Ever since the dawn of modern science in the ancient civilisations of Greece and the Middle East, people have tried to explain the universe in simple terms.
[Pythagoras pic]
The discovery of geometry in Ancient Greece led philosophers such as Pythagoras to suggest that there was some natural mathematical order that described the universe.
[perfect solids pic]
Pythagoras discovered the five perfect solids, each of them described simply in geometrical terms. He found natural counterparts to his perfect solids in the mineral world.
[perfect solids / minerals pic]
However, while these crystals replicated the simple beauty of geometry these perfect crystals are rare. Pythagoras must have at some time picked up a pebble and wondered how its irregular, rough surface could fit into his geometric universe.
[Dark ages pic]
The fall of the classical civilisations and the rise of religious bigotry during the dark ages prevented the advancement of science for over fifteen hundred years.
[Kepler pic]
Johanes Kepler was born in 1571. From his youth he was fascinated with the stars. He devoted his life to the search for a natural geometrical order that described the positions of the six known planets in the solar system.
[Kepler circle-triangle]
Whilst teaching mathematics he discovered that by inscribing an equilateral triangle inside a circle representing the orbit of Saturn, another circle drawn inside the triangle represented the orbit of Jupiter.
[Kepler thingy]
Kepler worked for the rest of his life trying to fit the orbits of the planets in between the five Pythagorean perfect solids. He drew diagrams 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 light inscribed on massive crystal spheres. Two thousand years previously Pythagoras had talked of the 'harmony of the spheres', thinking that the orbits of the planets were in some way related to the geometric progression of the notes on a musical scale.
Kepler himself said "Geometry provided God with a model for the Creation, Geometry is God Himself".
[Picture needed]
As time passed Kepler became increasing frustrated with the fact that however hard he tried he could not get the planets to match his model exactly. He thought that his limited astronomical data, collected from a time before the telescope, must be incorrect.
[Tycho Brahe]
Eventually he obtained the observations of the Dutch Astronomer Tycho Brahe. From Kepler's point of view this made things worse. Not only did the new data not fit his geometrical model of the universe, it also proved that the orbits of the planets were not, as previously assumed circular, but were elliptical.
[elliptical orbit diagram]
Kepler was devastated. The universe could not be explained purely in terms of geometry. The fact that he had discovered one of the most important discoveries in astronomy in two thousand years was little consolation. Geometry had failed him.
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Chapter 2 - The early fractal pioneers
[newton]
Isaac Newton, in his investigations into Gravity carried on with Kepler's work on the motion of the planets. Whilst studying this he came across a problem.
[two planets]
With two planets of the same mass, working the gravitational attraction between them was easy.
[two planets with gravitational attraction shown]
At any point the gravitational field would attract an object to one of the two planets, the area of gravitational attraction for each planet is 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 immediately around each planet a point would be attracted to that planet, but in the space between the three planets there appeared to be a chaotic region where the final outcome could not be predicted.
[newtbasin picture]
Now, with modern computers we can calculate the appearance of these chaotic zones, and we find infinitely complex patterns in the gravitational fields.
[fade]
[picture]
Helge von Koch was a Swedish mathematician. In 1904 he described a mathematical structure that he called his 'coastline'.
[kock coastline]
Koch realised when he discovered this structure that it had some unusual properties. Firstly, the line is infinitely long. Despite this the line fits into a finite space. A triangle can be drawn around the shape, a finite geometrical shape with an infinite complex line inside it.
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 is 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 centuries, described several similar shapes, fantastic mathematical curios with which to amuse their fellow mathematicians, but the true importance of their discoveries were not to be known in their life times.
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Chapter 3 - Julia and the Complex Plane
[picture needed]
In 1914 two French mathematicians, Gaston Julia and Pierre Fatou were investigating the properties of iterative formulae. An iterative formula is simply a formula where the result is put back through the formula several times until a final result is obtained or until infinity is reached.
[z^2+c]
The particular formula that Julia was interested in was very simple. Take a number, square it, and add a constant value. Now take this value, 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 it can be shown that all initial values for z above one rapidly rise towards infinity, values below one sink towards zero, but with z set to 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 minus infinity to zero to infinity. Complex numbers can best be described as points on a plane, like co-ordinates on graph paper, they have an x value, called the 'real' component, and a y value, called the 'imaginary' component.
Complex numbers can be added, subtracted, multiplied and divided just as any traditional number. Complex numbers allow results for previously unsolvable equations, like the square root of minus one.
[0,0]
Julia tried out the formula with a constant of (0,0). using various values for the initial Z value, he was not surprised to find that initial values that did not drift off to infinity could all be plotted on 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 to find that the shapes he was getting were irregular. Everything had to be meticulously calculated by hand, and although he saw fragments of whorls and spirals it is only with the advent of modern computers that the true beauty of these images can be displayed.
But 1914 was not the best time to be studying mathematics in Paris. Julia found little interest in his discoveries whilst Europe was in the midst of the most destructive war in its history.
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Chapter 4 - Mandelbrot and the birth of Fractal science
[Benoit piccy]
Benoit Mandelbrot was born in Poland in 1924. His family was Jewish, his 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 Paris where the young Benoit came under the influence of his mathematician uncle. When Hitler advanced into France the family broke up, and Benoit spent most of the war years living in Tulle, a town in central France. He had little formal schooling but was guided by teachers often themselves anxious to avoid the attention of the Nazis. He was never in one place for long.
[Ecole Polytechnique pic if possible]
After the war, Benoit returned to Paris and passed the entrance examination for the prestigious Ecole Polytechnique. One of his teachers 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 research department. Thereafter, he followed various careers mostly as a lecturer teaching engineering, economics, physiology an d mathematics. In none of these was he regarded very highly by his colleagues, they did not like his informal and unorthodox methods.
[Koch coastline]
In the early 1970's Mandelbrot started thinking about Koch's work, in particular his 'coastline'. He thought about the coastline of Great Britain.
[Britain from satellite picture]
Mandelbrot wondered, "How long is the coast of Great Britain?". He soon realised that the length of the coast depends on the length of your ruler.
[it depends]
The island viewed from space can be measured approximately. A survey team measuring at 100 meter intervals around the coast will give a different, higer value. Crawling around the coast with a 6" rule will give a higer value still. A snail traversing every small pebble would, if it could live long enough and give an answer, arrive at a very much larger result.
[infinitely long]
Rather than settling down to an approximate value for the length of the coastline, Mandelbrot found that the smaller the lines used to measure the coastline, the larger the value he got. The coastline of Britain was infinitely long.
[Sierpinski triangle]
Mandelbrot looked again at the Sierpinski triangle. Here was a finite triangle, one of the most stable forms in mathematics. A triangle is removed 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 to think in terms of dimensions.
[dimensions]
In mathematical terms a dot has no dimensions, a straight line one, a square has two, and a cube has three. But what about the Sierpinski triangle?
[sierpinski]
How many dimensions does this structure have. Not 0, not 1, not completely two either because it contains zero area. It appeared to be somewhere inbetween one and two. Mandelbrot could immediately see that he was on the right lines. The idea was a brilliant solution, and he went on to define ways of working out the dimensions of all such figures. They all had fractional dimensions.
[Fractus]
One day in 1975 while helping his sone with his latin homework, he was struct by the latin root Fractus, leading to such english words as fracture and fraction. He invented the word 'Fractal' to cover his new geometry and another branch of mathematics was born.
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Chapter 5 - The Mandelbrot Set
[b&w julia sets]
In the late seventies Mandelbrot remembered his old professor, Gaston Julia, and his formula. He began plotting Julia sets using a computer at Harvard and outputting to an old printer. They becaue fascinated with the results, which were far less detailed that those seen here. One day in 1979 he slightly altered the way they were generated and saw the first unclear outlines of the set that bears his name today. They rushed to the mainframe computer at IBM to confirm their discovery and saw the Mandelbrot set being generated for the first time.
[picture mandelbrot set]
Even the mainframe computer at that time was not able to produce it to the same level of detail you see here.
There are an infinite number of different Julia patterns, but only one Mandelbrot set. Mandelbrot discovered his set was a guide to all the Julia patterns.
[ Guide to Julia shapes]
Every point on this picture of the Mandelbrot set can be used as the constant value for the Julia formula. All the points within the black area of the Mandelbrot produce solid Julias, points outside of it produce fragmentated Julias.
