Cluster dimensionality in the game of Go

Abstract

Go is a very popular and highly sophisticated game invented by Chinese thousands of years ago. Two players (playing black and white) have a game on a two-dimensional 19 × 19 lattice board. The players alternate and put their stones one by one on the Go board. Each of the players tries to capture his opponent's stones, and defends his own stones by constructing his territories and “eyes”. Thus Go is also an invading and self-organizing game. By using box-counting, the cluster dimensionalities of some typical completed games, performed by professional players, have been calculated and reported in this letter. Differing from the clusters of random site percolation (RSP), the clusters of the Go games show highly connected and correlated behavior. Usually, a spanning cluster can be constructed at the end for a completed game even though the concentration p (∼0.5 defined as the quotient of the number of the black (white) stones in controlled areas divided by 361) is very far from the site percolation threshold p c (=0.592746…). The present results show that, for the completed games, the cluster dimensionality of the Go is in a range of 1.8–1.9, which is close to the fractal dimensionality of the largest RSP cluster (∼1.85 at p c ) obtained in 2D 20 × 20 lattices.