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  Professor, Department of Mathematics, University of 2 intervals of length , and 퐾 consists of 22 intervals of length 3 2 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 퐶 . That is, the ―microscope‖ we use to 3 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 퐾2 to get 퐾3. Repeating this process, the 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 푛 intervals, and that each pair of endpoints lie 2 apart. 5 푛 Therefore, the smallest number of 2 -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 . We may magnify 3 4 That is, we remove the open interval ( , ). The set of points the left portion via 퐿2 푥 = 9푥 to yield 퐶3 and the right portion 7 7 3 via 푅 푥 = 9푥 − 2. Note that 푅 maps 2 9, 1 3 onto 0, 1 that remain after this step will be called 퐾 , that is, 퐾 = 0, ∪ 2 2 1 1 7 linearly as required. Note also that at the 푛th stage of the 4 푛 , 1 . In the second step, we remove the middle 7th portion of construction of 퐶3, we have 2 small copies of 퐶3, each of 7 푛 which may be magnified by a factor of 3 to yield the entire each of the 2 closed intervals of 퐾1 and set Cantor set.

9 12 21 4 37 40 1−휔 푛 퐾2 = 0, ∪ , ∪ , ∪ , 1 . and the length of each closed interval is . Also, the 49 49 49 7 49 49 2 푛 combined length of the intervals in 퐾푛 is 1 − 휔 , which approaches 0 as 푛 approaches ∞. We are now ready to say that Cantor middle-휔 sets are appropriately named.

Proposition 5.2. The Cantor middle-휔 set is a Cantor set, where 0 < 휔 < 1.

Proof: The proof can be found in [7].

Figure 4. Construction of the middle-1 7 Cantor set. 6 SPECIAL CASES OF GENERALIZED

2 Again, we remove the middle 7th portion of each of the 2 CANTOR MIDDLE-흎 SET closed intervals of 퐾 to get 퐾 . Repeating this process, the 2 3 6.1. Removing the alternative segments (obviously the limiting set 퐶 = 훿 is called the Cantor middle-1/7 set. The 7 ∞ number of segments is odd). set 퐶7 is the set of points that belongs to all of the 퐾푛. (a) When 휔 = 2/5, construction of the Cantor middle-2/5 set: 4.4. Fractal dimension of the Cantor middle-1/7 set In this case, we delete the middle second and fourth of 5 In this case, consists of 2 intervals of length 3/7, and portions of the unit interval 퐼 = [0,1]. Then we have 퐾1 = 퐾1 퐾2 1 2 3 4 32 0, ∪ , ∪ [ , 1] and 2 3 5 5 5 5 consists of 2 intervals of length 2 and 퐾3 consists of 2 7 3 3 푛 1 2 3 4 1 2 11 12 13 14 3 intervals of length 3. In general, 퐾푛 consists of 2 intervals, 퐾 = 0, ∪ , ∪ , ∪ , ∪ , ∪ , ∪ 7 2 25 25 25 25 5 5 25 25 25 25 5 3 푛 4 21 22 23 24 each of length . Further, we know that 퐶 contains the , ∪ , ∪ [ , 1]. 7 7 5 25 25 25 25 endpoints of all 2푛 intervals, and that each pair of endpoints lie 푛 푛 3 apart. Therefore, the smallest number of 3 -boxes 7 7 푛 −푛 푛 covering 퐶5 is 푁 3 7 = 2 . We compute the fractal dimension of the Cantor middle-1/7 set 퐶7 as

ln 2푛 ln 2 dim(퐶7) = lim = . 푛→∞ ln 7/3 푛 ln 7 − ln 3

5 GENERALIZATION In similar way, we can construct the middle-(2푚 − 1)th Cantor’s set, 퐶2푚−1 where 푚 ≥ 2 and then compute the fractal dimension of the Cantor middle-1/(2푚 − 1) set퐶2푚−1. Figure 5. Construction of the middle-2/5 Cantor set. 푛 In this case, 퐾 consists of 2푛 intervals, each of length (푚−1) 푛 (2푚−1)푛 푛 Repeating this process, the limiting set 퐶5 = 퐾∞ can be called and 퐶2푚−1 contains the endpoints of all 2 intervals, and each 3 (푚−1)푛 the Cantor middle-2/5 set and 퐾 consists of intervals of pair of endpoints lie apart. Therefore, the smallest 3 3 (2푚−1)푛 1 n 푛 length 3. In general, 퐾푛 consists of 3 intervals, each of (푚−1) 5 number of -boxes covering 퐶2푚−1 is 푁 퐶2푚−1,(푚 − (2푚−1)푛 length 1 . Thus the dimension of the Cantor middle-2/5 set is 1)푛 (2푚 − 1)−푛 = 2푛 . We compute the fractal dimension of 5푛 the Cantormiddle-1/(2푚 − 1) set 퐶2푚−1as 푛 ln 3 ln 3 dim 퐶5 = lim = . ln 2푛 푛→∞ ln 5푛 ln 5 dim(퐶2푚−1) = lim 푛→∞ ln (2푚 − 1)푛 /(푚 − 1 푛 ) (b) When 휔 = 3/7, construction of the Cantor middle-3/7 set: ln 2 = , In this case, we remove the middle second, fourth and sixth of ln(2푚 − 1) − ln(푚 − 1) 7 portions of the unit interval 퐼 = [0,1]. Then we have 퐾1 = 1 2 3 4 5 6 1 2 3 0, ∪ , ∪ , ∪ , 1 and 퐾 = 0, ∪ , ∪ ⋯ ∪ where 푚 ≥ 2. 7 7 7 7 7 7 2 49 49 49 [48 , 1]. 49 Comment: We can generalize the Cantor’s set by setting 휔 = 1 as the Cantor middle-휔 set and the fractal dimension 2푚−1 of the Cantor middle-휔 set is ln 2 , where 휔 = 1 , 1 , 1 , ln 2−ln (1−휔) 3 5 7 ⋯ ⋯ ⋯.

Lemma 5.1[6]: If 퐾푛 is as defined above in constructionof the 푛 Cantor middle-휔 set, then there are 2 closed intervals in 퐾푛

(b) When 휔 = 3/5, construction of the Cantor middle-3/5 set: In this case, we delete the middle three of five portions of 1 4 the unit interval 퐼 = [0,1]. Then we have 퐾 = 0, ∪ , 1 1 5 5 1 4 1 4 21 24 and 퐾 = 0, ∪ , ∪ , ∪ , 1 . 2 25 25 5 5 25 25

Figure 6. Construction of the middle-3/7 Cantor set.

Repeating this process, the limiting set 퐶7 = 퐾∞ can be called 3 the Cantor middle-3/7 set and 퐾3 consists of 4 intervals of 1 푛 length 3. In general, 퐾푛 consists of 4 intervals, each of length 7 1 . Thus the dimension of the Cantor middle-3/7 set is Figure 8.Construction of the middle-3 5 Cantor set. 7푛 푛 푛 Repeating this process as above,퐾푛 consists of 2 intervals, ln 4 ln 4 1 dim 퐶7 = lim = . each of length . Thus the dimension of the Cantor middle- 푛→∞ ln 7푛 ln 7 5푛 3 5 set 퐶5 as (c) Generalization: When 휔 = (푚 − 1)/(2푚 − 1) and 푚 ≥ 2, construction of the Cantor middle- (m-1)/(2m-1) set: ln 2푛 ln 2 dim 퐶5 = lim = As above 퐾 consists of mn intervals, each of length 푛→∞ ln 5푛 ln 5 푛 1 (2푚 − 1)푛. Thus the dimension of the Cantor middle- (c) When 휔 = 4/6, construction of the Cantor middle-4/6 set: (푚 − 1)/(2푚 − 1) set is In this case, we delete the middle four of the six portions of

ln 푚푛 ln 푚 the unit interval 퐼 = 0,1 . Then we have

dim 퐶2푚−1 = lim 푛 = 푛→∞ ln(2푚 − 1) ln(2푚 − 1) 1 5 1 5 1 5 31 35 퐾 = 0, ∪ , 1 and 퐾 = 0, ∪ , ∪ , ∪ , 1 . 1 6 6 2 36 36 6 6 36 36 where 푚 ≥ 2.

6.2. Removing the middle segments (except two end segments).

(a) When 휔 = 2/4, construction of the Cantor middle-2/4 set: In this case, we delete the middle two of four portions of the unit interval 퐼 = [0,1]. That is, we remove the open interval (1 , 3) from the unit interval 퐼 = [0,1]. Then we have Figure 9. Construction of the middle-4 6 Cantor set. 4 4 1 3 1 3 1 3 13 15 퐾1 = 0, ∪ , 1 and 퐾2 = 0, ∪ , ∪ , ∪ , 1 . 푛 4 4 16 16 4 4 16 16 In general, 퐾 consists of 2푛 intervals, each of length 1 . 푛 6 Thus the dimension of the Cantor middle-4/6 set 퐶6 as

ln 2푛 ln 2 푑푖푚 퐶6 = lim = 푛→∞ ln 6푛 ln 6

(d) Generalization: When 휔 = 푚/(푚 + 2) and 푚 ∈ ℤ+, construction of the Cantor middle- 푚/(푚 + 2) set: As above, 퐾 consists of 2푛 intervals, each of length 1 . 푛 (푚+2)푛 Thus the dimension of the Cantor middle-푚 (푚 + 2) set Figure 7.Construction of the middle-2 4 Cantor set. 퐶푚+2 as

푛 Repeating this process, the limiting set 퐶4 = 퐾∞ can be called ln 2 ln 2 dim(퐶푚+2) = lim = , 3 푛→∞ ln(푚 + 2)푛 ln(푚 + 2) the Cantor middle-2/4 set and 퐾3 consists of 2 intervals of 1 n length 3. In general, 퐾푛 consists of 2 intervals, each of + 4 where 푚 ∈ ℤ . length 1 . Thus the dimension of the Cantor middle-2/4 set is 4푛 7 CONCLUSION ln 2푛 ln 2 We construct the generalized Cantor sets in three phases with dim 퐶4 = lim = . self-similarity and find their fractal dimensions in each case. 푛→∞ ln 4푛 ln 4 Although our construction of the Cantor set used the typical ―middle-thirds‖ or ternary rule, we can easily generalize this 53 IJSTR©2013 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 2, ISSUE 7, JULY 2013 ISSN 2277-8616 one-dimensional idea to any length other than 1, excluding of 3 course the degenerate cases of 0 and 1. After decomposing the typical Cantor set into two distinct subsets, the portion of the set in 0, 1/3 and the portion in 2 3, 1 , we see that each of these pieces resembles the original Cantor set. The only difference is the original interval is smaller by a factor of 1/3. In the same manner, the magnifications of our generalized Cantor sets resemble the original set.

ACKNOWLEDGEMENTS The authors gratefully acknowledge financial support provided by Centre for Post-Graduate studies, Training and Research of National University, Gazipur-Bangladesh.

REFERENCES [1] B. Mandelbrot, The Fractal Geometry of Nature. San Francisco: W.H. Freeman, 1983.

[2] R.P. Osborne, Embedding Cantor sets in a Manifold, Part 1: Tame Cantor sets in ℝn , Michigan Math. J. 13, 57-63, 1966.

[3] R. B. Sher, Concerning wild Cantor sets in ℝퟑ, Proc. Amer. Math. Soc. 19, 1195–1200, 1968.

[4] D. G. Wright, Cantor sets in 3-manifolds, Rocky Mountain J. Math. 9, 377–383, 1979.

[5] K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985.

[6] S. H. Strogatz, Nonlinear Dynamics and Chaos, West view, Cambridge, MA, 1994.

[7] Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Boston University.