Modification of Chaos Game with Rotational Variation on a Square


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(1) (2019), Pages 27-33 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 
 
 
 

Submitted: May 10, 2019 Reviewed: September 17, 2019 Accepted: November 30, 2019 
DOI: http://dx.doi.org/10.18860/ca.v6i1.6936  

Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo1, Indy Larasati2, Ika Hesti Agustin3, Firdaus Ubaidillah4 

Department of Mathematics, University of Jember, Indonesia 
Email: 1) kosala.fmipa@unej.ac.id, 2) indylarasati92@gmail.com,  

3) ikahesti.fmipa@unej.ac.id, 4) firdaus_u@yahoo.com 

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 n-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 1/2 of the fulcrum, that is α = 1/r = 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. 

INTRODUCTION 

The original chaos game is a game of drawing a number of points on an equilateral 
triangle with certain rules that are repeated iteratively. Sierpinski triangle is an example 
of a fractal object formed by chaos game. Devaney [1] has modified the Sierpinski triangle 
with rotational variations that form fractal "films". The modification of Sierpinski triangle 
can be done by varying the angle of rotation, compression ratio, or location of the vertices 
for chaos game problems. Based on the research conducted by Devaney, this study will 
discuss the formation of chaos game on a square which are rotated by using an affine 
transformation with a certain compression ratio. The production rules are by using 
random and non-random rules, where random rules are built by selecting random points 
that can be modified with certain conditions, while non-random rules for point selection 
are carried out regularly and sequentially as many points as desired. The study was 
conducted to analyze whether the chaos game on a square varied by rotation with certain 
rules will still make fractals form such as Sierpinski triangle. 

Fractals are defined as complex forms without limits because they have infinite 
details in a limited volume of space. This form will always look identical like an infinite 
pattern. However, if an object is made from elementary particles, then the details of the 
number of geometric shapes are limited. This amount is determined by the number of 
elementary particles and their relative position. The amount may be very large, but it is 

http://dx.doi.org/10.18860/ca.v6i1.6936


Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo 28 
 
 
 

always limited [2]. Some examples of fractal forms include Cantor's middle third set, Von 
Koch curve, Sierpinski triangle, and Cantor dust [3]. 

Chaos game with three vertices is one method to generate Sierpinski triangle. The 
shape of vertices in chaos games can also be rectangles, pentagons, hexagons, and n-
facets. Chaos games are generated using the specified compression ratio, namely the 
boundary between random points and vertices. The compression ratio is written with the 
symbol r and the result is the new distance point 𝑍𝑛+1 from the right point is  1/𝑟 from 
the previous distance 𝑍𝑛+1 to the vertex [4]. Chaos game is a point game where the 
production rules can still be developed, for example what if changing the number of 
vertices or changing the distance between the starting point and the specified vertex [5]. 
If the starting point in game chaos is located in a triangle, then the next marked point will 
always be located in the Sierpinski triangle. If the starting point is outside the triangle, 
then in some iteration a marked point will occur located in the triangle. Therefore, for 
some iteration the marked points are guaranteed to always be in the triangle [6]. 

Sierpinski triangle is a linear type of fractal that has identical self-similarity from 
one iteration to the next iteration. Sierpinski triangle is a triangle consisting of other 
triangles which are in the same shape but with half size. If a small part of the triangle is 
enlarged, it will look like another larger part. Affine transformation can generate 
Sierpinski triangle on chaos game with three vertices [7]. 

Transformation can be interpreted as a method that can be used to manipulate the 
location of a point. If the transformation is imposed on a set of points, then the object will 
change. Changes in this case are changes from the initial object location to the new 
location. The most commonly used method in graphical computer is the affine 
transformation method. An affine transformation is a transformation that uses a matrix 
to calculate the position of a new object. Transformation on two-dimensional objects can 
be in the form of translation, scale, and rotation.  

Translation can be interpreted by changing locations from the initial location to 
the new location. The point 𝑃(𝑥, 𝑦) is shifted by a number of  𝑇𝑟𝑥 on the x-axis and shifted 
by a number of  𝑇𝑟𝑦 on the y-axis. Therefore the general formula will be obtained as 

follows 
𝑄(𝑥, 𝑦) = 𝑃(𝑥, 𝑦) + 𝑇𝑟  = 𝑃(𝑥 + 𝑇𝑟𝑥 , 𝑦 + 𝑇𝑟𝑦 ). 

Scale can be interpreted by an initial location multiplied by the desired quantity. The 
original location is multiplied by the amount 𝑆𝑥 on the x-axis and 𝑆𝑦 on the y-axis. So that 

the general formula will be obtained as follows 
𝑄(𝑥, 𝑦) = 𝑃(𝑥, 𝑦) x 𝑆(𝑥, 𝑦) = 𝑃(𝑥 x 𝑆𝑥 , 𝑦 x  𝑆𝑦 ). 

Rotation can be interpreted by moving the initial location of an object (object) by rotating 
at an angle θ to the x-axis or y-axis. It is assuming that the centre point (0,0) with a positive 
angle is counter-clockwise and a negative angle is in the direction clockwise. The rotation 
formula is as follows 

𝑄(𝑥, 𝑦) = 𝑃(𝑥, 𝑦) x [
cos 𝜃 − sin 𝜃
sin 𝜃 cos 𝜃

]. 

Two-dimensional transformation uses two axes, namely the x-axis and the y-axis. 
The x-axis represents the length, while the y-axis represents the width [8]. We will modify 
the original chaos game on a square/quadrilateral by using geometric transformations, 
namely rotation, as Devaney done. The rules of affine transformation with linear algebraic 
principles for building a "fractal" film are formulated 

𝑇 (
𝑥
𝑦) = (

1/𝑟 0
0 1/𝑟

) (
cos 𝜃 − sin 𝜃
sin 𝜃 cos 𝜃

) (
𝑥 − 𝑥0
𝑦 − 𝑦0

) + (
𝑥0
𝑦0

). 

The affine transformation combines the calculations of translation, scales and rotation 



Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo 29 
 
 
 

matrices with a compression ratio written as r, and α is a distance that its value 1/r. The 
centre of rotation is (𝑥0, 𝑦0) with distance α, and θ is the angle of rotation [9]. 
 

RESULTS AND DISCUSSION  

Generation of modified chaos game on a square/quadrilateral will be done using 
random and non-random rules. The generation using random rules is done with some 
modifications. The compression ratio used is r=2, which means that the distance from the 
formation point is 1/2 of the weight point (that is α=1/r). There are four modifications 
here: 
1. The last point that has been chosen cannot be chosen again;  
2. The facing point cannot be chosen; 
3. The left entry point cannot be chosen; 
4. The right side point cannot be chosen.  

The left and right points are seen from sequential patterns 𝐴1− 𝐴2 − 𝐴3 − 𝐴4, where 
the left point is the previous point, and the right point is the point after it. The exception 
is point  𝐴3 where the left point is point 𝐴4, then after choosing point  𝐴3 it is not allowed 
to choose point 𝐴4 and otherwise. Facing points also apply only to rectangles. The rules 
will be different if applied to pentagon and other aspects. 

Non-random rules are also carried out with several modifications, namely non-
random 1 with 4 regular and sequential point selection patterns (𝐴1− 𝐴2 − 𝐴3 − 𝐴4), non-
random 2 with 16 point selection patterns, that is in multiples of 4 repeat from the last 
chosen point (𝐴1 − 𝐴2 −  𝐴3 −  𝐴3 −  𝐴4  −  𝐴1 −  𝐴2 − 𝐴2 −  𝐴3  −  𝐴4  −  𝐴1 −  𝐴1  −  𝐴2  −
 𝐴3  − 𝐴4  −  𝐴4), non-random 3 with 20 point selection patterns, that is in multiples of 5 
jumping over one point after that and in 𝑡 iterations namely 𝑡 ≡ 4 𝑚𝑜𝑑 (5) the chosen 
point is the point facing the last point (𝐴1 − 𝐴2 − 𝐴3 − 𝐴1 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴4 − 𝐴2 −
𝐴3 − 𝐴4 − 𝐴1 − 𝐴3 −  𝐴1 −  𝐴2 −  𝐴3 − 𝐴4 − 𝐴2 − 𝐴4), non-random 4 with 24 point selection 
patterns, that is in multiples of 6 and in 𝑡 iterations namely 𝑡 ≡ 5 𝑚𝑜𝑑 (6) repeats once like 
the previous point, then at 𝑡 ≡ 4 𝑚𝑜𝑑 (6) jumps over one point after its last point (𝐴1 −
𝐴2 −  𝐴3 −  𝐴3 −  𝐴3 −  𝐴1 − 𝐴2 − 𝐴3 − 𝐴4 −  𝐴4− 𝐴4 −   𝐴2 − 𝐴3 − 𝐴4 − 𝐴1 − 𝐴1 − 𝐴 1 − 𝐴3 −
𝐴4 −  𝐴1 −  𝐴2 −  𝐴2 −  𝐴2 − 𝐴4), and non-random 5 with 28 point selection patterns, in 𝑡 
iterations namely 𝑡 ≡ 4 𝑚𝑜𝑑 (7) jumps over one point from the last point, in the iteration 
𝑡 ≡ 5 𝑚𝑜𝑑 (7) repeats like the last point selected, at the iteration 𝑡 ≡ 6 𝑚𝑜𝑑 (7) jumps 
over one point from the last point again, and in multiples of 7 repeats one point from the 
last point chosen (𝐴1 − 𝐴2 − 𝐴3 − 𝐴1 − 𝐴1 − 𝐴3 − 𝐴3  − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴4 − 𝐴1 − 𝐴2 − 𝐴4 − 𝐴4 −

𝐴2 − 𝐴2 − 𝐴3 −  𝐴4 −  𝐴1 − 𝐴3 − 𝐴3 − 𝐴1 −  𝐴1 − 𝐴2 − 𝐴3 − 𝐴4 − 𝐴2 − 𝐴2 − 𝐴4 − 𝐴4).  

Steps to make a chaos game: 
a. Make a rectangle and give a name to each rectangular vertex, for example  𝐴1,   𝐴2,   𝐴3,  
    and 𝐴4; 
b. Choose a random starting point and is named 𝑇0; 
c. Select one vertex in the square to be associated with the starting point 𝑇0 to make the  
    game chaotic; 
d. Determining the new point named 𝑇1 and  the  distance  (α)  is  half  the  distance  of  the  

vertex to the starting point according to the predetermined compression ratio, namely        
𝑟 = 2; 

e. Repeating steps (c) with the starting point is a new point  obtained  from  the  previous  
    iteration and may be repeated by selecting the vertex as a reference point; 
f. Do the desired number of points to iterations; 

 



Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo 30 
 
 
 

The rotation rule used is by using an affine transformation. Suppose the rotation 
is done with the rotation angle given 30° and the fulcrum is 𝐴4. Then the new point formed 
will shift by 30° towards the chosen fulcrum 𝐴4. The difference in random and non-
random game chaos formation lies in the selection of random points where points may be 
chosen randomly according to the desired modification rules, while the selection of points 
on non-random rules is determined by the desired pattern, in this study 4 patterns were 
selected, 16 patterns, 20 patterns, 24 patterns and 28 point repetition patterns.  

The results of the forms obtained by random rules and non-random chaos games 
with positive and negative angle values will be compared, and the fulcrum of rotation is 
at the quadrilateral angle. The visualization results in chaos game on a square by random 
rules and their modification is done by rotation of angles every 10°, 𝛼=1/2 and iterations 
of 10,000. All objects generated in the rotation of positive angles are random rules and 
their modifications can be said to be fractal objects because they have similar shapes to 
each other (self-similarity). The rotation at the negative angle is also a fractal object. The 
difference is in the direction of rotation, where the rotation of the positive angle rotates 
counter-clockwise while the rotation of the negative angle rotates clockwise. The 
visualization for the positive angle are shown in Figure 1 to Figure 5. 

 
Figure 1. Rotational variation with random rules  

 

       
Figure 2. Rotational variation with modification 1 

 

               
Figure 3. Rotational variation with modification 2  

 



Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo 31 
 
 
 

 
Figure 4. Rotational variation with modification 3 

 

 
Figure 5. Rotational variation with modification 4 

 
The visualization of the rules of non-random chaos games is done with 𝛼 = 1/2, 

5,000 iterations, and at positive angles of rotation every 30° it does not produce a clear 
shape, but only like far apart points. The visualization of non-random rules simulation and 
their modifications shown in Figure 6 to Figure 7. 

          
Figure 6. Rotational variation with non-random 1 

         

 
Figure 7. Rotational variation with non-random 2 

 
The resulted points of non-random chaos game at a certain coordinate are due to 

the continuous selection of vertices that are patterned regularly. The points of the 
formation are spread unevenly and only "convergent" to a certain point as much as the 
pattern specified, and not converging on the four points of the support point as stated in 
previous studies. This causes the results of the chaos game with non-random rules not a 



Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo 32 
 
 
 

fractals form because they have no similarity to each other (have no self-similarity). The 
coordinates of the new points generated from the last few iterations of non-random rules 
1 to non-random rules 2 shown in Figure 8. 

           
Figure 8. New coordinate points of non-random 1 (left) and non-random 2 (right) 

 
It is different from random rules, where all the modifications form fractal objects because 
the selection of points is done randomly. According to [3] this is because the main 
principle of game chaos is to use chaos theory, which is a description of the complex 
movements and behavior of certain non-linear dynamic systems whose movements 
depend heavily on the initial state. Chaos formation of games with point selection is done 
non-randomly with a regular pattern that is determined as much as anything but still will 
not form a fractal due to the selection of regular and recurring points, so that the point 
will gather at a certain point. Previous studies that combined random and non-random 
rules also did not show fractal objects due to the involvement of non-random patterns, so 
they did not form fractals, although there were some patterns that were almost fractal, 
but less identical and still said that the object was not is a fractal.  

 

CONCLUSIONS 

Rotational variation of chaos game on a square/quadrilateral are built by random rules 
and all their modifications produce fractal objects because they have similar shapes (have 
self-similarity). Whereas the chaos game with non-random rules which the pattern is 
repeating points every 4 patterns, 16 patterns, 20 patterns, 24 patterns are not fractal 
objects because the formation points are far apart so they do not have similarity in form 
to one another points, but they have the characteristic of "convergent" repeating at certain 
coordinate points as much as the pattern of selection of a specified point. 
Based on the results of the study, we get an open problem that can be done for further 
research: 
1. Shows other modifications such as a combination of rotation variations and  
    compression ratio; 
2. Show analytic studies to prove that a chaos game object is a fractal 



Modification of Chaos Game with Rotational Variation on a Square 

Kosala Dwidja Purnomo 33 
 
 
 

ACKNOWLEDGEMENTS 

We gratefully acknowledge the support from Keris Gerbang Mata and LP2M – University 
of Jember of year 2019. 
 

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