuse their fellow mathematicians, but the true Šimportance of their discoveries were not to be known in their life Štimes. [fade out] Chapter 3 - Julia and the Complex Plane [picture needed] In 1914 two French mathematicians, Gaston Julia and Pierre Fatou were Šinvestigating the properties of iterative formulae. An iterative Šformula is simply a formula where the result is put back through the Šformula several times until a final result is obtained or until Šinfinity is reached. [z^2+c] The particular formula that Julia was interested in was very simple. ŠTake a number, square it, and add a constant value. Now take this Švalue, square it and add the constant to it again, and so on. [calculations] The results are fairly straight forward. With the constant set to zero Šit can be shown that all initial values for z above one rapidly rise Štowards infinity, values below one sink towards zero, but with z set Što an initial value of one the result remains one. [complex plane] Next Julia tried the formula with complex numbers. Normal numbers can be described as points on a line stretching from Šminus infinity to zero to infinity. Complex numbers can best be Šdescribed as points on a plane, like co-ordinates on graph paper, they Šhave an x value, called the 'real' component, and a y value, called Šthe 'imaginary' component. Complex numbers can be added, subtracted, multiplied and divided just Šas any traditional number. Complex numbers allow results for Špreviously unsolvable equations, like the square root of minus one. [0,0] Julia tried out the formula with a constant of (0,0). using various Švalues for the initial Z value, he was not surprised to find that Šinitial values that did not drift off to infinity could all be plotted Šon the plane inside a circle centred at the origin with a radius of 1. [0,-.5] Next Julia tried some other values for the constant, and was surprised Što find that the shapes he was getting were irregular. Everything had Što be meticulously calculated by hand, and although he saw fragments Šof whorls and spirals it is only with the advent of modern computers Šthat the true beauty of these images can be displayed. But 1914 was not the best time to be studying mathematics in Paris. ŠJulia found little interest in his discoveries whilst Europe was in Šthe midst of the most destructive war in its history. [fade] Chapter 4 - Mandelbrot and the birth of Fractal science [Benoit piccy] Benoit Mandelbrot was born in Poland in 1924. His family was Jewish, Šhis father worked in the Clothing trade and his mother was a dentist. [Refugees in WW2 pic] In 1936, sensing the coming of the Nazi threat the Mandebrots moved to ŠParis where the young Benoit came under the influence of his Šmathematician uncle. When Hitler advanced into France the family broke Šup, and Benoit spent most of the war years living in Tulle, a town in Šcentral France. He had little formal schooling but was guided by Šteachers often themselves anxious to avoid the attention of the Nazis. ŠHe was never in one place for long. [Ecole Polytechnique pic if possible] After the war, Benoit returned to Paris and passed the entrance Šexamination for the prestigious Ecole Polytechnique. One of his Šteachers was an elderly French mathematician, Gaston Julia. [picture of early IBM computer system] In 1954 Mandelbrot emigrated to the USA where he began work at IBM's Šresearch department. Thereafter, he followed various careers mostly as Ša lecturer teaching engineering, economics, physiology an d Šmathematics. In none of these was he regarded very highly by his Šcolleagues, they did not like his informal and unorthodox methods. [Koch coastline] In the early 1970's Mandelbrot started thinking about Koch's work, in Šparticular his 'coastline'. He thought about the coastline of Great ŠBritain. [Britain from satellite picture] Mandelbrot wondered, "How long is the coast of Great Britain?". He Šsoon realised that the length of the coast depends on the length of Šyour ruler. [it depends] The island viewed from space can be measured approximately. A survey Šteam measuring at 100 meter intervals around the coast will give a Šdifferent, higer value. Crawling around the coast with a 6" rule will Šgive a higer value still. A snail traversing every small pebble would, Šif it could live long enough and give an answer, arrive at a very much Šlarger result. [infinitely long] Rather than settling down to an approximate value for the length of Šthe coastline, Mandelbrot found that the smaller the lines used to Šmeasure the coastline, the larger the value he got. The coastline of ŠBritain was infinitely long. [Sierpinski triangle] Mandelbrot looked again at the Sierpinski triangle. Here was a finite Štriangle, one of the most stable forms in mathematics. A triangle is Šremoved from the centre to leave three triangles. From each of these the centre portion is removed, and so on, ad infinitum. [Sierpinski...] Mandelbrot tacked the subject from a different point of view. He began Što think in terms of dimensions. [dimensions] In mathematical terms a dot has no dimensions, a straight line one, a Šsquare has two, and a cube has three. But what about the Sierpinski Štriangle? [sierpinski] How many dimensions does this structure have. Not 0, not 1, not Šcompletely two either because it contains zero area. It appeared to be Šsomewhere inbetween one and two. Mandelbrot could immediately see that Šhe was on the right lines. The idea was a brilliant solution, and he Šwent on to define ways of working out the dimensions of all such Šfigures. They all had fractional dimensions. [Fractus] One day in 1975 while helping his sone with his latin homework, he was Šstruct by the latin root Fractus, leading to such english words as Šfracture and fraction. He invented the word 'Fractal' to cover his new Šgeometry and another branch of mathematics was born. [fade out] Chapter 5 - The Mandelbrot Set [b&w julia sets] In the late seventies Mandelbrot remembered his old professor, Gaston ŠJulia, and his formula. He began plotting Julia sets using a computer Šat Harvard and outputting to an old printer. They becaue fascinated Šwith the results, which were far less detailed that those seen here. ŠOne day in 1979 he slightly altered the way they were generated and Šsaw the first unclear outlines of the set that bears his name today. ŠThey rushed to the mainframe computer at IBM to confirm their Šdiscovery and saw the Mandelbrot set being generated for the first Štime. [picture mandelbrot set] Even the mainframe computer at that time was not able to produce it to Šthe same level of detail you see here. There are an infinite number of different Julia patterns, but only one ŠMandelbrot set. Mandelbrot discovered his set was a guide to all the ŠJulia patterns. [ Guide to Julia shapes] Every point on this picture of the Mandelbrot set can be used as the Šconstant value for the Julia formula. All the points within the black Šarea of the Mandelbrot produce solid Julias, points outside of it Šproduce fragmentated Julias. [junction Julia] Those