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An Exploration of the Generalized Cantor Set INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 7, JULY 2013 ISSN 2277-8616 An Exploration Of The Generalized Cantor Set Md. Shariful Islam Khan, Md. Shahidul Islam Abstract: In this paper, we study the prototype of fractal of the classical Cantor middle-third set which consists of points along a line segment, and possesses a number of fascinating properties. We discuss the construction and the self-similarity of the Cantor set. We also generalized the construction of this set and find its fractal dimension. Keywords: Cantor set, Dimension, Fractal, Generalization, Self-similar. ———————————————————— 1 INTRODUCTION 2 BASIC DEFINITIONS A fractal is an object or quantity that displays self-similarity. The object needs not exhibit exactly the same structure but Definition 2.1: A set 푆 is self-similar if it can be divided into 푁 the same type of structures must appear on all scales. congruent subsets, each of which when magnified by a Fractals were first discussed by Mandelbrot[1] in the 1970, but constant factor 푀 yields the entire set 푆. the idea was identified as early as 1925. Fractals have been investigated for their visual qualities as art, their relationship to Definition 2.2: Let 푆 be a compact set and 푁(푆, 푟) be the explain natural processes, music, medicine, and in minimum number of balls of radius r needed to cover 푆. Then Mathematics. The Cantor set is a good example of an the fractal dimension[4] of 푆 is defined as elementary fractal. The object first used to demonstrate fractal dimensions is actually the Cantor set. The process of log푁(푆, 푟) dim 푆 = lim . generating this fractal is very simple. The set is generated by 푟→0 log(1/푟) the iteration of a single operation on a line of unit length. In each iteration, the middle third from each lines segment of the previous set is simply removed. As the number of iterations In the line (ℝ1), a ball is simply a closed interval. increases, the number of separate line segments tends to infinity while the length of each segment approaches zero. 3 CONSTRUCTION OF CANTOR SET Under magnification, its structure is essentially indistinguishable from the whole, making it self-similar[2]. We 3.1. Cantor middle-1/3 set studied the dimension of the Cantor set that its magnification To construct this set (denoted by 퐶3), we begin with the factor is three, or the fractal is self-similar if magnified three 1 2 interval 0,1 and remove the open set , from the closed times. Then we noticed that the line segments decompose 3 3 into two smaller units. We also studied the fractal dimensions interval 0,1 . The set of points that remain after this first step 1 2 of the generalized Cantor sets. We explore the generalization will be called 퐾1, that is, 퐾1 = 0, ∪ , 1 . In the second step, of the Cantor set with fractal dimension and demonstrate the 3 3 we remove the middle thirds of the two segments of K1, that is, diagram of self-similarity of this generalized Cantor set in 1 2 7 8 1 2 3 6 7 8 remove , ∪ , and set 퐾 = 0, ∪ , ∪ , ∪ , 1 three phases. Although we used the typical middle-thirds or 9 9 9 9 2 9 9 9 9 9 9 ternary rule[3] in the construction of the Cantor set, we be what remains after the first two steps. Delete the middle generalized this one-dimensional idea to any length other than thirds of the four remaining segments of K2 to get K3. 1 , excluding the degenerate cases of 0 and 1. Repeating this process, the limiting set 퐶3 = 퐾∞ is called the 3 Cantor middle 1/3 set. ———————————————— Figure 1. The Cantor set, produced by the iterated process of removing the middle third from the previous segments. The Cantor set has zero length, and non-integer dimension. PhD Research fellow (Working as Assistant Professor under MOE, BD), Natural Science Group, National 3.2. Fractal dimension of the Cantor middle-1/3 set University, Gazipur 1704, Bangladesh. E-mail: [email protected] The set 퐶3 is contained in 퐾푛 for each 푛. Just as 퐾1 consists of 1 2 Professor, Department of Mathematics, University of 2 intervals of length , and 퐾2 consists of 2 intervals of length 3 Dhaka, Dhaka 1000, Bangladesh. 1 3 1 2 and 퐾3 consists of 2 intervals of length 3. In general, 퐾푛 E-mail: [email protected] 3 3 50 IJSTR©2013 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 7, JULY 2013 ISSN 2277-8616 consists of 2푛 intervals, each of length 1 . Further, we know 4 GENERALIZED CANTOR MIDDLE-흎 SETS 3푛 that 퐶 contains the endpoints of all 2푛 intervals, and that each (ퟎ < 흎 < 1) pair of endpoints lie 3−푛 apart. Therefore, the smallest number −푛 −푛 푛 of 3 -boxes covering 퐶3 is 푁 퐶3, 3 = 2 . Compute the 4.1. Cantor middle-1/5 set dimension of the Cantor middle-1/3 set 퐶3 as To build this set (denoted by 퐶5) we can follow the same procedure as construction of the middle-third Cantor’s set. ln 2n n ln 2 ln 2 First we delete the open interval covering its middle fifth from dim (C3 ) lim lim 0.6309. the unit interval 퐼 = [0,1]. That is, we remove the open interval n n n n ln3 ln3 2 3 ln3 ( , ). The set of points that remain after this step will be called 5 5 2 3 퐾 . That is, 퐾 = 0, ∪ , 1 . In the second step, we remove 3.3. Self-similarity of the Cantor middle-1/3 set 1 1 5 5 One of the most important properties of a fractal is known as the middle fifth portion of each of the 2 closed intervals of 퐾1 self-similarity[5]. Roughly speaking, self-similarity means if we and set examine small portions of the set under a microscope, the 4 6 2 3 19 21 퐾 = 0, ∪ , ∪ , ∪ , 1 . image we see resembles our original set. To see this let us 2 25 25 5 5 25 25 look closely at 퐶3. Note that 퐶3 is decomposed into two distinct subsets, the portion of 퐶3 in 0, 1/3 and the portion in 2 3, 1 . If we examine each of these pieces, we see that they resemble the original Cantor set 퐶3. Indeed, each is obtained by removing middle-thirds of intervals. The only difference is the original interval is smaller by a factor of 1/3. Thus, if we magnify each of these portions of 퐶3 by a factor of 3, we obtain the original set. More precisely, to magnify these portions of 퐶3, we use an affine transformation. Let 퐿 푥 = 3푥. If we apply 퐿 to the portion of 퐶3 in 0, 1/3 , we see that 퐿 maps this portion onto the entire Cantor set. Indeed, 퐿 maps 1 9 , 2/9 to 1 3 , 2/3 , 1 27 , 2/27 to 1 9 , 2/9 , and so forth (Fig. 2). Each of the gaps in the portion of 퐶 in 0, 1/3 is 3 Figure 3. Construction of the Cantor middle-1 5 set. taken by 퐿 to a gap in 퐶3. That is, the ―microscope‖ we use to magnify 퐶3 ∩ 0, 1/3 is just the affine transformation 퐿 푥 = 2 3푥. Again, we remove the middle fifth portion of each of the 2 closed intervals of 퐾 to get 퐾 . Repeating this process, the 2 3 limiting set 퐶5 = 퐾∞ is called the Cantor middle-1 5 set. The set 퐶5 is the set of points that belongs to all of the 퐾푛. 4.2. Fractal dimension of the Cantor middle-1/5 set The set 퐶5 is contained in 퐾푛 for each n. Just as 퐾1 consists 2 of 2 intervals of length 2/5, and 퐾2 consists of 2 intervals of 2 3 2 3 2 length 2 and 퐾3 consists of 2 intervals of length 3. In 5 5 푛 general, 퐾 consists of 2푛 intervals, each of length 2 . 푛 5 푛 Further, we know that 퐶5 contains the endpoints of all 2 2 푛 intervals, and that each pair of endpoints lie apart. 5 2 푛 Therefore, the smallest number of -boxes covering 퐶 is Figure 2. Self-similarity of the Cantor middle-thirds set 5 5 푛 −푛 푛 푁 퐶5, 2 5 = 2 . We compute the fractal dimension of the To magnify the other half of 퐶3, namely 퐶3 ∩ 2 3 , 1 , we use Cantor middle-1/5 set 퐶5 as another affine transformation, 푅 푥 = 3푥 − 2. Note that 푅 2 3 = 0 and 푅 1 = 1 so 푅 takes 2 3 , 1 linearly onto ln 2푛 ln 2 dim( 퐶5) = lim = . 0, 1 . As with 퐿, 푅 takes gaps in 퐶3 ∩ 2 3 , 1 to gaps in 퐶3, so 푛→∞ ln 5/2 푛 ln 5 − ln 2 푅 again magnifies a small portion of 퐶3 to give the entire set. Using more powerful ―microscope‖, we may magnify arbitrarily 4.3. Cantor middle-1/7 set small portions of 퐶 to give the entire set. For example, the 3 To build this set (denoted by 퐶7) we first delete the open portion of 퐶3 in 0, 1/3 itself decomposes into two self-similar interval covering its middle 7th from the unit interval 퐼 = [0,1]. pieces: one in 0, 1/9 and one in 2 9 , 1 3 .
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