## On the Iota-Delta Function: Mathematical Representation of Two-Dimensional Cellular Automata

**Luan C. de S. M. Ozelim*****André L. B. Cavalcante ^{†}**

*Department of Civil and Environmental Engineering*

*University of Brasilia*

*Brasilia, DF, 70910-900, Brazil*

**luanoz@gmail.com*

^{†}abrasil@unb.br#### Abstract

Even though the patterns appearing in the evolution of two-dimensional cellular automata have been deeply studied, the evolution rules themselves have not received the same amount of attention. In the present paper, the evolution rules of totalistic and outer totalistic two-dimensional cellular automata for a set of neighborhood templates have been expressed in terms of the iota-delta function. Additionally, the idea of iota-delta function bases for the rule space of two-dimensional cellular automata evolution is introduced. By means of suitable bases, every two-dimensional cellular automaton can have its evolution rule described in terms of the iota-delta function. This approach enables investigating the evolution rules as mathematical functions and the evolution itself as a function composition process.