Algebraic Construction of The Cantor Set William Obeng-Denteh, Valentine Achegbe Hornuvo Department of Mathematics Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Email:
[email protected] [email protected]
Abstract The middle-third stages of the Cantor set and others were constructed step-by-step. In this paper, two algorithms together with the algebraic method, were derived as a faster way to construct the Cantor set.
Keywords:
Cantor set, general case algorithm, middle case algorithm , line segment, algebraic methods, direct substitution, unit interval, dimension moment, fractal dimension, self-similarity, fractal dimension, Hausdorff dimension
1
Introduction
Georg F. L. P. Cantor (1845 − 1918) in 1883, formed the sequence 0, 13 , 23 , 1 by dividing the unit 1 2 , interval B0 = [0, 1] into three equal segments. He removed the open middle set to form 3 3 h i h i the union B1 = 0, 31 ∪ 32 , 1 of two closed sets which are similar to the unit interval. Cantor h i h i again, removed the open middle sets 19 , 92 and 79 , 89 from 0, 31 ∪ 23 , 1 to get another union h i h i h i h i B2 = 0, 19 ∪ 29 , 31 ∪ 23 , 79 ∪ 89 , 1 which is similar to B1 [4][5]. Cantor continued the recursive process which led to the formation of the Cantor dust [4]. The set left after the recursive removal of middle open segments is the Cantor set because, Cantor first introduced it in 1883 after Henry J. S. Smith (1826 − 1883) discovered it in 1874. It is called Cantor ternary set because, it consists of all real numbers in the unit interval that can be expressed as a ternary fraction using 00 s and 20 s only. n
The Cantor set has deep and remarkable properties such as self-similarity. That is, it produces 2 th copies of itself at the n stage (self-similarity). The Cantor set has a fractal dimension and Hausdorff ln(2) th dimension of = 0.631 and its dimension moment at the n stage is ln(3) n P 1 df x2i = 1, df x2i = n , i = 1, 2, . . . , n (conservation law) [3][4]. It has topological and analytical 3 i=0 1
ln(2) = logr 2, where r ln(r) represents the open segment that is being removed. Dimension moment of a general Cantor set is n P df 1 df x2i = 1, x2i = n , i = 1, 2, . . . , n, where j is the number of segments on the unit given as j i=0 interval [0, 1]. properties as well. Generally, a Cantor set has a Hausdorff dimension of
Cantor sets appear in fields such as mechanics where quantum mechanics is defined on a Cantor set using a stochastic motion. They also appear in fields where where sequential search methods are used to optimize objective functions. It is also useful in in decision-making processes tied to collection development. In the following section we used two algorithms with an algebraic approach to construct elements of the Cantor set based on the formulae in [1] and [2].
2 2.1
The Cantor Set Formula General Case Algorithm th
For predetermined values 2 ≤ i ≤ j − 1, k = j + 1 we want to construct the i − j Cantor set using the General Case Algorithm for k points in the interval [0, 1]. th
Problem: To construct the i − j Step 1: Set B0 = [0, 1], Step 2: Compute x =
th
th
stages of the
stages of the Cantor set
t = n∗0
i i i−1 , y= and z = 1 − j j j
Step 3: Do until n = t ∞
{Bn }n=1 = xBn−1 ∪ [y + zBn−1 ] Exit Do t
{Bn }n=1 = union xBn−1 , [y + zBn−1 ]
Loop Step 4: Stop Hence, if B0 = [0, 1], then B1 = xB0 ∪ [y + zB0 ] = x[0, 1] ∪ [y + z[0, 1]] = [0, x] ∪ [y, (y + z) = 1]
(1)
B2 = xB1 ∪[y+zB1 ] = x[[0, x] ∪[y, 1]] ∪[y+z[[0, x] ∪[y, 1]]] = [0, x2 ]∪[xy, x]∪[y, y+zx] ∪[y+zy, 1] (2)
B3 = xB2 ∪[y+zB2 ] = x[[0, x2 ]∪[xy, x]∪[y, y+zx] ∪[y+zy, 1]]∪[y+z[[0, x2 ]∪[xy, x]∪[y, y+zx] ∪[y+zy, 1]]] = [0, x3 ] ∪ [x2 y, x2 ] ∪ [xy, xy + zx2 ] ∪ [xy + xyz, x] ∪ [y, y + zx2 ] ∪ [y + xyz, y + xz]∪ 2
[y + yz, y + yz + xz 2 ] ∪ [y + yz + yz 2 , 1]
(3)
The process continues, culminating into ∞
{Bn }n=4 = x(B(n−1) ) ∪ [y + z(B(n−1) )], n ∈ N 2.1.1
(4)
Construction By Direct Substitution
By substituting the values of x, y and z obtained at Step 2 directly in to the equations (1), (2), (3) th the general j sets can be constructed. That is 3
(x, y, z) → [(1), (2), (3)] = {Bn }n=1 = x(B(n−1) ) ∪ [y + z(B(n−1) )], n ∈ N 2.1.2
Example 1 th
Problem: To construct the 85 Step 1: B0 = [0, 1],
i = 85,
th
− 999
j = 999
set for t = 3 (i.e. 3 iterations) t=3
Step 2: Set x=
28 85 85 914 85 − 1 = , y= and z = 1 − = 999 333 999 999 999
Step 3: Since n = 1 < t = 3, B1 =
28 B 333 0
∪
h
85 999
+
914 B 999 0
i
=
28 [0, 1] 333
∪
h
85 999
+
i
914 [0, 1] 999
h i h i 28 85 = 0, 333 ∪ 999 ,1
Since n = 2 < t = 3, i h ii h hh i h iii 28 85 85 914 28 85 0, 333 ∪ 999 , 1 ∪ 999 + 999 0, 333 ∪ 999 ,1 h i h i h i h i 784 28 2380 85 53897 162605 =⇒ B2 = 0, 110889 ∪ 332667 , 333 ∪ 999 , 332667 ∪ 332667 , 1
B2 =
28 333
hh
Since n = 3 = t, we shall stop after this iteration. B3 = h
28 B 333 2
∪
h
85 999
+ h
914 B 999 2
i
h i i i h i 66640 2380 4552940 21952 784 1509116 28 =⇒ B3 = 0, 36926037 ∪ 110778111 , 110889 ∪ 332667 , 110778111 ∪ 332334333 , 333 ∪ h i h i h i h i 85 10142141 30452015 53897 162605 77538553 233451055 , ∪ , ∪ , ∪ , 1 999 110778111 332334333 332667 998001 332334333 997002999 Step 4: Stop the operation since n = t
3
2.2
Middle Case Algorithm
We want to construct the middle-j for k points in the interval [0, 1]. th
Problem: To construct the i − j
th
th
stages of the Cantor set using the Middle Case Algorithm
stages of the Cantor set.
Step 1: For a predetermined odd value j, Set k = j + 1, Step 2: Compute x =
B0 = [0, 1],
t = n∗0
k−2 k and y = 2j 2j
Step 3: If j is even, go to Step 4 (And use the general case) Else, Do until n = t ∞
{Bn }n=1 = xBn−1 ∪ [y + xBn−1 ] Exit Do
t
{Bn }n=1 = union xBn−1 , [y + xBn−1 ]
Loop End if Step 4: Stop Therefore if B0 = [0, 1], B1 = xB0 ∪ [y + xB0 ] = x[0, 1] ∪ [y + x[0, 1]] = [0, x] ∪ [y, 1]
(5)
2
B2 = xB1 ∪ [y + xB1 ] = [0, x2 ] ∪ [xy, x] ∪ [y, x + y] ∪ [xy + y, 1] 3
2
2
3
2
(6)
3
B3 = xB2 ∪ [y + xB2 ] = [0, x ] ∪ [x y, x ] ∪ [xy, x + xy] ∪ [x y + xy, x] ∪ [y, x + y] ∪ 2
2
3
2
[x y + y, x + y] ∪ [xy + y, x + xy + y] ∪ [x y + xy + y, 1]
(7)
This culminates into {Bn }∞ n=4 = xBn−1 ∪ [y + xBn−1 ], n ∈ N 2.2.1
(8)
Construction By Direct Substitution
By substituting the values of x and y obtained at Step 2 directly in to the equations (5), (6), (7) th the middle j sets can be constructed. That is (x, y) → [(5), (6), (7)] = (8) 2.2.2
Example 2 th
To construct the middle 999
sets (self-similar), Set Consider the following Problem.
Problem: To construct the middle 999
th
set for t = 3 (i.e. 3 iterations) 4
Step 1: B0 = [0, 1],
j = 999,
k = 1000
t=3
Step 2: Set 499 1000 500 1000 − 2 = and y = = 1998 999 1998 999
x=
Step 3: Since n = 1 < t = 3, and j is odd, B1 =
499 B 999 0
∪
h
500 999
+
499 B 999 0
i
=
499 [0, 1] 999
∪
h
500 999
+
i
499 [0, 1] 999
h
= 0,
499 999
i
∪
h
i
500 ,1 999
Since n = 2 < t = 3, i h ii h hh i h iii 500 500 499 499 500 0, 499 ∪ , 1 ∪ + 0, ∪ , 1 999 999 999 999 999 999 h i h i h i h i 249500 499 500 748501 749000 =⇒ B2 = 0, 249001 ∪ , ∪ , ∪ , 1 998001 998001 999 999 998001 998001
B2 =
499 999
hh
Since n = 3 = t, we shall stop after this iteration.
h
124251499 997002999
i
B3 = h
499 B 999 2
∪
h
124500500 249001 , 997002999 998001
500 999
i
+ h
499 B 999 2
i
249500 373501999 , 998001 997002999
i
h
373751000 499 , 997002999 999
i
∪ ∪ ∪ ∪ i h i h i h i 623501000 748501 749000 872502499 872751500 500 623251999 ∪ 997002999 , , 998001 ∪ 998001 , 997002999 ∪ 997002999 ,1 999 997002999
=⇒ B3 = 0, h
Step 4: Stop the operation since n = t
3
Concluding Remarks
The Middle Case algorithm and General case Algorithm can be applied using algebraic methods to construct the Cantor Set. The Middle Case Algorithm can be applied to delete open middle segments for any k points on [0, 1] while the General Case Algorithm can be applied to delete any open subinterval, except the first and the last for k points in [0, 1]. The approach used is one of the fastest way to construct the Cantor set. The two algorithms can be advanced in order to optimize accuracy (min. inaccuracy or max. accuracy) in construction due to multiple results obtained at higher iterations. We want to conclude with the following question. Was fission imposed on [0, 1] to obtain the Cantor set? What kind of fission?
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References [1] Hornuvo, V., & Obeng-Denteh, W. (2018 Aug 27). THE GENERAL KTH CONCEPT OF THE CANTOR SET FORMULA. Journal of Mathematical Acumen and Research 3(2):1-4 ISSN:2467-8929 [2] Hornuvo, V., & Obeng-Denteh, W. (2018 Jan 31) EXTENSION OF THE CANTOR SET: THE MIDDLE KTH CONCEPT. Journal of Mathematical Acumen and Research 3(1):1-3 ISSN:2467-8929 [3] Kohavi,Y., Davdovich, H. (May 2006) Topological dimensions, Hausdorff dimensions & fractals [4] Kolman, B., Hill, D. R. (2004). Introductory Linear Algebra: An Applied first Course (8th Edition) ISBN-13:978-0131437401 ISBN-10: 0131437402 [5] Obeng-Denteh, W., Amoako-Yirenkyi, P., & Owusu Asare, J. (2016). Cantor Ternary Set Formula-Basic Approach, British Journal of Mathematics and Computer Science, 13(1):1-6, DOI: 10.9734/BJMCS/2016/21435
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