Dimension of Self-Similar Fractals

What is the dimension of a set of points For familiar geometric objects, the answer is clear—lines and smooth curves are one-dimensional, planes and smooth surfaces are two-dimensional, solids are three-dimensional, and so on. If forced to give a definition, we could say that the dimension is the minimum number of coordinates needed to describe every point in the set. For instance, a smooth curve is one-dimensional because every point on it is determined by one number, the arc length from some fixed reference point on the curve. 

What is the dimension of the von Koch curve Since it s a curve, you might be tempted to say it s one-dimensional. But the trouble is that K has infinite arc length. To see this, observe that if the length of So is To, then the length of S, is j = f Lj, because 5j contains four segments, each of length j. The length increases by a 

Moreover, the arc length between any two points on is infinite, by similar reasoning. Hence points on K aren t determined by their arc length from a particular point, because every point is infinitely far from every other  

With this paradox as motivation, we now consider some improved notions of 

If we shrink the square by a factor of 2 in each direction, it takes four of the small squares to equal the whole. Or if we scale the original square down by a factor of 3, then nine small squares are required. In general, if we reduce the linear dimensions of the square region by a factor of r, it takes r of the smaller squares to equal the original. 

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