Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

2D generalizations [of entropies]

Above 1D no systematic method seems to exist for finding exact formulas for entropies (as expected from the discussion at the end of Chapter 5). Indeed, even working out for large n how many of the 2^(n^2) possible configurations of a n × n grid of black and white squares contain no pair of adjacent black cells is difficult. Fitting the result to 2^(h n^2) one finds h0.589, but no exact formula for h has ever been found. With hexagonal cells, however, the exact solution of the so-called hard hexagon lattice gas model in 1980 showed that h0.481 is the logarithm of the largest root of a degree 12 polynomial. (The solution of the so-called dimer problem in 1961 also showed that for complete coverings of a square grid by 2-cell dominoes h=Catalan/(Pi Log[2]) 0.421.)

From Stephen Wolfram: A New Kind of Science [citation]