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In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus could be applied. Sets or functions that are not sufficiently smooth or regular have tended to be named as " pathological" and not worthy of study. They were regarded as individual curiosities and only rarely were thought of as a class to which a general theory might be applicable. However, in recent years this attitude has changed. Irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provides a general framework for the study of such irregular sets. (Falconer) The word ‘fractal’ was coined in 1975 by mathematician Benoit B. Mandelbrot to describe an intricate-looking set of curves, many of which were not yet seen before the creation of the computer. Fractals often exhibit self-similarity, which means that various copies of an object can be found in the original object at smaller size scales. This continues for many magnifications – like an endless nesting of Russian dolls within dolls. (Pickover) Fractals appear everywhere in nature, in galaxies and landscapes, in earthquakes and geological cracks, in aggregates and colloids, and even in the human body. Fractal geometry is an important tool in the analysis of phenomena, ranging from rhythms in music melodies to the human heartbeat and DNA sequences. Many professions including, mathematics, astronomy, physics, chemistry, engineering, and biology use fractal geometry. (Bunde)
Waclaw Sierpinski was born on March 14, 1882, in Warsaw, Poland. Sierpinski attended the University of Warsaw in 1899, when all classes were taught in Russian. He graduated in 1904 and went on to teach mathematics and physics at a girl's school in Warsaw. He left teaching in 1905 to get his doctorate at the Jagiellonian University in Cracow. After receiving his doctorate in 1908, Sierpinski went on to teach at the University of Lvov. During his years at Lvov, he wrote three books and many research papers. These books were The Theory of Irrational numbers (1910), Outline of Set Theory (1912), and The Theory of Numbers (1912). In 1919, Sierpinski accepted a job as a professor at the University of Warsaw, and this is where
Waclaw Sierpinski (The Mactutor)
he would spend the rest of his life. Throughout his career, Sierpinski wrote 724 papers and an amazing 50 books. Sierpinski studied many areas of mathematics, including, irrational numbers, set theory, fractal geometry, and theory of numbers.
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In 1915, Sierpinski thought of a great variation for the ternary tree. The Sierpinski gasket is obtained by starting with an equilateral triangle thought of as a solid object. You then divide it into four smaller equilateral triangles, of which the middle one is removed. The middle one is removed so a triangular hole is produced. With the three remaining solid equilateral triangles, you proceed in the same way so three smaller triangular holes appear, and so on. The fractal dimension of the Sierpinski gasket is df= log3/log2. (Lauwerier) We denote by L the length of a fixed side, say the side of the triangle. M(L) denotes the mass of the gasket when the length of the side is L. So M(L) is the mass of the whole gasket. Because of self-similarity the whole gasket is again in one of the sub-triangles. Its mass is now M(L/2). Since 3 of the smaller triangles make up the whole gasket we get M(L) = 3 M(L/2). On the other hand the relation between the linear measure L and the mass of any geometric object is M = A L^d where A is some constant depending on the shape of the object, and d is the dimension. Combining the equations we get A L^d = 3A (L/2)^d, which simplifies to 1 = 3(1/2)^d or 2^d = 3. Taking logarithms on both sides and solving for the dimension we get d = log(3) / log(2) =1.5850….(Daepp)
Sierpinki gasket (The MacTutor)
There are two similar fractal concepts that Sierpinski created, which are extremely fascinating. The Sierpinki carpet is generated in close analogy to the Sierpinski gasket. We first pick a positive integer n and a second one k with k < n^2.(Daepp) Out of those squares, k squares are chosen and removed. This procedure is then repeated by dividing each of the smaller squares left into n^2 smaller squares and removing the k squares that are located at the same positions as in the first iteration. This procedure is repeated again and again. (Bunde)
Sierpinski Carpet (Bunde)
The Sierpinski carpet for the above example has n = 5 and k = 9. It is possible for the carpets to look different when k is changed. Using the same idea as for the calculation of the dimension of the gasket, we denote by L the length of a square side and by the M mass of the carpet. Considering that n^2 – k smaller squares with side length L/n make up the whole carpet with side length L we get M(L) = (n^2 – k) M (L/n). Combining this with the general formula M = A L^d for some constant A we get (n^2 – k) A (L/n)^d = A L^d which simplifies to n^2 – k = n^d. Taking logarithms on both sides and solving for the dimension we get d = log(n^2 –k) / log n. In the example above, the fractal dimension is log(16) / log(5) = 1.7227….(Daepp)
The Sierpinski sponge is constructed by starting from a cube, and subdividing it into 3*3*3 = 27 smaller cubes. Using the same arguments as for the gasket and the sponge, we get the dimension df = log(20) / log(3) = 2.7268….(Daepp) You take out the central cube and its six nearest neighbors. Each of the remaining twenty small cubes are processed in the same way. After each iteration, the volume of the sponge is reduced by a factor of 20/27, while the total surface area increases. M(1/3L) = 1/20M(L), which makes the fractal dimension for this example, df = log20/log3. (Bunde)
Sierpinski Carpet (Bunde)
Today, trees are very important to the field of fractal geometry. Pythagoras (569 – 500 BC) studied the famous figure in which squares are placed on the sides of a right triangle. He proved that the sum of the areas of the squares on two of the sides equaled the area of the square on the hypotenuse. In our present day, the figure that Pythagoras created has grown into the fractal term "tree." (Lauwerier)
In the Pythagoras tree, you find that a square with a given square number n supports an isosceles right triangle from which two smaller squares sprout. The one on the left has a number 2n, an even number, while the one on the right, 2n + 1, is an odd number. Together, these three form a geometrical representation of the Pythagoras Theorem, which states that the smaller squares together, have the same area as the larger square. The smaller squares are equal to the larger square, which makes each one half the size of its predecessor. If the area of the original square is 1, then the total of squares 2 and 3 will also equal 1. The same holds true for each of the smaller squares in the series. In the example below, the eight squares, 8,9,10,…,15, together have the same area as the original base square. (Lauwerier)
Pythagoras Tree (Lauwerier)
Fractal geometry is not only fun to play around with, but it is also amazing to look at. Almost everywhere you look, you find a fractal of some sort. The Sierpinski gasket and Pyhtagoras’ tree interest me the most because they are very easy to work with. The only problems that I would have with fractal geometry, is trying to create them on the computer! To conclude, fractals are a big part of everything around us, such as the human body and the galaxy. Where would we be if people like Sierpinski and Archimedes hadn’t made their discoveries?
1] Bunde, Armin and Havlin, Shlomo, Fractals in Science, NewYork:Springer-Verlas, 1994.
2] Falconer, Kenneth, Fractal Geometry, Chichester, Englend: Jonh Wiles anf Sons, 1990.
3] Lauwerier, Hans, Fractals, Princeton, NJ: Princeton University Press, 1996.
4] Pickover, Clifford, Fractal Horizons, New York: St. Martin’s Press, 1996.
5] The MacTutor History of Mathematics, Biography of Waclaw Sierpinski, Dec. 6,1999, http://www-groups.dcs.st-and.ac.uk/history/Mathematics/Sierpinski.html