This paper presents a study to analyze and modify the Islamic star pattern using digital algorithm, introducing a method to efficiently modify and control classical geometric patterns through experiments and applications of computer algorithms. This will help to overcome the gap between the closeness of classical geometric patterns and the influx of design by digital technology and to lay out a foundation for efficiency and flexibility in developing future designs and material fabrication by promoting better understanding of the various methods for controlling geometric patterns.
With the advance of digital technology, the development of surfaces in modern structures enjoys an unprecedented freedom of expression. The various ability of computer programs, in tune with the will of designers to discover a new design, accelerates the speed of “limitless” design through proliferation, modification, and trajectory tracking. The rapid development of computer technology results in a tendency to perform unpredictable calculations with the computer using an algorithm beyond the control of the artist. The development of manufacturing technology has also enabled the construction of various experimental shapes, which provides a good justification as meaningful construction work. These phenomena try to differentiate themselves from the rules of classical geometry by using terminologies such as “Digital Geometry” and “Digital Materiality.” To counter this rapid trend, some architects severely limit the role of the computer, refuse designs made by digital programs, and instead produce designs based on the tradition and history of the sense of geometry. This study will focus on the disparity of such an extreme position regarding the use of computer algorithms in design. The purpose of this study is to identify a connection point of classic geometry and algorithmic design. In other words, to overcome the closeness of classic patterns through studies on the patterns produced by designers and also overcome the influx of design by digital technology, the objective of this study is to introduce a method to efficiently modify and control classical geometric patterns through experiments and applications of computer algorithms.
As the analysis object of this study, we used the Islamic star pattern. Specifically, this study selected the 4.8.8 pattern among the modified star pattern examples used in Hankin’s method. For analysis and experiment control of this pattern, we utilized “Grasshopper” and “Rhinoscript,” which are plugins for the Rhinoceros program by Robert McNeel and Associates, and “Processing” developed by Ben Fry and Casey Reas. The purpose of the study is presented in Section
“Tessellation” can be defined as a pattern of more than one shape which completely covers a certain plane. The regular splitting method of a plane is a method which leaves no gaps by using a certain shape, completely fills out the space without overlapping, and does not allow for overlapping of shapes or gaps [
Basic formation and combination formula of a tessellation.
If the range of an equilateral polygon is narrowed as in Figure
Equilateral polygon combinations.
The patterns which appear on the Islamic buildings and tiles of the Middle Ages started from simple designs and developed into complex designs with mathematical symmetry over centuries. These complex patterns were modified by the strapwalk method (Figure
Islamic strapwork.
Such Islamic patterns have had great influence on modern artists (Figures
Owen Jones, 1856.
Arabic pattern development (Lewis F. Day).
An algorithm is a group of welldefined rules and a finite number of steps used to solve a mathematical problem. It is a set of welldefined rules and commands in a finite number, and it can solve a problem by applying its limited rules [
Algorithm.
The Islamic star pattern using geometric figures is one of the world’s greatest ornament design traditions. Its expansion including architectural application significantly grew in the Middle East and Central Asia [
Hankin’s pattern developing process.
Variations of patterns using Hankin’s method (Craig S. Kaplan).
As mentioned in Section
Unit module.
4.8.8 combination.
According to Hankin’s method, the shape of stars can transform with a change in the angle of the sides of each basic geometric figure (Figure
Transformation of 4.8.8 combination.
Types of controlling method.
Point on segment control diagram (Type A).
Radius control diagram (Type B).
Point on segment control diagram (Type C).
Side on segment control diagram (Type D).
To create a simple 4.8.8 geometric figure on a plane, we have to split sides first. For the formation of geometric 4.8.8 figures, we have to start with a plane with a rectangular grid system (Figure
Basic geometry from diagrid.
An algorithm formula can be used to create the surface of a rectangle (Figure
Algorithmic formula.
The sidesplitting method can be divided into the point on side control and the radius control methods. In the point on side control method, a point on a segment (Figure
Sidesplitting control method diagram.
If each point in Type A is controlled (Figure
Examples of basic variations: Type A.
Examples of additional variations: Type A.
Examples of basic variations: Type B.
If two types of the sidesplitting control method, Type A and Type B, are combined, the patterns will be as follows (Figures
Type A patterning.
Type B patterning.
The expansion of a single module can be divided into two methods: the point on segment control and the side on segment control. This method is based on the idea that an octagon can be divided into eight triangles of the same shape (Figure
Single module expansion method diagram.
Because Type C can control two points (Figure
Examples of basic variations: Type C.
Examples of additional variations: Type C.
Type C patterning.
Examples of basic variations: Type D.
Type D patterning.
This study will look into the Islamic star pattern formation of a geometric figure combination of 4.8.8, which appears in Hankin’s method, depending on the four types of parameter control, and then will analyze the modification possibility and the convenience of algorithm designs of each of the four types of parameter control.
Modification possibility means how diversely a basic geometric figure can be modified. Basically, by appropriately controlling the parameter of each of the four types of parameter control, we can modify patterns into diverse shapes [Types A~D1~3]. However, whether additional modifications are possible or not will depend on whether the parameters are individually controlled or integrated. If the parameters are individually controlled [Type A, Type C], it is possible to modify them into other patterns. In contrast, if the parameters are integrated [Type B, Type D], it is relatively difficult to produce additional modifications.
In this paper, the convenience of design is analyzed from an algorithm perspective. In terms of the implementation of mathematical formula algorithms, if a program has a relatively simple mathematical formula and is convenient to control, implementation of the program is easy and its mathematical formula algorithm has an outstanding design convenience (Figure
Typespecific evaluation analysis.
Side splitting  Single module expansion  

Type  
Surface type  Point on segment control 
(Type B) 
Point on segment control 
Side on segment control 
Conceptual diagram 




Characteristics  Controls each line individually: creates different module shapes in a finite number  (i) Controls only the radius: creates a greater number of regular modules than the point on segment control 
(i) Easy modulation for construction: allows for an economical production system 
(i) Easy modulation for construction: allows for an economical production system 


Evaluation items  
Modification possibility  ○  △  ○  △ 
Expandability  ○  ○  △  △ 
Construction convenience  △  ○  ○  
Module diversity  ○  △  ○  △ 
Additional modification types 




Possible  Difficult  Possible  Difficult  


Design convenience  Low < high  
(program implementation supremacy)  Type A < Type C < Type D < Type B 
Pattern variations using Attractor Algorithm.
See Table
This study aims to analyze and modify the Islamic star pattern using digital algorithm, introducing a method to efficiently modify and control classical geometric patterns through experiments and applications of computer algorithm. This study reveals that with the help of algorithmic design strategy we can analyze and undermine the rigidity of the classical geometry of Islamic star pattern and expand its design potentials. Clear understandings of the classical geometry and proper experiments with digital algorithm can contribute to overcoming the gap between the classical geometry and the digital technology and to laying out a solid foundation for efficiency and flexibility in developing future designs and material fabrication.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was supported by the National Research Foundation of Korea no. NRF2013R1A1A2058553.