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

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-similarity. The complexity of fractals and the property of self-similarity have a large set of real world applications. Fractals can be generated using construction procedure/algorithms/simply by repetition of mathematical formula that are often recursive and ideally suited to computer. In this paper a new transformation named as Carry Value Transformation (CVT) is defined in binary 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 selfsimilar-fractals, the algorithm using CVT also produces periodic and chaotic patterns. The underlying development of CVT is same as the concept of Carry Save Adder (CSA) [3] where carry or overflow bits generated in the addition process of two integers are saved in the memory. Here we perform the bit wise XOR operation of the operands to get a string of sum-bits (ignoring the carry-in) and simultaneously the bit wise ANDing of the operands to get a string of carry-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 fractal, Sierpinski triangle. Section V deals with the various ways like L-System, Cellular Automata and Tillings by which the same fractal in binary number system can be constructed. CVT can also be used for the production of periodic as well as chaotic patterns are shown in section VI. The analytical and algebraic properties of CVT are discussed in section VII. The definition of CVT in two-dimension is slightly 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].

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 complexity of nature. Starting from an initial seed he studied the space-time diagram of all the 256 three-neighborhood elementary CA rules and classified the rules into four distinct classes according to the complexity of the pattern. According to him the Class 2 rules deals with the periodic and fractal patterns. In 1-D the global state or simply 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 state” of an n-bit CA (at time t) is X t , its “next state” (at time t+1), denoted by X t+1 , is in general given by 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 respectively the left boundary of x 1 and right boundary of x n incase of periodic boundary CA and those values are 0 in case of null boundary CA. If the same local mapping (rule) determines the “next” bit in each cell of a CA, the CA will be called a Uniform CA, otherwise it will be called a Hybrid CA. For our purpose, we have used one-way CA, which allows only one-way communication, i.e., in a 1-D array 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 by 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.

The carry or overflow bits are usually generated at the time of addition between two n-bit strings. In the usual addition process, carry value is alwa

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