Modification of Chaos Game with Rotation Variation on a Square

Abstract

Chaos game is a game of drawing a number of points in a geometric shape using certain rules that are repeated iteratively. Using those rules, a number of points generated and form some pattern. The original chaos game that apply to three vertices yields Sierpinski triangle pattern. Chaos game can be modified by varying a number of rules, such as compression ratio, vertices location, rotation, and many others. In previous studies, modification of chaos games rules have been made on triangles, pentagons, and -facets. Modifications also made in the rule of random or non-random, vertex choosing, and so forth. In this paper we will discuss the chaos game of quadrilateral that are rotated by using an affine transformation with a predetermined compression ratio. Affine transformation is a transformation that uses a matrix to calculate the position of a new object. The compression ratio r used here is 2. It means that the distance of the formation point is  of the fulcrum, that is  = 1/2. Variations of rotation on a square or a quadrilateral in chaos game are done by using several modifications to random and non-random rules with positive and negative angle variations. Finally, results of the formation points in chaos game will be analyzed whether they form a fractal object or not.