{"id":581,"date":"2016-10-28T23:30:18","date_gmt":"2016-10-29T03:30:18","guid":{"rendered":"http:\/\/aristotle2digital.blogwyrm.com\/?p=581"},"modified":"2016-10-23T19:36:57","modified_gmt":"2016-10-23T23:36:57","slug":"game-of-life","status":"publish","type":"post","link":"https:\/\/aristotle2digital.blogwyrm.com\/?p=581","title":{"rendered":"Game of Life"},"content":{"rendered":"

Last month\u2019s column<\/a> explored the surprising complexity that results from the repeated application of the simple set of rules called the Half or Triple-Plus-One<\/strong> (HTPO).\u00a0 Despite the fact that the rules are easy to understand and apply, composing their application iteratively led to rich patterns that defy the ability of the current state of mathematical logic to fully predict or prove.\u00a0 So it should come as no surprise that there similar systems exhibit \u2018big things in small packages\u2019 behavior.<\/p>\n

One class of systems where the application of simple local rules leads to complex global behavior is known as cellular automata<\/a>.\u00a0 These systems are usually modeled as living on a grid or lattice usually with a rectangular topology, although other connection schemes exist.<\/p>\n

The state of the system is specified by a discrete state variable living at the grid point, with the most common state being specified by the integers 0 or 1, corresponding to \u2018on\u2019 or \u2018off\u2019.\u00a0 The transition rules for deciding whether a grid point turns from on to off or vice versa are usually based on the local neighborhood of the grid point and are usually specified by relatively simple rules.<\/p>\n

As in the HTPO case, the fun begins when the entire system dynamics for extended systems are taken into account.\u00a0 Even though a specific grid point only interacts directly with its neighbors (the precise definition of what these are differ from system to system) its future evolution is determined by its neighbor\u2019s interactions with their neighbors and so on.\u00a0 In a real sense, local rules propagate out globally, and this is how the complexity results.<\/p>\n

Perhaps the best known and popular cellular automaton system is Conway\u2019s Game of Life<\/a>.\u00a0 This system takes place on a rectangular grid with the state of the system specified by 0 for off (\u2018dead\u2019) and 1 for on (\u2018alive\u2019). \u00a0Each cell has 8 neighbors, the closest cells to the north, north-east, east, south-east, south, south-west, west, and north-west.\u00a0 The rules for how a particular grid cell evolves in time are:<\/p>\n