Extending Conway’s Game of Life

Abstract

AbstractWe introduce an extension to Conway’s Game of Life (GoL) that allows more than 2 states (dead/alive; 0/1) to be specified. State 0 is reserved for dead or lifeless cells, and states 1 to n − 1 for cells that are “alive.” Associated with state σ is the value vσ of cells such that v0 = 0, v1 = 1, and for σ = 2, …, n − 1 the cell value vσ is free to be assigned any positive integer value. The value Vij of the Moore neighborhood surrounding cells at array locations (i, j) (i = 1, …, r rows, j = 1, …, c) is the sum of state values of the 8 cells that touch the cell at location (i, j) (4 north, south, east and west side cells and 4 NE, NW, SE, and SW corner cells; except for boundary cells, which have 5 neighbors on an edge or 3 neighbors on a corner, and noting that torroidal arrays have no boundaries). The game is based on a set of rules involving the values Vij that determine whether cells at locations (i, j) in state σ, stay in this state or progress to state σ + 1, σ = 0, …, n − 2, or die (remain dead in the case of dead cells). Specifically, for two non-negative constants 0 ≤ pσL ≤ pσU, if Vij< pσL, Vij ∈ [pσL, pσU] or Vij> pσU then a cell in state σ at location (i, j) respectively stays in state σ, progresses to state σ + 1, or dies. In this paper, we illustrate some of the properties of such games of life with three, four and six states, identifying still-life, oscillator, glider and replicator pattern sequences. We also examine the long term behavior of patterns arising from random and regular (stripped diagonals) starting configurations, as well as set-piece motifs. Most importantly, we introduce two freely downloadable, Numerus runtime alterable model platforms (Numerus RAMPs) that can be used to simulate the three and four state GoLs discussed in this paper, and provide a guide on how these RAMPs can be used to explore the behavior of our GoLs.