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Figure 6. Construction of the middle-3/7 Cantor set. Repeating this process, the limiting set C, = K,, can be called the Cantor middle-3/7 set and K; consists of 4° intervals of length a In general, K,, consists of 4” intervals, each of length — Thus the dimension of the Cantor middle-3/7 set is

Figure 6 Construction of the middle-3/7 Cantor set. Repeating this process, the limiting set C, = K,, can be called the Cantor middle-3/7 set and K; consists of 4° intervals of length a In general, K,, consists of 4” intervals, each of length — Thus the dimension of the Cantor middle-3/7 set is

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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. To construct this set (denoted by C3), we begin with the interval [0,7] and remove the open set (3) from the closed interval [0,7]. The set of points that remain after this first step will be called K,, that is, K, = [o, ;| U [=, 1]. In the second step, we remove the middle thirds of the two segments of K;, that is, 1 2 7 8 1 2 3 67 8 remove (5,5) U(5,5) and set k= [0,5]u[5.5[u[5.¢]u [>| be what remains after the first two steps. Delete the middle thirds of the four remaining segments of K> to get Kz. Repeating this process, the limiting set C3 = K., is called the 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. To construct this set (denoted by C3), we begin with the interval [0,7] and remove the open set (3) from the closed interval [0,7]. The set of points that remain after this first step will be called K,, that is, K, = [o, ;| U [=, 1]. In the second step, we remove the middle thirds of the two segments of K;, that is, 1 2 7 8 1 2 3 67 8 remove (5,5) U(5,5) and set k= [0,5]u[5.5[u[5.¢]u [>| be what remains after the first two steps. Delete the middle thirds of the four remaining segments of K> to get Kz. Repeating this process, the limiting set C3 = K., is called the Cantor middle 1/3 set.
One of the most important properties of a fractal is known as self-similarity[5]. Roughly speaking, self-similarity means i examine small portions of the set under a microscope, we the image we see resembles our original set. To see this let us look closely at C3. Note tha subsets, the portion of C3 in [0, 1/3 [2/3, 1]. If we examine eac C3 is decomposed into two dis h of these pieces, we see that inct and the portion in hey resemble the original Cantor set C;. Indeed, each is obtained by removing middle-thirds the original interval is smal magnify each of these po obtain the original set. More precise ler by a factor of 7/3. Thus, i rtions of C3 by a factor of 3, portions of C3, we use ana of intervals. The only difference is we we y, to magnify these fine transformation. Let L(x) = 3x. If we apply L to the portion of C3 in [0 7/3], we see that L maps this portion onto the entire Cantor set. Indeed, L 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 C3 in [0, 1/3] is taken by L to a gap in C3. That is, the “microscope” we use to magnify C3 [0,1/3] is just the affine transformation L(x) = 3x. To magnify the other half of C3, namely C3N [2/3,1], we use another affine transformation, R(x) =3x-—2. Note that R(2/3) =0 and R(1)=1 so R takes [2/3, 7] linearly onto [O, 1]. As with L, R takes gaps in C3 Nn [2/3,1] to gaps in C3, so R again magnifies a small portion of C3 to give the entire set. Using more powerful “microscope”, we may magnify arbitrarily small portions of C3; to give the entire set. For example, the portion of C3 in [0, 1/3] itself decomposes into two self-similar pieces: one in [0, 7/9] and one in [2/9, 1/3]. We may magnify the left portion via L2(x) = 9x to yield C3 and the right portion via R,(x) = 9x — 2. Note that R, maps [2/9, 1/3] onto [0 7] linearly as required. Note also that at the nth stage of the construction of C3, we have 2” small copies of C3, each of which may be magnified by a factor of 3” to yield the entire Cantor set.
One of the most important properties of a fractal is known as self-similarity[5]. Roughly speaking, self-similarity means i examine small portions of the set under a microscope, we the image we see resembles our original set. To see this let us look closely at C3. Note tha subsets, the portion of C3 in [0, 1/3 [2/3, 1]. If we examine eac C3 is decomposed into two dis h of these pieces, we see that inct and the portion in hey resemble the original Cantor set C;. Indeed, each is obtained by removing middle-thirds the original interval is smal magnify each of these po obtain the original set. More precise ler by a factor of 7/3. Thus, i rtions of C3 by a factor of 3, portions of C3, we use ana of intervals. The only difference is we we y, to magnify these fine transformation. Let L(x) = 3x. If we apply L to the portion of C3 in [0 7/3], we see that L maps this portion onto the entire Cantor set. Indeed, L 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 C3 in [0, 1/3] is taken by L to a gap in C3. That is, the “microscope” we use to magnify C3 [0,1/3] is just the affine transformation L(x) = 3x. To magnify the other half of C3, namely C3N [2/3,1], we use another affine transformation, R(x) =3x-—2. Note that R(2/3) =0 and R(1)=1 so R takes [2/3, 7] linearly onto [O, 1]. As with L, R takes gaps in C3 Nn [2/3,1] to gaps in C3, so R again magnifies a small portion of C3 to give the entire set. Using more powerful “microscope”, we may magnify arbitrarily small portions of C3; to give the entire set. For example, the portion of C3 in [0, 1/3] itself decomposes into two self-similar pieces: one in [0, 7/9] and one in [2/9, 1/3]. We may magnify the left portion via L2(x) = 9x to yield C3 and the right portion via R,(x) = 9x — 2. Note that R, maps [2/9, 1/3] onto [0 7] linearly as required. Note also that at the nth stage of the construction of C3, we have 2” small copies of C3, each of which may be magnified by a factor of 3” to yield the entire Cantor set.
Figure 3. Construction of the Cantor middle- 7/5 set. Again, we remove the middle fifth portion of each of the 2 closed intervals of K, to get K3. Repeating this process, the limiting set C; = K,, is called the Cantor middle- 71/5 set. The set C; is the set of points that belongs to all of the K,,.
Figure 3. Construction of the Cantor middle- 7/5 set. Again, we remove the middle fifth portion of each of the 2 closed intervals of K, to get K3. Repeating this process, the limiting set C; = K,, is called the Cantor middle- 71/5 set. The set C; is the set of points that belongs to all of the K,,.
Figure 4. Construction of the middle- 1/7 Cantor set. Again, we remove the middle 7th portion of each of the 2 closed intervals of K, to get K3. Repeating this process, the limiting set C; = 6,, is called the Cantor middle-1/7 set. The set C; is the set of points that belongs to all of the K,,.
Figure 4. Construction of the middle- 1/7 Cantor set. Again, we remove the middle 7th portion of each of the 2 closed intervals of K, to get K3. Repeating this process, the limiting set C; = 6,, is called the Cantor middle-1/7 set. The set C; is the set of points that belongs to all of the K,,.
Figure 5. Construction of the middle-2/5 Cantor set. Repeating this process, the limiting set C; = K,, can be called the Cantor middle-2/5 set and K; consists of 3° intervals of
Figure 5. Construction of the middle-2/5 Cantor set. Repeating this process, the limiting set C; = K,, can be called the Cantor middle-2/5 set and K; consists of 3° intervals of
Figure 7.Construction of the middle-2/4 Cantor set. 6.2.Removing the middle segments (except two end segments). Repeating this process, the limiting set C, = K,, can be called the Cantor middle-2/4 set and K3 consists of 2° intervals of 7 CONCLUSION
Figure 7.Construction of the middle-2/4 Cantor set. 6.2.Removing the middle segments (except two end segments). Repeating this process, the limiting set C, = K,, can be called the Cantor middle-2/4 set and K3 consists of 2° intervals of 7 CONCLUSION
Figure 8.Construction of the middle-3/5 Cantor set. Repeating this process as above,K,, consists of 2” intervals, each of length = Thus the dimension of the Cantor middle- 3/5 set C- as
Figure 8.Construction of the middle-3/5 Cantor set. Repeating this process as above,K,, consists of 2” intervals, each of length = Thus the dimension of the Cantor middle- 3/5 set C- as
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