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EADEM MUTATA RESURGO |  |
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The word fractal was coined by Benoit Mandelbrot in the early 1970's. It is derived from the Latin word ``fractus'' which aptly means ``broken'', i.e., fragmented or irregular. Mandelbrot observed that certain natural geometries, e.g., coastlines, terrain and clouds, exhibited a simplifying invariance under scale, i.e., their geometries possessed similarities that were invariant to changes in magnification or resolution. He discovered that this invariance to scale existed in a large variety of artificial and natural phenomena. This invariance to scale, i.e, ``self-similarity'', is central to ``Fractal Geometry''. A wide class of natural geometries appear to possess this underlying fractal character within a range of scale. Fractal geometry, thus, provides a means by which to express and simulate both simple and complex, natural geometry. A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale.
There are many mathematical structures that are fractals; e.g.
Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and
Lorenz attractor. Fractals also describe many real-world objects, such
as clouds, mountains, turbulence, coastlines, roots, branches of
trees, blood vesels, and lungs of animals, that do not correspond to
simple geometric shapes.
In what follows, we briefly describe a few of the so called Mathematical
Monsters, which are description of classical strictly self-similar fractal
sets. We then introduce the notion of fractal dimension via the Koch
construction set.
Georg Cantor, a German mathematician, in 1883, introduced a set, now called the Cantor set that had some exceptional properties. Following is a simple geometric construction scheme to visualize the Cantor set. The construction of the Cantor set (C) is done by starting with a unit line [0,1]. Divide the unit line segment into three equal parts and then remove the middle third (leaving the end points)to form new segments that exist at [0, 1/3] and [2/3, 1]. This is the first stage in the construction of the C set. In the next stage repeat the above process of removing the middle third of the each of the two line segments that was obtained in the last step to get four smaller line segments. This process of removing the middle third from the remaining segments from the previous stage, is continued adinfinitum. The figure below shows the first six stages in the construction of the C set. The points marked by the white vertical line (in stage 0 and stage 1) are the line segments that are removed.
So what is a Cantor set? From the intuitive geometric way of construction, it is clear that the points (the end points of the segments) 0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 1/27, 2/27, 7/27, 8/27, 19/27, ..., must be a part of the C set. There is a slight danger in this geometric metaphor. The way the points in the C set are listed seems to suggest that the set of points in the C set are countably infinite (like the set of rational numbers). But it is known from set theory that a C set is a set of uncountably infinite points (like the set of irrational numbers). Before, a more definite definition is given, we calculate the total length of the line segments, at the nth stage of construction. The total number of disjoint segments at the nth stage is 2n with each segment of length (1/3)n. It follows that the total length at the nth stage is (2/3)n which in the limit of n going to infinity goes to zero. The fact that C set is a set of uncountably infinite points with zero measure, is a perplexing thing. It should be noted that the definition of the C set about to be stated doesn't follow from the geometric construction process rather the geometric process described above is a practical algorithm to approximate the C set. Thus, the C set is the set of points in the interval [0,1] whose ternary representation (base 3, like base 10 for decimal representation) do not contain 1.
This produces the set of real numbers x such that
The definition of C set given by the equation above resolves the issue regarding the cardinality (whether the set of points are denumerable or enumerable) and the related question -- if the elements of the C set are only the end points of the segments generated in the geometric construction process. Thus, a point like 2/9, 2/3 we know is in the C set and is readily verified by expressing it in the ternary form - 0.02 and 0.2. Considering points like 1/3, 1/9 which is known to be in C set when expressed in ternary base looks like 0.1 and 0.01 !! Now, this is easily resolved by noting that there is an ambiguity in the representation. The point 1/3 can also be represented as 0.0`2 and 1/9 as 0.`2, where the bar on top implies nonterminating zero or two. Done!! To see there are infinitely many points in C set that are not the end points, note end points just correspond to numbers which have ternary expansion ending with nonterminating zeros or twos. All other combinations of zeros and twos (and there are infinitely many) give points from the C set. Thus, if we are to pick at random a point from the C set, we will with probability 1 pick a point which is not an end point. From this, one can see that any point in the C set can be approximated arbitrarily closely by other points from the C set and yet the set is a dust of points. If one takes the part of C which lies in the interval [0, 1/3], then one can regard that part as a scaled down version of the entire set. This can be seen by the fact that for every point in C we find a corresponding point in [0, 1/3] by diving the original point in C by 3. This in the ternary representation means, if we have a point in C represented as 0.220202... then the corresponding one in the interval [0,1/3] is found by dividing the number by 3 which like adding a zero after and decimal point, ie., 0.0220202... This is correspondence is ofcourse true for every point. This implies that the part of the Cantor set present in [0,1/3] is an exact copy of the entire Cantor set scaled down by the factor 1/3.
Helge von Koch, a Swedish mathematician, in 1904, introduced what is now called a Koch curve. Koch curve is a fractal curve characterized by such properties as - a curve that is infinitely long, contained within a finite region, not differentaible at any point (they just have corners). A geometric construction scheme for the Koch curve is the following. Start with a straight line segement, say, between [0, 1]. Partition the unit line in three equal parts and replace the middle third by an equilateral triangle and take the base of the triangle away.
This is the first satge of the construction as shown in the figure above (top left). This basic construction process is repeated with each of the remaining four segemnents ad infinitum. The figure above shows the construction stage 1, 2 ,3 and 6. Notice that the self-similarity is built-in feature of the construction process -- each part of the 4 parts in the kth step is again a scaled down version, by a factor of 3, of the entire curve in the previous (k-1)st stage. The length of the Koch curve is infinite. Assuming that we started with a unit line segment the length of the curve in the kth stage is equal to the number of line segments times the length of each segment. The number of pieces in the kth stage is 4k and the length of a segment is (1/3)k. Thus, the the total length of the curve is (4/3)k at the kth stage of construction. Since, the Koch curve, by definition, is the limiting set of the above geometric construction scheme, when k goes to infinity. The length of the Koch curve is infinite.
In 1890, Giuseppe Peano and David Hilbert,, in 1891, introduced curves which 'fill' a plane, ie., given a region in plane, there is a curve which passes through every point in that region! Such curves are now called space filling curves. The Peano curve is obtained by a modified version of the Koch curve construction scheme. Start with a unit line and replace it with a figure shown below (top left). Thus, at each stage one line segment is replaced by 9 line segments scaled down by a factor of 3. Repeat the procedure with the remaining 9 line segments ad infinitum.
The figure above shows the Peano curve through the first four stages of construction.
Dimensions form the main mathematical tool for the study of fractal sets. They are non-integer indices that can be measured approximately by experiment. They quantify the static geometry of an object, an objective means by which to compare fractal sets and quantify our perception of how densely a fractal set occupies the space in which it exists. Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. The differences come in what is exactly meant by öbject size" and what is meant by "measurement scale" and how to get an average number out of many different parts of a geometrical object. In one dimension consider a line segment. If the linear dimension of the line segment is doubled then obviously the length (characteristic size) of the line has doubled. In two dimensions, if the linear dimensions of a rectangle, for example, is doubled then the characteristic size, the area, increases by a factor of 4. In three dimensions if the linear dimension of a box are doubled then the volume increases by a factor of 8. This relationship between dimension D, linear scaling L and the resulting increase in size S can be generalised and written as
This is just telling us mathematically what we know from everyday experience. If we scale a two dimensional object for example then the area increases by the square of the scaling. If we scale a three dimensional object the volume increases by the cube of the scale factor. Rearranging the above gives an expression for dimension depending on how the size changes as a function of linear scaling, namely
In the examples above the value of D is an integer, either 1, 2, or 3, depending on the dimension of the geometry. This relationship holds for all Euclidean shapes. There are however many shapes which do not conform to the integer based idea of dimension given above in both the intuitive and mathematical descriptions. That is, there are objects which appear to be curves for example but which a point on the curve cannot be uniquely described with just one number. If the earlier scaling formulation for dimension is applied the formula does not yield an integer. There are shapes that lie in a plane but if they are linearly scaled by a factor L, the area does not increase by L squared but by some non integer amount. These geometries are called fractals! Consider the simple Koch snowflake fractal shape. The method of creating this shape, as described above, is to repeatedly replace each line segment with 4 line segements which are 1/3rd the original line segment. The process of selecting line segemnt and replacing it by four reduced copies of itself in the prescribed form continues for ever. Unlike Euclidean shapes this object has detail at all levels. If one magnifies an Euclidean shape such as the circumference of a circle it becomes a different shape, namely a striaght line. If we magnify this fractal more and more detail is uncovered, the detail is self similar, in fact, it is strictly self similar. At each iteration the length of the curve increases by a factor of 4/3. Thus, the limiting curve is of infinite length! This curve manages to compress an infinite length into a finite area of the plane without intersecting itself! Considering the intuiitve notion of 1 dimensional shapes, although this object appears to be a curve with one starting point and one end point, it is not possible to uniquely specify any position along the curve with one number as we expect to be able to do with Euclidean curves which are 1 dimensional. Although the method of creating this curve is straightforward, there is no algebraic formula that describes the points on the curve. To calculate the dimension of Koch curve, we proceed in the following way. With a scale factor L = 3, we obtain the first third of the whole curve. We need S = 4, such pieces to cover the original set by repeated translations and rotations of this scaled-down piece. We can also scale with a factor L = 3n, using S = 4n pieces to cover the original set. Thus for the Koch curve, using 2 we obtain the dimension D = log(4)/log(3). Similar, argument shows the dimension of Cantor set is D = log(2)/log(3) and that of the Peano curve D = log(4)/log(2) = 2.
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