Act of CVT and EVT in The Formation of Number Theoretic Fractals Pabitra Pal Choudhury, Sk. Sarif Hassan Applied Statistics Unit, Indian Statistical Institute, Kolkata, 700108, INDIA Email: pabitrapalchoudhury@gmail.com sarimif@gmail.com Sudhakar Sahoo, Birendra Kumar Nayak P.G. Department of Mathematics, Utkal University, Bhubaneswar-751004, INDIA Email: sudhakar.sahoo@gmail.com bknatuu@yahoo.co.uk Abstract — In this paper we have defined two functions that have been used to construct different fractals having fractal dimensions between 1 and 2. More precisely, we can say that one of our defined functions produce the fractals whose fractal dimension lies in [1.58, 2) and rest function produce the fractals whose fractal dimension lies in (1, 1.58]. Also we tried to calculate the amount of increment of fractal dimension in accordance with base of the number systems. And in switching of fractals from one base to another, the increment of fractal dimension is constant, which is 1.58, it‘s quite surprising! Keywords-Carry Value Transformation, Extreme Value Transformation, Fractals, Fractal dimension. We have constructed two functions namely Carry Value transformation and Extreme Value Transformation; and on using these we have generated different fractals of fractal dimension lying between the interval (1, 2). Indeed, for any given number in between (1, 2) this functions could give one fractal having fractal dimension nearer to the given number. The maximum error could be 0.12 in approximating fractal dimension. Finally our observation centers on the fact that when we switch from one fractal to another fractal on using our said methodology, the amount of fractal dimension (amount of chaos) is remaining unaltered as 1.58. I. INTRODUCTION In this paper we have tried to make an association between natural number and fractals, with the help of two defined transformations. Numbers corresponding to different number systems with base b (b=2, 3, 4…etc) signifies different fractals. Here, we have defined two new transformations called as ―Carry Value Transformation (CVT) and Extreme Value Transformation (EVT)‖. With these two mapping we have generated fractals whose dimension lying in between the open interval (1, 2) [figure 7, 14]. It should be noted that we are traversing this interval discretely, but very densely also. That is for any given number from (1, 2) we could be able to give a fractal whose dimension is nearer to that given number. In our journey we have got a fractal whose dimension is 1.68, fortunately this very fractal dimension is the fractal dimension of music (Sri Lankan, Chariots of Fire) [7]. So this fractal could be a frame of lyrics for music. And undoubtedly there are a lot of fractals, which are of fractal dimension 1.68, and possibly these fractal-frames make new lyrics. In general, the fractal dimension of music is around the number 1.65, and our generated fractals could interpolate the number 1.65, as fractal dimension. In this paper also we have tried to calculate the amount of increment of fractal dimension in accordance with base of the number systems. And in switching of fractals from one base to another, the increment of fractal dimension is constant, which is 1.58, fractal dimension of Sierpinski Gasket. The organization of the paper is as follows. Section 2 discusses some of the basic concepts on fractals, fractal dimension, which are used in the subsequent sections. The concept of CVT is defined in section 3. In section 4, we have explored the formation of fractals in different bases of the number systems. And then we have generalized the concept of formation of fractals in any arbitrary bases of the number system. In section 5, we have explained the amount of increment of fractal dimension in accordance with base of the number systems. In the next sections, we have defined another transformation to generate fractals having fractal dimension lying in between (1, 1.58]. In addition, we have discussed formation of fractals in different bases of the number system and ultimately we have made it-generalized concept in any base of the number system as like for CVT we have done it. On highlighting other possible applications of CVT and some future research directions a conclusion is drawn in section 8. II. REVIEW OF SOME FUNDAMNETALS OF FRACTALS For quite a long time the scientific community was very much worried due to our inability to describe the shape of cloud, a mountain, a coastline or a tree on using the traditional Euclidean Geometry. In nature, clouds are not really spherical, mountains are not conical, coastlines are not circular, even the lightning doesn‘t travel in a straight line. More generally, we would be able to conclude that many patterns of nature are so irregular and fragmented, that, compared with Euclid Geometry –a term, can be used in this regard to denote all of the standard geometry. Mathematicians have over the years disdained this challenge and have increasingly chosen to flee