Fuzzy hexagonal automata and snowflakes

There is a common perception that snowflakes are 6-sided and a common quip that "no two snowflakes are alike". The 6-fold symmetry suggests that the growth is deterministic, while the differences suggest the growth is so sensitive to conditions that remarkable variety appears. We investigate fuzzy automata that give a great diversity of growth patterns, have sensitivity to background conditions, and which maintain the symmetry of snowflakes. Introduction As early as 1611 Kepler wrote about the mystery of the six-symmetry of snowflakes, suggesting that it was related to the close packing of spheres [1]. Modern atomic theory makes it clear that the hexagonal packing does play a key role in ice crystal shape [2]; however, there remains much mystery in the details of the intricate designs we see in snowflakes. Cellular automata have been viewed as models of growth for many years. Perhaps the most famous automaton, the Game of Life [3], yields rich and complex patterns that continue to generate ongoing study [4]. Mackay suggested in 1976 that automata could be used to simulate snowflake growth [5]. He illustrated with a simple branching tree structure created by Ulam using an automaton on a square lattice. He went on to suggest ideas for rules and that vapor pressure and convexity should play important roles. Wolfram has much more recently written at length on automata [6]. He suggests that simple automata are responsible for much of the complexity that we observe in nature. In particular, he describes the simple and impressive snowflake automaton in the style of Packard [7,8]. However, that automaton leads to limited diversity. They suggest that using larger neighborhoods or accounting for temperature rises near recently added ice might enhance the reality. We recently implemented this Packard-Wolfram snowflake automaton in J and generalized it to various Boolean automata on a hexagonal array [9]. While the behavior was in some cases wild, there did not seem to be diverse, complex behavior resembling growth in those Boolean automata. Here we consider some generalized automata that maintain six-fold symmetry and which exhibit diverse growth patterns. The cells of the automata we will consider are arranged in a hexagonal lattice and each cell will be allowed to have a fuzzy value; that is, any value between 0 and 1. The automata use simple local arithmetic combinations, depending upon the neighborhood configuration, so that the six-fold symmetry persists. We are interested in exploring the diversity of growth patterns that these automata generate. In particular, we are interested in sensitivity to background conditions. While we have an eye towards the kind of diversity seen in real snowflake growth, we are not modeling the physics of snowflake ↑ [email protected]