Theory of Carry Value Transformation (CVT) and its Application in Fractal formation

Theory of Carry Value Transformation (CVT) and it’s Application in Fractal formation Pal Choudhury Pabitra 1 , Sahoo Sudhakar 1 , Nayak Birendra Kumar 2 , and Hassan Sk. Sarif 2 1, Applied Statistics Unit, Indian Statistical Institute, Kolkata, 700108, INDIA 2. P.G. Department of Mathematics, Utkal University, Bhubaneswar- 751004 Email: pabitrapalchoudhury@gmail.com, sudhakar.sahoo@gmail.com, bknatuu@yahoo.co.uk & sarimif@gmail.com Abstract . In this paper the theory of Carry Value Transformation (CVT) is designed and developed on a pair of n-bit strings and is used to produce man y interesting patterns. One of them is found to be a self- similar fractal whose dimension is same as the dimension of the Sierpinski triangle. Different construction procedures like L-system, Cellular Automata rule, Tilling for this fractal are obtained which signifies that like other tools CVT can also be used for the formation of self-similar fractals. It is shown that CVT can be used for the production of periodic as well as chaotic patterns. Also, the analytical and algebraic properties of CVT are discussed. The definition of CVT in two-dimension is slightly modified and its mathematical properties are highlighted. Finally, the extension of CVT and modified CVT (MCVT) are done in higher dimensions. Keywords - Carry Value Transformation, Fractals, L-S ystem, Cellular Automata and Tilling, Discrete Dynamical System. 1. Introduction Benoit Mandelbrot coined the word fractal from the Latin adjective fractus. The corresponding Latin verb frangere means ‘to break’ to create irregular fragments. The precise definition of “Fractal” according to Benoit Mandelbrot is as a set for which the Hausdroff Besicovitch dimension strictly exceeds the topological dimension [1]. Many things in nature are very complex, chaotic but exhibit some self-similarit y. The complexity of fractals and the property of self-similarit y have a large set of real world applications. Fractals can be generated using construction procedure/algorithms/simply b y repetition of mathem atical formula that are often recursive and ideally suited to computer. In this paper a new transfor mation named as Carr y Value Transformation (CVT) is defined in binar y number system and using CVT we are trying to explore that behind the complexity of nature there remains a simple methodology about which most often we are ignorant. Earlier, in [2] we have used CVT with Cellular Automata in efficient hardware design of some basic arithmetic operations. But in this paper although we emphasize the formation of self- similar-fractals, the algorithm using CVT also produces periodic and chaotic patterns. The underlying develop ment of CVT is same as the concept of Carr y Save Adder (CSA) [3] where carr y or overflow bits generated in the addition process of two integers are saved in the memory . Here we perfor m the bit wise XOR operation of the operands to get a string of sum-bits (ignoring the carry-in) and simultaneousl y the bit wise ANDing of the operands to get a string of carr y-bits, the latter string is padded with a ‘0’ on the right to signify that there is no carry-in to the LSB. The organization of the paper is as follows. Section II discusses some of the basic concepts on fractals, L-systems, tilling problem, Cellular Automata etc. which are used in the subsequent sections. The concept of CVT is defined in section III. It can be found in section IV that CVT generates a beautiful self-similar fractal whose dimension is found to be same as that of monster f ractal, Sierpinski triangle. Section V deals with the various way s like L-System, Cellular Automata and Tillings by which the same fractal in binar y number system can be constructed. CVT can also be used for the production of pe riodic as well as chao tic patterns ar e shown in section VI. The a nal ytical and algebraic properties of CVT are discussed in section VII. The definition of CVT in two-dimension is slightl y modified and its mathematical properties are highlighted in section VIII. Section IX deals with the extension of CVT and modified CVT (MCVT) in higher dimensions. On highlighting other possible applications of CVT and some future research directions a conclusion is drawn in section X. It should be noted that a preliminary version of this paper has been published in an international conference [13]. 2. Fractal Basics This sections deals with the basics of Fractals [1, 4, 5] and the various wa ys the fractals may be constructed like L- Systems [6], Cellular Automata [7, 8] and s y nthesis of Tilling [9] etc. I t is intended for readers who are not conversant with the fundamentals of these concepts. 2.1 Why we study fractals We feel ver y much worried due to our inabilit y to describe using the traditional Euclidean Geo metry , the shape of cloud, a mountain, a coa stline or a tree. In n ature, clouds are not rea lly spherical, mountains are not conical, coa stlines are not circular, even the lightning doesn’t travel in a straight line. More generally, we could be able to conclude that many patterns of natu re are so irregular and fragm ented, that, co mpared with Euclid Geometry –a ter m, can be used in this regard to denote all of standard geometr y. Mathematicians have over the y ears disdained this challenge and have increasingly chosen to flee nature b y devising theories unrelated to natural objects we can see or feel. After a long time, responding to this challenge, Benoit Mandelbrot developed a new geometr y of nature and implemented its use in a number of diverse arenas of science such as Astronomy, Biology, Mathematics, Physics, and Geography and so on [1, 4, 5, 10, 11]. This new-born geometry can describe many of the irregular and fragmented (chaotic) patterns around us, and leads to full-fledged theories, by identif y ing a family of shapes, now-a-da ys which we people call ‘FRACTALS’. 2.2 Measuring Fractal dimension The fractal dimension alone does not give an idea of what “fractals” are really about Mandelbrot founded his insights in the idea of self similarit y, requiring that a true fractal “fracture” or break apart into smaller pieces that resemble the whole. This is a special case of the idea that there should be a d yn amical system underl ying the geometry of the set. This is partly wh y the idea fractals have become so popular throughout science; it is a fundamental aim of science to seek to understand the underl ying dynamical properties of any natural phenomena. It has now beco me apparent that relatively simple d y namics, more precisel y d yn amical system can produce the fantasticall y intricate shapes and behavior that occur throughout nature. Now let us tr y to define what fractal dimension (self-similarity dimension) is. Given a self-similar structure [1], there is a relation between the reduction factor (scaling factor) ‘S’ and the number of pieces ‘N’ into which the structure can be divided; and that relation is… N =1/S D , equivalently, D =log (N)/log (1/S) This ‘D’ is called the Fractal di mension (Self-similarity dimension) 2.3 Ways to construct fractals 2.3.1 Lindenmayer Systems (L-system) produces fractals As a biologist, Ar istid Lindenmayer [6] studied growth patterns in various types of algae. In 1968 he developed Lindenmay er s y stems (or L-Systems) as a mathematical formalism for describing the growth of simple multi-cellular organisms. The central concept of L-System is that of rewriting. In g eneral rewriting is a technique for defining complex object by successively replacing parts of a simple initial object using a set of production rules. Definition of an L-System: An L-system is a formal grammar consisting of 4 parts: A set of variables : s y mbols that can be replaced b y production rules. A set of constants : symbols that do not get replaced. An axiom , which is a string, composed of some number of variables and/or constants. The axiom is the initial state of the system. A set of production rules defining the way variables can be replaced with combinations of constants and other variables. A production consists of two strings - the predecessor and the successor. 2.3.1 Fractals by Cellular Automata rules The scientific output of Wolfram’s [7, 8] work played a central role in launching Cellular Automata (CA) as a new field of science to understand the complexit y of nature. Starting from an initial seed he studied the space-time diagram of all the 256 three-neighborhood ele mentary CA rules and classified the rules into four distinct classes according to the complexity of the pattern. According to him the Cla ss 2 rules deals with the periodic and fractal patterns. In 1-D the global state or si mply state of a CA at any time-instant t is represented as a vector X t = ( x 1 t , x 2 t ,…., x n t ) where x i t denotes the bit in the i th cell x i at time-instant t . If the “present st ate” of an n -bit CA (at time t ) is X t , its “ne x t state” (at ti me t+1 ), denoted by X t+1 , is in general given b y the global mapping F( X t ) = ( f 1 ( lb t , x 1 t , x 2 t ), f 2 ( x 1 t , x 2 t , x 3 t ),…, f n ( x n -1 t , x n t , rb t ) ), where f i is a local mapping to the i th cell and lb and rb denote respectivel y the left boundary of x 1 and right boundary of x n incase of periodic boundar y CA and those values are 0 in case of null boundar y CA . If the same local mapping (rule) determines the “ne x t” bit in each cell of a CA, the CA will be called a Unifor m CA, otherwise it will be called a H yb rid CA. For our purpose, we have used one-way CA , which allows only one-way co mmunication, i.e., in a 1-D arra y each cell depends only on itself and its left neighbor. One can also consider dependence on the cell and its right neighbor. However both sides dependence is not allowed. Just like L-system and Cellular Automata, fractals can also be obtained by Iterated Function Systems [4] and using different Tiles [9]. A complex figure can be easily (in most of the cases) synthesized b y using of tiles. Next section discusses a new and efficient construction tool named as CVT by which uncountable number of fractal patterns can be generated. 3. Carry Value Transformation (CVT) The carry or overflow bits are usuall y generated at the time of addition between two n-bit strings. In the usual addition process, carr y value is always a single bit and if generated then it is added column wise with other bits and not saved in its own place. But the car ry value defin ed here are the usual car ries generated bi t wise and stored in their respective places as shown in “Fig. 1”. 1 1 1 1 1 1 1 1 1 1 ............................... 0 ....................... ....................... ............ n n n n n n n n n n carry value c c c a a a a b b b b a b a b a b a b − − − − − = = = ⊕ = ⊕ ⊕ ⊕ [Figure 1: Carry generated in i th column is saved in (i-1) th column] Thus to find out the carr y value we perfor m the bit wise XOR operation of the operands to get a string of sum-bits (ignoring the carr y -in) and simultaneously the bit wise ANDing of the operands to get a string of carr y-bits, the latter string is padded with a ‘0’ on the right to signify that there is no carry-in to the LSB. Thus the corresponding decimal value of the string of carry bits is alwa ys an even integer. Now we can give a precise definition of CVT as follows: Let { 0 ,1 } B = and CVT is a mapping defined as 1 : ( ) n n n CVT B B B + × → where n B is the set of strings of length n on { 0,1 } B = . More specifically, if 1 1 1 1 ( , , ..., ) ( , , ..., ) n n n n a a a a and b b b b − − = = then 1 1 1 1 ( , ) ( , , ..., , 0) n n n n CVT a b a b a b a b − − = ∧ ∧ ∧ is an (n+1) bit string, belonging to set of non-negative integers, and can be computed bit wise by logical AND operation followed b y a 0, which denotes no carr y is generated in the LSB at the ti me of addition procedure. In other words, CVT is a mapping fromwhere is set of non-negative integers. Illustration: Suppose, we want the CVT of the nu mbers (13) 10 ≡ (1101) 2 and (14) 10 ≡ (1110) 2 . Both are 4-bit nu mbers. The carr y value is computed as follows: Carry: 1 1 0 0 0 Augend: 1 1 0 1 Addend: 1 1 1 0 XOR: 0 0 1 1 [Figure 2: Carry generated in i th column is saved in (i-1) th column] Conceptually, in the general addition process the carry or overflow bit fro m each stage (if any) goes to the next stage so that, in each stage after the first (i.e. the LSB position with no carry-in), actually a 3-bit addition is performed instead of a 2-bit addition by means of the full adder. Instead of going for this traditional method, what we do is that we perfor m the bit wise XOR operation of the operands (ignoring the carr y -in of each stage from the previous stage) and simultaneously the bit wise ANDing of the operands to get a string of carr y-bits, the latter string is padded with a ‘0’ on the right to signify that there is no carr y -in to the LSB (the overflow bit of this ANDing being alwa ys ‘0’ is simply ignored). In our example, bit wise XOR gives (0011) 2 ≡ (3) 10 and bit wise ANDing followed by zero-padding gives (11000) 2 ≡ (24) 10 . Thus ( 1101 ,1110) 11000 CVT = and equivalently in decimal notation one can write ( 13 ,14) 24 CVT = . In the next section we have used the carr y value in decimal to construct the CV table. 4. Generation of Self-fractal using CVT A table is constructed that contains only the carr y values (or even terms) defined above between all possible integers a’s and b’s arranged in an ascending order of x and y-axis respectively. We observe some interesting patterns in the table. We would like to make it clear how the CV-table i s constructed. Step 1 . Arrange a ll the integers 0 1 2 3 4 5 6 ... (as long as we want) in ascending order and place it in both, uppermost row and leftmost column in a table. Step 2 . Compute ( , ) CVT a b as mentioned in section III and store it in deci mal form in the (a, b) position. Step 3 . Then we look on the pattern of an y integer, and we have made it color. This shows a very beaut iful consistent picture, which we see as a fractal as shown in table I followed by “Fig. 3”. Choosing different sequence of rows and columns in the CV- table in step 1, one can obtain uncountable nu mber of patterns only b y co mputing a single operation ( , ) CVT a b in each entr y position (a, b) of the table. Thus one of the advantages of CVT over other construction tools lies in the number of cell evolutions for the for mation of fractals. In this case only one uniform evolution is required at each cell position to obtain the required fractal pattern where as in case of L-System, Cellular Auto mata, Iterated Function System etc many iterations are required. Illustration: [Figure 3: A fractal structure on using CVT of different integer values] [Figure 4: Shows the fractal generated b y the CVT] [Figure 5: Shows the hierarchy to generate the above fractal] We have analyzed the table I and found some interesting pa tterns as follows: 1. Diagonal values are having one type of pattern contains all possible even integers 0, 2, 4, 6, 8…etc. 2. Starting from (0,0) position one can construct a (2x2) matrix b y filling the values in its right, bottom and diagonal (bottom-right) positions and recursivel y this will leads to (4x4), (8x8), (16x16) m atrices…etc that can be seen in “Fig. 4”. 3. A recursive pattern exists in each square block of size (2 k x2 k ) for k=0, 1, 2, 3…etc. starting from (0, 0) position and if this block is partitioned in the middle (both in row and column) into 4 equal sub blocks of size (2 k-1 x2 k-1 ) then also the same pattern can be observed. If the 4-blocks are treated as 4 quadrants (1 st , 2 nd , 3 rd and 4 th ) in the two dimensional Euclidian space then the patterns in first three quadrants are exactly same and in the 4 th quadrant the pattern is same only values are dif ferent and that can also be easily constructed by adding 2 k to each element of an y of the three quadrants. That is if ij a denote the entry of th i row and th j column of the CV-table of size (2 k x2 k ), then { 1 1 1 ( mod 2 )( mod 2 ) 1 1 1 ( mod 2 )( mod 2 ) 0 , 2 2 2 , 2 k k k i j k k k k k i j a for i j ij a for i j a − − − − − − ≤ < + ≤ < = 4. One can partition a square block of size (2 k x2 k ) for k=0, 1, 2, 3…etc. into two different c lasses. Where the elements of 1 st , 2 nd and 3 rd are in Class-I and the elements of 4 th quadrant is in Class-II. Further Class-I elements can be obtained from Class-II by element wise modulo 2 k operations ( ( ) mod 2 k ij b a = ). Thus 2 k can act as a pivot. 5. Take, a sequence of non-negative integers from 2 n Z (n≥0), find the CV table. Observe that for any n≥0, after infinitely many steps we will be having the same fractal what we have got earlier as shown above. 6. Take, an odd sequence i.e. {1, 3, 5, 7, 9, 11, 13, 15…} and try to find the CV table. Observe the pattern of zeros. This pattern is same as what we have found earlier in above. Now, if we multipl y the sequence by 2 n (n≥0), then this sequence also lead to the sa me pattern. Literally, one can say that Sierpinski gasket (fractal) is a ver y stable fractal. Dimension of this fractal For this fractal, N=3, S=1/2, where l is the initial l ength. Fractal dimension D is given b y … 3=1/(1 / 2) D Or D= log3/log2 ≈ 1.585 This is same as the dimension of Sierpinski triangle. Thus CVT fractal as obtained b y us can be regarded as a relative to Sierpinski triangle [5]. In [12], eleven lists of ways are given for the construction of Sierpinski gasket (also, the Sierpinski triangle) and the author conjectured that there are undoubtedly more. Thus CVT defined in this paper is another way for the construction of this kind of fractal. Next section discusses other various way s to construct the self-similar CVT fractal as obtained in “Fig 3”. 5. Various ways to construct the CVT fractal 5.1 L- System to generate CVT fractal Let F 1, F 2 denote horizontal and vertical line segments respectivel y . Starting from the axiom as shown in table II and “Fig. 5” two rules are defined which produces the above fractal gene rated by CVT, after infinitel y m any iterations. Table 1. A table with fractal dimension of fractals according as base of the number s ys tem L-System for CVT fractal Variables: F 1 , F 2 Constants: - Axiom: F 2 –F 1 Rules: F 1 = F 1 -F 2 - - F 2 -F 1 F 2 =F 2 -F 1 - - F 1 -F 2 Angle increment: 90 degrees [Figure 6: Ssyntactic representation of Axio m and Rules] [Figure 7: L-System is used to generate the above fractal] 5.1 Cellular Automata rules to construct CVT fractal In [4] we found that the fractal generated from one-dimensional, two-neighborhood, binar y Cellular Automata rule starting from an initial sheet, if rotated by an angle of 180 degree is sa me as our CVT fractal. According to Wolfram naming convention [10], [11] this is rule number 6, whose truth table and initial sheet on which rule 6 would be applied are given below… [Figure 8: The rules and the initial seed of CA to generate the CVT fractal] After finitely many steps, the space-time pattern for the above CA rule is shown in “Fig. 8”. [Figure 9: CA evolution from the initial seed generates t he CVT fractal] “Fig. 9” shows the same CA evolution by replacing 1 by a black cell and 0 by a white cell as usual done b y Wolfram. Now by flipping verticall y (same as the vertical rotation) this CA evolution we get the same CVT fractal as shown in “Fig. 3”. [Figure 10: The rules and the initial seed of CA to generate the CVT fractal] 5.1 Synthesis of CVT fractal by Tilling The constituent parts on synthesis can give rise to the fractal picture. Here we have used four ke y tiles those are used to generate CVT fractal shown in “Fig. 10”. (a) (b) [Figure 11: (a) shows four key tiles those are used to generate the CVT fractal and (b) shows the arrangements of these keys.] Different ways of construction of the self-si milar CVT fractal signifies that like other tool s CVT can also be used as a construction tool for the formation of different kinds of self-similar fractals. Next section shows that not onl y the CV- table produces self-similar fractals but CVT can also be used to generate periodic as well as chaotic patte rns. 5. CVT for generating Periodic and Chaotic patterns Already we have shown in section 4, that CVT can produce a self-similar pattern (fractal). Here we would like to produce another type of source to have a periodic pattern. For t hat, first of all we need a periodic sequence of nu mbers where we can be able to apply CVT to obtain the pattern. And we have a renowned domain of such sequences those are obtained considering decimal representation of rational numbers. Let us consider an example of rational number 1/7. The decimal representation of 1/7 is 0. 142857142857142857142857 142857 … (142857 is being repeated). Taking the above decimal representation as a sequence of integers, let us construct the CV- Table in binary number system as follows… 0 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 4 0 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 2 0 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 8 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 5 0 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 1 0 2 8 0 0 10 10 7 0 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 1 4 2 8 4 0 10 14 1 0 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 4 0 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 2 0 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 8 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 1 6 0 0 5 0 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 1 0 2 8 0 0 10 10 7 0 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 1 4 2 8 4 0 10 14 1 0 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 4 0 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 2 0 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 8 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 1 6 0 0 5 0 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 1 0 2 8 0 0 10 10 7 0 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 1 4 2 8 4 0 10 14 1 0 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 4 0 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 2 0 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 8 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 1 6 0 0 5 0 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 1 0 2 8 0 0 10 10 7 0 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 1 4 2 8 4 0 10 14 1 0 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 4 0 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 2 0 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 8 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 1 6 0 0 5 0 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 1 0 2 8 0 0 10 10 7 0 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 1 4 2 8 4 0 10 14 1 0 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 2 0 0 0 2 2 4 0 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 0 8 0 0 8 8 2 0 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 8 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 6 0 0 0 0 0 1 6 0 0 5 0 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 10 2 8 0 0 10 1 0 2 8 0 0 10 10 7 0 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 14 2 8 4 0 10 1 4 2 8 4 0 10 14 [Figure 12: a fractal structure on using CVT of different integer values] Here we have considered t he pattern of zeros, which we have made it colored. Clearl y this pattern of zeros i s periodic. Here, we can conclude that for every rational number we can be able to have periodic pattern using CVT, sometimes it may be followed by some chaotic patterns as if starts f rom an explosion then things becomes stable. That is, here we can encounter a source of countabl y infinite number of almost periodic patterns corresponding to rational numbers because we know that the set of rational numbers is countable and infinite. 5.1 Formation of Chaotic Pattern We have not yet seen that whether CVT can produce a non-periodic or chaotic pattern or not! Now we are ready to demonstrate an example where we could have a chaotic patt ern using CVT. First of all we need a rando m sequence of integers, and then we will be appl y ing CVT as we have applied earlier. Here also we have a good domain of such sequence s, which can be considered the decimal representation of irrational numbers. Let us consider an example of an irrational number √2. The decimal representat ion of √2 is 1.414213562373095048801688 724229698078568671875376948073176679737990…(Non-periodic, non-recurring). Considering the above sequence let us try to construct the CV-tab le in binary number s ys tem as shown in table IV . 1 4 1 4 2 1 3 5 6 2 3 7 3 0 9 5 0 4 8 8 0 1 6 8 8 7 2 4 2 2 9 6 9 8 0 7 8 5 6 8 6 7 1 8 7 5 3 7 6 9 4 8 0 7 3 1 7 6 6 7 9 7 3 7 9 9 0 7 3 2 4 7 5 4 6 2 1 0 7 0 3 9 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 4 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 4 0 8 0 8 0 0 0 8 8 0 0 8 0 0 0 8 0 8 0 0 0 0 8 0 0 8 0 8 0 0 0 8 0 0 0 8 0 8 8 0 8 8 0 0 8 8 0 8 8 0 8 0 0 8 0 0 8 8 8 8 0 8 0 8 0 0 0 8 0 0 8 8 8 8 8 0 0 0 8 0 0 0 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 0 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 5 2 8 2 8 0 2 2 10 8 0 2 10 2 0 2 1 0 0 8 0 0 0 2 8 0 0 10 0 8 0 0 2 8 2 0 0 10 0 10 8 0 8 10 2 0 10 1 0 2 10 8 2 8 0 0 1 0 2 2 10 8 8 10 2 10 2 10 2 2 0 10 2 0 8 10 10 8 8 0 2 0 10 0 2 2 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 4 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 5 2 8 2 8 0 2 2 10 8 0 2 10 2 0 2 1 0 0 8 0 0 0 2 8 0 0 10 0 8 0 0 2 8 2 0 0 10 0 10 8 0 8 10 2 0 10 1 0 2 10 8 2 8 0 0 1 0 2 2 10 8 8 10 2 10 2 10 2 2 0 10 2 0 8 10 10 8 8 0 2 0 10 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 8 0 8 0 0 0 8 8 0 0 8 0 0 0 8 0 8 0 0 0 0 8 0 0 8 0 8 0 0 0 8 0 0 0 8 0 8 8 0 8 8 0 0 8 8 0 8 8 0 8 0 0 8 0 0 8 8 8 8 0 8 0 8 0 0 0 8 0 0 8 8 8 8 8 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 0 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 4 0 8 0 8 0 0 0 8 8 0 0 8 0 0 0 8 0 8 0 0 0 0 8 0 0 8 0 8 0 0 0 4 0 0 0 8 0 8 8 0 8 8 0 0 8 8 0 8 8 0 8 0 0 8 0 0 8 8 8 8 0 8 0 8 0 0 0 8 0 0 8 8 8 8 8 0 0 0 8 0 0 0 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 8 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 0 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 1 8 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 1 8 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 5 2 8 2 8 0 2 2 10 8 0 2 10 2 0 2 1 0 0 8 0 0 0 2 8 0 0 10 0 8 0 0 2 8 2 0 0 10 0 10 8 0 8 10 2 0 10 1 0 2 10 8 2 8 0 0 1 0 2 2 10 8 8 10 2 10 2 10 2 2 0 10 2 0 8 10 10 8 8 0 2 0 10 0 2 2 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 8 0 0 0 0 0 0 0 0 0 0 0 16 0 0 16 0 0 0 1 6 16 0 0 0 16 1 6 1 6 0 0 0 0 16 0 0 16 0 16 16 0 0 16 0 16 0 16 16 0 0 16 0 1 6 0 16 0 16 0 0 16 0 0 16 1 6 16 0 16 1 6 16 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0 16 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 5 2 8 2 8 0 2 2 10 8 0 2 10 2 0 2 1 0 0 8 0 0 0 2 8 0 0 10 0 8 0 0 2 8 2 0 0 10 0 10 8 0 8 10 2 0 10 1 0 2 10 8 2 8 0 0 1 0 2 2 10 8 8 10 2 10 2 10 2 2 0 10 2 0 8 10 10 8 8 0 2 0 10 0 2 2 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 4 0 8 0 8 0 0 0 8 8 0 0 8 0 0 0 8 0 8 0 0 0 0 8 0 0 8 0 8 0 0 0 8 0 0 0 8 0 8 8 0 8 8 0 0 8 8 0 8 8 0 8 0 0 8 0 0 8 8 8 8 0 8 0 8 0 0 0 8 0 0 8 8 8 8 8 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 4 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 4 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 4 0 8 0 8 0 0 0 8 8 0 0 8 0 0 0 8 0 8 0 0 0 0 8 0 0 8 0 8 0 0 0 8 0 0 0 8 0 8 8 0 8 8 0 0 8 8 0 8 8 0 8 0 0 8 0 0 8 8 8 8 0 8 0 8 0 0 0 8 0 0 8 8 8 8 8 0 0 0 8 0 0 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 5 2 8 2 8 0 2 2 10 8 0 2 10 2 0 2 1 0 0 8 0 0 0 2 8 0 0 10 0 8 0 0 2 8 2 0 0 10 0 10 8 0 8 10 2 0 10 1 0 2 10 8 2 8 0 0 1 0 2 2 10 8 8 10 2 10 2 10 2 2 0 10 2 0 8 10 10 8 8 0 2 0 10 0 2 2 0 4 0 8 0 8 0 0 0 8 8 0 0 8 0 0 0 8 0 8 0 0 0 0 8 0 0 8 0 8 0 0 0 8 0 0 0 8 0 8 8 0 8 8 0 0 8 8 0 8 8 0 8 0 0 8 0 0 8 8 8 8 0 8 0 8 0 0 0 8 0 0 8 8 8 8 8 0 0 0 8 0 0 0 0 6 0 8 0 8 4 0 4 8 12 4 4 12 4 0 0 8 0 8 0 0 0 0 12 0 0 1 2 4 8 4 4 0 12 0 0 0 12 0 8 12 0 1 2 12 0 0 12 8 4 12 12 0 8 0 0 1 2 4 0 12 12 12 1 2 0 12 4 12 0 0 0 12 4 4 8 12 8 8 12 4 0 0 12 0 4 0 0 2 0 0 0 0 4 0 4 0 4 4 4 4 4 0 0 0 0 0 0 0 0 0 4 0 0 4 4 0 4 4 0 4 0 0 0 4 0 0 4 0 4 4 0 0 4 0 4 4 4 0 0 0 0 4 4 0 4 4 4 4 0 4 4 4 0 0 0 4 0 4 0 4 0 0 4 4 0 0 4 0 4 0 0 1 2 0 2 0 0 2 2 2 0 0 2 2 2 0 2 2 0 0 0 0 0 2 0 0 0 2 0 0 0 0 2 0 2 0 0 2 0 2 0 0 0 2 2 0 2 2 2 2 0 2 0 0 0 2 2 2 2 0 0 2 2 2 2 2 2 2 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 8 2 8 4 2 6 10 1 2 4 6 14 6 0 2 1 0 0 8 0 0 0 2 12 0 0 1 4 4 8 4 2 2 12 2 0 0 14 0 10 12 0 1 2 14 2 14 10 6 14 12 2 8 0 0 1 4 6 2 14 12 12 1 4 2 14 6 14 2 2 0 14 6 4 8 14 10 8 12 4 2 0 14 0 6 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 0 2 0 4 2 6 2 4 4 6 6 6 0 2 2 0 0 0 0 0 2 4 0 0 6 4 0 4 4 2 0 2 0 0 6 0 2 4 0 4 6 2 0 6 2 6 6 4 2 0 0 0 6 6 2 6 4 4 6 2 6 6 6 2 2 0 6 6 4 0 6 2 0 4 4 2 0 6 0 6 2 0 9 2 0 2 0 0 2 2 2 0 0 2 2 0 0 18 2 0 0 1 6 16 0 2 0 16 16 2 0 0 0 0 18 0 18 16 0 2 16 2 0 16 0 2 2 16 2 2 2 2 0 18 0 16 0 2 2 2 2 0 0 2 1 8 2 2 2 18 18 0 2 2 0 0 2 2 0 0 0 2 0 2 0 2 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [Figure 13: a fractal structure on using CVT of different integer values] Here we have considered the pattern of zeros, which we have made it colored. Clearl y this pattern of zeros is a non- periodic (chaotic). Here, we can conclude that for every irrational number we can be able to have non-periodic (chaotic) pattern using CVT. That is here from we can be able to conclude that an uncountable number of periodic and non periodic pattern corresponding to an y real numbers can be obtained through CVT. As we know that the set o f real numbers is uncountable. 5. Analytical and Algebraic properties of CVT It is shown in section V that, the fractal obtained by CVT can also be constructed by 1-D CA rules, which is basicall y a sub class of Discrete Dynamical S ys tem. This motivates us to stud y about the dy namical properties o f CVT. Interestingly (Z, CVT) is a Discrete Dynamical S ys tem. Let us make you recalled the definition of a Dynamical system. Dynamical system : A dynamical system is a semi-group G acting on a space M, i.e. there is a map …………………….. (1) If is a group, then is called Invertible dynamical s ystem. In another form we can formalize the condition – (1) as follows. A dynamical system is semi-group G acting on a space M, i.e. there is a map …………………….. (2) Let us come to in our arena…. Let us define ; Theorem 1 : (Z, CVT) is a Discrete Dynamical System. Proof: To show (Z, CVT) is a Dynamical system we have to find a function such that Let us consider the map be the CVT. Clearl y, is a semi -group. Let a, b, x Є Z and a=(a n, a n-1, a n-2, …,a 1 ) 2 ,b=(b n, b n-1, b n-2, …,b 1 ) 2 ,x=(x n, x n-1, x n-2, …,x 1 ) 2" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ( ), ( )) ( , , . .., , 0) ( , , ..., , 0) (( ) ( ), ( ) ( ), ..., ( ) ( ), 0), 0) (( ), ( ), ..., ( ), 0) (( a b n n n n n n n n n n n n n n n n n n n n n n CVT CVT x CVT x a x a x a x b x b x b x a x b x a x b x a x b x a b x a b x a b x CVT − − − − − − − − − − − = ∧ ∧ ∧ ∧ ∧ ∧ ∧ = ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ = ∧ ∧ ∧ ∧ ∧ ∧ = 1 1 1 1 1 1 ( , ) , , ..., , 0), ( , , ..., )) ( ( , ), ) ( ) n n n n n n CVT a b a b a b a b x x x CVT CVT a b x CVT x − − − ∧ ∧ ∧ = = Hence, (Z, CVT) is a Discrete Dynamical S ystem. It can be noted that this definition of CVT when treated as a binary oper ation doesn’t satisfy algebraic properties such as Closure, Associative, Exist ence of identity and Existence of inverse except Commutativit y . But as we have seen in section IV CVT could produce fractals. Now we can slightl y modify our CVT definition and redefine it so that without affecting the fractal for mation it also satisfies some of those interesting algebraic properties. 6. Modified Carry Value Transformation (MCVT) Formal definition of MCVT MCVT i s a mapping defined as : ( ) n n n MCVT B B B × → where n B is the s et o f strings of length n on { 0 ,1 } B = . More specifi cally, if 1 1 1 1 ( , , ..., ) ( , , ..., ) n n n n a a a a and b b b b − − = = then 1 1 1 1 ( , ) ( , , ..., ) n n n n MCVT a b a b a b a b − − = ∧ ∧ ∧ is an n bit string and can be compu ted bit wise by lo gical A ND opera tion, wh ich denotes no carry, is generated in th e LSB at the t ime of addit ion pro cedu re. 1 1 1 1 1 1 1 1 1 1 ...................... ...................... ...................... .............. n n n n n n n n n n carry value c c c a a a a b b b b a b a b a b a b − − − − − = = = ⊕ = ⊕ ⊕ ⊕ [Figure 14: Carry generated in i th column is saved in the same i th column.] Theor em 2 : (MCVT, n B ) is a commutative monoid. Proof: As the range of MCVT is n B so it satisfies Closure property . All other properties can be proved as follows. For Associative Property Claim : ( od ( , ), ) ( , od ( , )) Modified CVT M ified CVT a b c Modified CVT a M ified CVT b c = L.H.S 1 1 1 1 1 1 1 1 1 1 1 1 ( ( , ), ) (( , , ..., ), ( , , ..., )) ( , , ..., ) n n n n n n n n n n n n Modified CVT Modified CVT a b c ModifiedCVT a b a b a b c c c a b c a b c a b c − − − − − − = ∧ ∧ ∧ = ∧ ∧ ∧ ∧ ∧ ∧ R.H.S 1 1 1 1 1 1 1 1 1 1 1 1 ( , ( , )) (( , , ..., ), ( , , ..., )) ( , , ..., ) n n n n n n n n n n n n Modified CVT a Modified CVT b c ModifiedCVT a a a b c b c b c a b c a b c a b c − − − − − − = ∧ ∧ ∧ = ∧ ∧ ∧ ∧ ∧ ∧ For Existence of identity 1 1 1 1 1 1 (( , , ..., ), ( 1 , 1 , ..., 1 )) ( 1 , 1 , ..., 1 ) ( , , ..., ) n n n n n n ModifiedCVT a a a a a a a a a − − − = ∧ ∧ ∧ = For Commutative 1 1 1 1 1 1 1 1 d ( , ) ( , , ..., ) ( , , ..., ) ( , ) n n n n n n n n Mo ified CVT a b a b a b a b b a b a b a ModifiedCVT b a − − − − = ∧ ∧ ∧ = ∧ ∧ ∧ = Hence, (MCVT, n B ) is a commutative monoid . In the modified CV table the decimal values in each entries are exactly half i.e. / 2 ij ij a a = because onl y the LSB position 0 is not padded in this type of vectors. Thus the fractals generated by both these CVT operations are exactl y same but the modified CVT has an additional advantage of getting so me algebraic properties than the conventional CVT. Following section deals with the extension of both CVT as well as MCVT in higher dimensions. 7. Extension of CVT and MCVT in higher dimensions Let us define CV T (or MC VT) recursi vely in higher dimen sional spac e k Z . CV T (or MCV T) is def ined a s 1 ( ) : ( ... ) , n n n n n CVT or MCVT B B B B B + × × × × → Where n B is the set o f strings o f leng th n on { 0,1 } B = . More s pecif ically, 1 2 1 2 1 ( , , ..., ) ( ( , , ..., ), ) k k k CVT x x x CVT CVT x x x x − = In a simi lar fashion, w e can write 1 2 1 2 1 ( , , ..., ) ( ( , , ..., ), ) k k k MCVT x x x MCVT MCVT x x x x − = Where is a positive integer for both CVT and MCVT. It is to be noted that 2 k = is the termi nating condit ion for the above recurs ive p rocedu res. In parti cular for 3 k = , 1 2 3 1 2 3 ( , , ) ( ( , ), ) CVT x x x CVT CVT x x x = where 1 2 ( , ) CVT x x could be evalua ted as defined abov e in section 3. This definit ion helps us to generate fractals in space, where as earlier we have got fracta ls in plane. On exploring these ideas w e can build up fr actals in n- dimen sional space. 8. Conclusion and Future Research directions This paper presents a new transformation named as Carry Value Transformation (CVT) applied on a pair of integers. Previously we have used this CVT for Efficient Hardware design of arith metic operations [2]. On further investigation of this transform in binar y nu mber system produces a beautiful pattern, which is found to be a fractal having dimension 1.585, same as that of Sierpinski triangle. Further CVT can be applied for the production of periodic and chaotic patterns. Interestingl y, it is proved that ( Z, CVT) is a Discrete Dynamical S y ste m. Further, t he definition of CVT is slightl y modified and its mathematical properties are highlighted where we have shown that (MCVT, n B ) is a commutative monoid. Finall y , the extension of CVT and modified CVT (MCVT) are done in higher dimensions. Authors are of fir m conviction that CVT/MCVT can be used in the way the other mathematical transforms (e.g., Fourier, Discrete Cosine, Laplace, Wavelet, Cellular Auto mata Transforms etc.) are used; only the domains will vary from one transform to another. Further studies of algebraic and anal y tical p roperties of these transformations are highly needed for the complete exploration of this potential area. Further, authors are expecting to get a close relationship between the CVT/MCVT with many different application areas like Computational geometry , Data compression, Quad trees, Defective chessboards, Pattern Classification, Theor y of Computation, and Analysis of Cellular Automata Rules etc. Exploring all these will be our immediate future research directions. References: [1] B.B. Mandelbrot, The fractal geometry of nature . New York, [2] P.P. Choudhury, S. Sahoo, M. Chakrabort y, 2008 I mplem entation of Basic Arith metic Operations Using Cellular Automaton, ICIT-08, IEEE CS Press , 11 th International Conference on Information Technology , pp 79-80, [3] J. L. Henness y and D. A Patterson, 1996“Computer Architecture: A Quantitative Approach” ( 2 nd edition ), Morgan Kaufmann, San Francisco. [4] M. 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