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In this paper the theory of Carry Value The organization of the paper is as follows. Section II Transformation (CVT) is designed and developed on a pair of ndiscusses some of the basic concepts on fractals, L-systems, bit strings and is used to produce many interesting patterns. One tilling problem, Cellular Automata etc. which are used in the of them is found to be a self-similar fractal whose dimension is subsequent sections. The concept of CVT is defined in section same as the dimension of the Sierpinski triangle. Different III. It can be found in section IV that CVT generates a beautiful construction procedures like L-system, Cellular Automata rule, self-similar fractal whose dimension is found to be same as thatTilling for this fractal are obtained which signifies that like other of monster fractal, Sierpinski triangle. Section V deals with the tools CVT can also be used for the formation of self-similar various ways like L-System, Cellular Automata and Tillings by fractals. Finally it is shown that CVT can also be used for the which the same fractal in binary number system can be production of periodic as well as chaotic patterns, constructed. CVT can also be used for the production of Keywords-Carry Value Transformation, Fractals, Lperiodic as well as chaotic patterns are shown in section VI. On System, CellularAutomata and Tilling.highlighting other possible applications of CVT and some future research directions a conclusion is drawn in section VII.

The notion of Carry Value Transformation (CVT) is a model of Discrete Deterministic Dynamical System. In this paper, we have studied some interesting properties of CVT and proved that (1) the addition of any two non-negative integers is same as the sum of their CVT and XOR values.(2) While performing the repeated addition of CVT and XOR of two non-negative integers "a" and "b" (where a ≥ b), the number of iterations required to get either CVT=0 or XOR=0 is at most the length of "a" when both are expressed as binary strings. A similar process of addition of Modified Carry Value Transformation (MCVT) and XOR requires a maximum of two iterations for MCVT to be zero. (3) An equivalence relation is defined in the set Z Z × which divides the CV table into disjoint equivalence classes.

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An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length [Formula: see text] has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient [Formula: see text] where [Formula: see text] is the length and [Formula: see text] is the weight of the binary string.