Classification of Two Dimensional Cellular Automata Rules for Symmetric Pattern Generation

• Main Authors: Nisha Vellarayil Mohandas, Lakshmanan Jeganathan
• Format: Article
• Language: English
• Published: MDPI AG 2018-12-01
• Series: Symmetry
• Subjects: two dimensional cellular automata; lattice; star graph; rules; symmetry
• Online Access: https://www.mdpi.com/2073-8994/10/12/772

Cellular automata (CA) are parallel computational models that comprise of a grid of cells. CA is mainly used for modeling complex systems in various fields, where the geometric structure of the lattices is different. In the absence of a CA model to accommodate different types of lattices in CA, an angle-based CA model is proposed to accommodate various lattices. In the proposed model, the neighborhood structure in a two dimensional cellular automata (2D-CA) is viewed as a star graph. The vertices of the proposed graph are determined by a parameter, angle <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>&#952;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. Based on the angle <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>&#952;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, the neighborhood of the CA, which is treated as the vertices of the graph, varies. So this model is suitable for the representation of different types of two dimensional lattices such as square lattice, rectangular lattice, hexagonal lattice, etc. in CA. A mathematical model is formulated for representing CA rules which suit for different types of symmetric lattices. The star graph representation helps to find out the internal symmetries exists in CA rules. Classification of CA rules based on the symmetry exists in the rules, which generates symmetric patterns are discussed in this work.