### Modification of Chaos Game with Rotation Variation on a Square

Kosala Dwidja Purnomo, Indry Larasati, Ika Hesti Agustin, Firdaus Ubaidillah

#### 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.

#### Keywords

chaos game, rotational variation, random, non-random.

PDF

#### References

Devaney, Robert L. 2003. Fractal Patterns and Chaos Game. Department of Mathematics: Boston University, Boston MA 02215.

Shenker, O. R. 1994. Fractal Geometry is not the Geometry of Nature. Jerusalem: The Hebrew University of Jerusalem.

Sulistiyantoko, D. 2008. Application of Sequence Row on Simple Fractal Geometry Calculation Formations. Essay. Mathematics Education Program, Faculty of Science and Technology, Sunan Kalijaga UIN Yogyakarta.

Miller, C. 2011. Communicating Mathematics III: Z-Corp 650. http://www.maths.dur.ac.uk.

Yunaning, F. 2018. Study of the Rules for Non-Random Chaos Games in the Triangle. Essay. Faculty of MIPA, University of Jember.

Zohuri, B. 2015. Dimensional Analysis and Self-Similarity Methods for Engineers and Scientist. Switzerland: Springer International Publishing.

Purnomo, Kosala D. 2016. Study of Sierpinski Triangle Establishment using Affine Transformation. Journal of Mathematics: Faculty of MIPA, Universty of Udayana, vol. VI No. 2:86-92.

Sulastri. 2007. Build Transformation in Three-dimensional Space Using Visual Basic 6.0. Journal of DYNAMIC Information Technology Volume XII, No. 1: 88-100. Stikubank University Faculty of Information Technology.

Sundbye, L. 1997. Chaos and Fractals on the TI Graphing Calculator. Department of Mathematical and Computer Sciences: Metropolitan State College of Denver.

DOI: https://doi.org/10.18860/ca.v6i1.6936

### Refbacks

• There are currently no refbacks.

Copyright (c) 2019 Kosala Dwidja Purnomo, Indry Larasati, Ika Hesti Agustin, Firdaus Ubaidillah