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

TL;DR

This paper introduces the Carry Value Transformation (CVT) theory, demonstrating its ability to generate self-similar fractals like the Sierpinski triangle and other complex patterns, with applications across various construction methods.

Contribution

The paper develops the CVT theory, explores its properties, and extends its application to higher dimensions, establishing it as a versatile tool for fractal pattern formation.

Findings

CVT can produce self-similar fractals like the Sierpinski triangle

CVT enables creation of periodic and chaotic patterns

Extended CVT to higher dimensions

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 many 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.