Acta Cryst. (2004). A60, 257±262 DOI: 10.1107/S0108767304007585 257 research papers Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Received 10 February 2004 Accepted 30 March 2004 # 2004 International Union of Crystallography Printed in Great Britain ± all rights reserved Crystal structures and cellular automata Sergey V. Krivovichev Department of Crystallography, St Petersburg State University, University Emb. 7/9, 199034 St Petersburg, Russia. Correspondence e-mail: skrivovi@mail.ru Cellular automata are used as dynamic topological models of crystal structures based upon frameworks consisting of fundamental building blocks. Structure is considered as an automaton that works in accord with a prescribed program. Cellular automata models are constructed for metal sul®de frameworks in minerals and compounds of the pentlandite±djer®sherite±bartonite series, for zeolite ACO and compounds with pharmacosiderite structure, leucophosphite, phosphovanadylite etc. 1. Introduction The theory of crystal growth is well elaborated in its ther- modynamic and kinetic aspects. However, the process of the growth of crystal structures in the crystal-chemical sense is poorly understood, i.e. how complex structures grow in terms of atoms, atomic clusters, molecules etc . On one hand, the crystal-chemical theory of crystal growth should be based upon direct observations of events happening on the surface of a crystal with necessary resolution at the atomic and molecular scales. On the other hand, such a theory must satisfy all present crystal-chemical knowledge acquired during the last 90 years. In this regard, construction of theoretical crystal- chemical models of the growth of complicated structures might be relevant for a future experiment-based crystal- chemical theory of crystal growth. The aim of this paper is to point out that the theory of automata may be used as a crystal-chemical model of construction of complicated structures step by step. Within this theory, the structure is thought of as a machine that works in accord with a prescribed program. In this regard, it can be compared to a computer that performs a certain sequence of a ®nite number of operations divided into single steps. A structure grows in three-dimensional space and has an active surface and a program working at this surface. The question is whether it is possible to outline general crystal-chemical rules for this program and to compare it with similar rules for chemically or structurally related crystals. As a ®rst model for a crystal-chemical model of growing structures based upon automata, the theory of cellular auto- mata may be used. Cellular automata are discrete determi- nistic dynamical systems ®rst introduced by von Neumann (1951) as models for self-reproductive biological systems. A version of a two-dimensional cellular automaton is known as a `Life game', invented by the mathematician John Conway. Application of cellular automata to modeling physical laws was considered in detail by Toffoli & Margolus (1987). Of recent interest to cellular automata is Stephen Wolfram's book A New Kind of Science (Wolfram, 2002), where he points out that cellular automata can be considered as universal models for physical, biological, sociological and other systems [for critical comments, see Krantz (2002) and Gray (2003)]. A comprehensive account of the theory of cellular automata may be found in Ilachinski (2001). We note that cellular automata have been applied in crystallography for modeling the growth of snow¯akes (Wolfram, 2002) and the growth of dendritic crystals in general (Kessler et al., 1990), and for modeling microstructures (Gandin & Rappaz, 1997) etc. As far as we know, cellular automata have never been used in crystal chemistry. In this paper, we concentrate mostly on inorganic crystal structures and, in particular, those that can be considered as composed from ®nite atomic clusters also known as funda- mental building blocks (FBBs). The FBB concept is similar to the concept of secondary building units (SBU) and is widely used for describing complex structural architectures observed in phosphates, sulfates, borates, silicates, sul®des, oxides etc . (Ban®eld & Veblen, 1992; Hawthorne, 1994; Burns et al. , 1995; Fe Ârey, 2000; Cahill & Parise, 2000). The FBB concept is especially convenient for cellular automata modeling since the growth of a structure can be thought of as consisting of single steps corresponding to an addition of a new FBB to the already existing piece of structure. 2. Cellular automata: a brief introduction A simple example of a cellular automaton is shown in Fig. 1(a). It consists of a line of square cells. Each cell takes on one of two possible states: black (1) or white (0). Each horizontal line of cells corresponds to a single step in time, t. At each step in time, each cell updates its current state according to a transition rule taking into account the states of cells in its neighborhood. The transition rule for the automaton shown in Fig. 1(a) is depicted in Fig. 1(b). This rule makes a cell white whenever both of its immediate neighbors were white on the step before. The initial condition (t = 0) is one black cell. The result of the work of this automaton is a periodic checker- board-like pattern of black and white cells.