1. INTRODUCTION TO FRACTALS 3 in Figure 3. What distinguishes fractals from the classical geometry objects is that their micro-sets may be as complex as the original sets. Consider, for example, the so-called Sierpinski gasket S. Figure 4. Sierpinski gasket. No matter how much you zoom in on it, it doesn’t become flat or gets simpler in any other respect. More precisely, if you start zooming in at some fixed point of S, the corresponding family of mini-sets has periodic behavior: whatever you see at the scale λ, you see again at the scales 2λ, 4λ, 8λ, and so on (provided that the magnifying factor λ is large enough so that you can see only a small piece of S at once). For more complicated fractals the behavior can be more subtle, so that when zooming in one observes a chaotic behavior of shapes. The proper tools to handle this chaos are no longer those of Linear Algebra but those of Ergodic Theory. Another difference between fractals and classical geometry objects is that frac- tals usually have non-integer (fractional) dimension. The classical notion of dimen- sion is, essentially, a Linear Algebra notion. The dimension of a classical object like a line or a surface in Rn is nothing but the linear dimension of its tangent plane, so a point has dimension 0, a line has dimension 1, a surface has dimension 2, and so on. However, as a rule, fractals do not have tangent planes and this linear algebra approach becomes meaningless for them. The dimension of fractals has to be defined in a different way and can be any non-negative real number up to the dimension of the ambient Euclidean space. We will return to this discussion in Chapter 2 and now we will look at how fractals arise in mathematics. The primary sources of fractal objects are infinite
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