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The universe of the Game of Life is an infinite, two-dimensional orthogonal grid of square cells , each of which is in one of two possible states, alive or dead , or populated and unpopulated , respectively.
Every cell interacts with its eight neighbours , which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:.
The initial pattern constitutes the seed of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed; births and deaths occur simultaneously, and the discrete moment at which this happens is sometimes called a tick.
Each generation is a pure function of the preceding one. The rules continue to be applied repeatedly to create further generations.
In late , John von Neumann defined life as a creation as a being or organism which can reproduce itself and simulate a Turing machine.
Von Neumann was thinking about an engineering solution which would use electromagnetic components floating randomly in liquid or gas.
Ulam discussed using computers to simulate his cellular automata in a two-dimensional lattice in several papers. In parallel, Von Neumann attempted to construct Ulam's cellular automaton.
Although successful, he was busy with other projects and left some details unfinished. His construction was complicated because it tried to simulate his own engineering design.
Over time, simpler life constructions were provided by other researchers, and published in papers and books. Motivated by questions in mathematical logic and in part by work on simulation games by Ulam, among others, John Conway began doing experiments in with a variety of different 2D cellular automaton rules.
Thus, he wanted some configurations to last for a long time before dying, other configurations to go on forever without allowing cycles, etc.
It was a significant challenge and an open problem for years before experts on cell automatons managed to prove that, indeed, Conway's Game of Life admitted of a configuration which was alive in the sense of satisfying Von Neumann's two general requirements.
While the definitions before Conway's Life were proof-oriented, Conway's construction aimed at simplicity without a priori providing proof the automaton was alive.
The game made its first public appearance in the October issue of Scientific American , in Martin Gardner 's " Mathematical Games " column.
Theoretically, Conway's Life has the power of a universal Turing machine: Since its publication, Conway's Game of Life has attracted much interest, because of the surprising ways in which the patterns can evolve.
Life provides an example of emergence and self-organization. Scholars in various fields, such as computer science , physics , biology , biochemistry , economics , mathematics , philosophy , and generative sciences have made use of the way that complex patterns can emerge from the implementation of the game's simple rules.
For example, cognitive scientist Daniel Dennett has used the analogy of Conway's Life "universe" extensively to illustrate the possible evolution of complex philosophical constructs, such as consciousness and free will , from the relatively simple set of deterministic physical laws, which might govern our universe.
The popularity of Conway's Game of Life was helped by its coming into being just in time for a new generation of inexpensive computer access which was being released into the market.
The game could be run for hours on these machines, which would otherwise have remained unused at night. In this respect, it foreshadowed the later popularity of computer-generated fractals.
For many, Life was simply a programming challenge: For some, however, Life had more philosophical connotations. It developed a cult following through the s and beyond; current developments have gone so far as to create theoretic emulations of computer systems within the confines of a Life board.
Many different types of patterns occur in the Game of Life , which are classified according to their behaviour. Common pattern types include: The earliest interesting patterns in the Game of Life were discovered without the use of computers.
The simplest still lifes and oscillators were discovered while tracking the fates of various small starting configurations using graph paper , blackboards , and physical game boards, such as those used in Go.
During this early research, Conway discovered that the R- pentomino failed to stabilize in a small number of generations. In fact, it takes generations to stabilize, by which time it has a population of and has generated six escaping gliders ;  these were the first spaceships ever discovered.
Some frequently occurring   examples of the three aforementioned pattern types are shown below, with live cells shown in black and dead cells in white.
Period refers to the number of ticks a pattern must iterate through before returning to its initial configuration. The pulsar  is the most common period 3 oscillator.
The great majority of naturally occurring oscillators are period 2, like the blinker and the toad, but oscillators of many periods are known to exist,  and oscillators of periods 4, 8, 14, 15, 30 and a few others have been seen to arise from random initial conditions.
Diehard is a pattern that eventually disappears, rather than stabilizing, after generations, which is conjectured to be maximal for patterns with seven or fewer cells.
Conway originally conjectured that no pattern can grow indefinitely—i. In the game's original appearance in "Mathematical Games", Conway offered a prize of fifty dollars to the first person who could prove or disprove the conjecture before the end of The prize was won in November by a team from the Massachusetts Institute of Technology , led by Bill Gosper ; the Gosper glider gun produces its first glider on the 15th generation, and another glider every 30th generation from then on.
For many years this glider gun was the smallest one known. Smaller patterns were later found that also exhibit infinite growth.
All three of the patterns shown below grow indefinitely. The first two create a single block-laying switch engine: The first has only ten live cells, which has been proven to be minimal.
Later discoveries included other guns , which are stationary, and which produce gliders or other spaceships; puffer trains , which move along leaving behind a trail of debris; and rakes , which move and emit spaceships.
It is possible for gliders to interact with other objects in interesting ways. For example, if two gliders are shot at a block in a specific position, the block will move closer to the source of the gliders.
If three gliders are shot in just the right way, the block will move farther away. This sliding block memory can be used to simulate a counter. It is possible to build a pattern that acts like a finite state machine connected to two counters.
This has the same computational power as a universal Turing machine , so the Game of Life is theoretically as powerful as any computer with unlimited memory and no time constraints; it is Turing complete.
Furthermore, a pattern can contain a collection of guns that fire gliders in such a way as to construct new objects, including copies of the original pattern.
A universal constructor can be built which contains a Turing complete computer, and which can build many types of complex objects, including more copies of itself.
This is the first new spaceship movement pattern for an elementary spaceship found in forty-eight years. Many patterns in the Game of Life eventually become a combination of still lifes, oscillators, and spaceships; other patterns may be called chaotic.
A pattern may stay chaotic for a very long time until it eventually settles to such a combination.
Life is undecidable , which means that given an initial pattern and a later pattern, no such algorithm exists that can tell whether the later pattern is ever going to appear.
This is a corollary of the halting problem: Indeed, since Life includes a pattern that is equivalent to a Universal Turing Machine UTM , this deciding algorithm, if it existed, could be used to solve the halting problem by taking the initial pattern as the one corresponding to a UTM plus an input, and the later pattern as the one corresponding to a halting state of the UTM.
It also follows that some patterns exist that remain chaotic forever. If this were not the case, one could progress the game sequentially until a non-chaotic pattern emerged, then compute whether a later pattern was going to appear.
On May 18, , Andrew J. Wade announced a self-constructing pattern dubbed Gemini that creates a copy of itself while destroying its parent.
These, in turn, create new copies of the pattern, and destroy the previous copy. Gemini is also a spaceship, and is the first spaceship constructed in the Game of Life that is an oblique spaceship, which is a spaceship that is neither orthogonal nor purely diagonal.
On November 23, , Dave Greene built the first replicator in Conway's Game of Life that creates a complete copy of itself, including the instruction tape.
From most random initial patterns of living cells on the grid, observers will find the population constantly changing as the generations tick by.
The patterns that emerge from the simple rules may be considered a form of mathematical beauty. Small isolated sub patterns with no initial symmetry tend to become symmetrical.
Once this happens, the symmetry may increase in richness, but it cannot be lost unless a nearby sub pattern comes close enough to disturb it.
In a very few cases the society eventually dies out, with all living cells vanishing, though this may not happen for a great many generations.
Most initial patterns eventually burn out, producing either stable figures or patterns that oscillate forever between two or more states;   many also produce one or more gliders or spaceships that travel indefinitely away from the initial location.
Because of the nearest-neighbour based rules, no information can travel through the grid at a greater rate than one cell per unit time, so this velocity is said to be the cellular automaton speed of light and denoted c.
Early patterns with unknown futures, such as the R-pentomino, led computer programmers across the world to write programs to track the evolution of Life patterns.
Most of the early algorithms were similar: Typically two arrays are used: Often 0 and 1 represent dead and live cells respectively.
A nested for loop considers each element of the current array in turn, counting the live neighbours of each cell to decide whether the corresponding element of the successor array should be 0 or 1.
The successor array is displayed. For the next iteration, the arrays swap roles so that the successor array in the last iteration becomes the current array in the next iteration.
A variety of minor enhancements to this basic scheme are possible, and there are many ways to save unnecessary computation. A cell that did not change at the last time step, and none of whose neighbours changed, is guaranteed not to change at the current time step as well.
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