Generation of 3D Fractal Images for Mandelbrot and Julia Sets

Fractals provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of self-similarity, scale invariance, and dimensionality. Images such as coastlines, terrains, cloud mountains, and most interestingly, random shapes composed of curves, sets of curves, etc. present a multivaried spectrum of fractals usage in domains ranging from multi-coloured, multi-patterned fractal landscapes of natural geographic entities, image compression to even modelling of molecular ecosystems. Fractal geometry provides a basis for modelling the infinite detail found in nature. Fractals contain their scale down, rotate and skew replicas embedded in them. Many different types of fractals have come into limelight since their origin. This paper explains the generation of two famous types of fractals, namely the Mandelbrot Set and Julia Set, the3D rendering of which gives a real-world look and feel in the world of fractal images

Geometry of knots in real projective 33-space

This paper discusses some geometric ideas associated with knots in real projective 3-space $\mathbb{R}P^3$. These ideas are borrowed from classical knot theory. Since Knots in $\mathbb{R}P^3$ are classified into three disjoint classes, - affine, class-$0$ non-affine and class-$1$ knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behaviour near the projective plane at infinity. We propose a surgery operation on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in $\mathbb{R}P^3$. We also define a notion of "genus" for knots in $\mathbb{R}P^3$ and compare its properties with the Seifert genus from classical knot theory. We prove that this genus detects knottedness in $\mathbb{R}P^3$ and gives some criteria for a knot to be affine and of class-$1$. We produce examples of class-$0$ non-affine knots with genus $1$. And finally we introduce a notion of companionship of knots in $\mathbb{R}P^3$ and using that we provide a geometric criteria for a knot to be affine. Thus we highlight that, $\mathbb{R}P^3$ admits a knot theory with a truly different flavour than that of $S^3$ or $\mathbb{R}^3$.Comment: 20 pages, 23 figure

Plat closures of spherical braids in RP3\mathbb{R}P^3

We develop a method for costructing links in $\mathbb{R}P^3$ as plat closures of spherical braids. This method is a generalization of the concept of \say{plats} in $S^3$. We prove that any link in $\mathbb{R}P^3$ can be constructed in this manner. We also develop a set of moves on spherical braids in the same spirit as the classical Markov moves on braids and show that two spherical braids can have isotopic plat closures if and only if they are related by a finite sequence of these moves. We introduce the notion of a new kind of permutation (called \textit{residual permutation}) associated to a spherical braid in $\mathbb{R}P^3$ and prove that the number of disjoint cycles in this residual permutation of a spherical braid is same as the number of components of the plat closure link of this braid.Comment: 20 Pages, 22 Figures, preliminary versio