Rice COMP 360 - Some Strange Properties of Fractal Curves

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Lecture 3: Some Strange Properties of Fractal CurvesI have been a stranger in a strange land. Exodus 2:221. Fractal StrangenessFractals have a look and feel that is very different from ordinary curves. Unlike commonplace curves such as lines or circles, there are no simple formulas for representing fractals like the Sierpinski gasket or the Koch curve. Therefore it should not be surprising that fractal curves also have geometric features that are unlike the properties of any other curves you have previously encountered. In this lecture we are going to explore some of the strangest peculiarities of fractals, including their dimension, differentiability, and attraction.2. DimensionYou may have been wondering about the origin of the term fractal. Many fractal curves have a non-integral dimension, a fractional dimension somewhere between one and two. Fractal refers to this fractional dimension. Standard curves like the line and the circle are 1-dimensional. To say that a curve is 1-dimensional means that the curve has no thickness; if the curve is black and the background is white, then when we look at the curve we see white on either side of a thin black curve. But fractals are different. Look at the Sierpinski gasket or the C-curve. There seem to be regions that are neither black nor white, but instead are gray. Such curves typically have dimension greater than one, but less than two; these curves do not completely fill up any region of the plane, so they are not 2-dimensional, but neither are these gray curves as thin as 1-dimensional curves. To calculate the actual dimensions of fractal curves, we first need to formalize the notion of dimension for some standard geometric shapes.The dimension of a line segment is one, the dimension of a square is two, and the dimension of a cube is three. There is a formal way to capture these dimensions. Suppose we split these objects by inserting new vertices at the centroids of their edges and faces. Then the line segment splits into 2 line segments, the square into € 4 = 22 squares, and the cube into € 8 = 23 cubes (see Figure 1). In each case the dimension appears in the exponent. There is nothing magical about splitting each edge into two equal parts. If we split each edge into N equal parts, then the line segment splits into N line segments, the square into € N2 squares, and the cube into € N3 cubes. Once again, the dimension appears in the exponent. Another name for an exponent is a logarithm, so we are going to formalize the notion of dimension in terms of logarithms.€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •€ •Figure 1: Inserting vertices at the centroids splits a line segment into 2 line segments, a square into € 4 = 22 squares, and a cube into € 8 = 23 cubes.Another way of thinking about what we have just done is that we have split the line, the square, and the cube into identical parts, where each part is a scaled down version of the original. This decomposition should remind you of the fractals that you encountered in Lecture 2, where each fractal is composed of several identical scaled down copies of the original fractal. Evidently, in this construction for the line, the square, and the cube, if D denotes dimension, then€ D = LogN(E) =Log(E)Log(N) , (1)where N is the number of line segments along each edge and E is the number of equal scaled down parts. But if N is the number of line segments along each edge, then € S = 1/ N is the scaling along each edge. Since € Log(S) = −Log(N ), we can rewrite Equation (1) by setting€ D = −Log(E)Log(S), (2)where S is the scale factor and E is the number of identical scaled down parts.Equation (2) has several important properties. First, notice that € S < 1, so € Log(S) < 0. Therefore, the minus sign on the right hand side of Equation (2) insures that the dimension D is positive. Second, since D is defined as the ratio of two logarithms, the base of the logarithm does not matter; dimension is the same in all bases. Finally, Equation (2) gives the same result as Equation (1) for the line, the square, and the cube, since in these cases € E = ND and € S = 1/ N, so€ −Log(E)Log(S)= −Log(ND)Log(1 / N)=DLog(N )Log(N)= D.Let’s see now what happens when we apply Equation (2) to fractal curves.2.1 Fractal Dimension. To apply our dimension formula, we need to consider self-similar curves. Recall that a curve is self-similar if it can be decomposed into a collection of identical curves each of which is a scaled version of the original curve. Most of the fractal curves we encountered in Lecture 2 such as the Sierpinski gasket and the Koch curve are self-similar curves. In fact, self-similarity is what allows us to write simple recursive turtle programs to generate these curves. Let’s look now at some examples.2Example 1: Sierpinski GasketThe Sierpinski gasket consists of three smaller Sierpinski gaskets, where the length of each edge of the smaller gaskets is one-half the length of an edge of the original gasket (see Figure 2, left). Thus € E = 3 and € S = 1/ 2, so€ D = −Log(E)Log(S)= −Log(3)Log(1/ 2)=Log(3)Log(2)≈1.585 ⇒1 < D < 2.Example 2: Koch CurveThe Koch curve consists of four smaller Koch curves, each one-third the size of the original curve (see Figure 2, right). Thus € E = 4 and € S = 1/ 3, so€ D = −Log(E)Log(S)= −Log(4)Log(1/ 3)=Log(4)Log(3)≈1.262 ⇒1 < D < 2.Figure 2: The Sierpinski gasket (left) and the Koch curve (right). The Sierpinski gasket is composed of three self-similar parts; the length of each edge of one of the smaller gaskets is one-half the length of an edge of the original gasket. The Koch curve is composed of four self-similar parts, each one-third the size of the original Koch curve.2.2 Computing Fractal Dimension from Recursive Turtle Programs. The fractal dimension of a self similar fractal curve can often be computed directly from its recursive turtle program: the number of recursive calls corresponds


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Rice COMP 360 - Some Strange Properties of Fractal Curves

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