1010:. Small isolated subpatterns 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 subpattern 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-neighbor 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
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975:. On May 18, 2010, Andrew J. Wade announced the first oblique spaceship, dubbed "Gemini", that creates a copy of itself on (5,1) further while destroying its parent. This pattern replicates in 34 million generations, and uses an instruction tape made of gliders oscillating between two stable configurations made of Chapman–Greene construction arms. These, in turn, create new copies of the pattern, and destroy the previous copy. In December 2015, diagonal versions of the Gemini were built.
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270:. Von Neumann's initial design was founded upon the notion of one robot building another robot. This design is known as the kinematic model. As he developed this design, von Neumann came to realize the great difficulty of building a self-replicating robot, and of the great cost in providing the robot with a "sea of parts" from which to build its replicant. Neumann wrote a paper entitled "The general and logical theory of automata" for the
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770:; 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. In 2015, a gun called the "Simkin glider gun", which releases a glider every 120th generation, was discovered that has fewer live cells but which is spread out across a larger bounding box at its extremities.
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number, or list of numbers, is what is required for a dead cell to be born. The second set is the requirement for a live cell to survive to the next generation. Hence B6/S16 means "a cell is born if there are six neighbors, and lives on if there are either one or six neighbors". Cellular automata on a two-dimensional grid that can be described in this way are known as
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805:: a configuration that leaves behind two-by-two still life blocks as it translates itself across the game's universe. The third configuration creates two such patterns. The first has only ten live cells, which has been proven to be minimal. The second fits in a five-by-five square, and the third is only one cell high.
1208:, there are three or more. State transitions are then determined either by a weighting system or by a table specifying separate transition rules for each state; for example, Mirek's Cellebration's multi-coloured Rules Table and Weighted Life rule families each include sample rules equivalent to the Game of Life.
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is the most common period-3 oscillator. The great majority of naturally occurring oscillators have a period of 2, like the blinker and the toad, but oscillators of all 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
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is a cross-platform (Windows, Macintosh, Linux, iOS, and
Android) open-source simulation system for the Game of Life and other cellular automata (including all Life-like cellular automata, the Generations family of cellular automata from Mirek's Cellebration, and John von Neumann's 29-state cellular
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The simultaneity means that when each cell counts the number of live neighbours around it, it uses its neighbours' old states before the update, not their new states after the update. If the cells are instead updated in reading order, so that each cell uses the old states of the cells to its right
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There are now thousands of Game of Life programs online, so a full list will not be provided here. The following is a small selection of programs with some special claim to notability, such as popularity or unusual features. Most of these programs incorporate a graphical user interface for pattern
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array. The result is that active areas that move across a field edge reappear at the opposite edge. Inaccuracy can still result if the pattern grows too large, but there are no pathological edge effects. Techniques of dynamic storage allocation may also be used, creating ever-larger arrays to hold
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To save memory, the storage can be reduced to one array plus two line buffers. One line buffer is used to calculate the successor state for a line, then the second line buffer is used to calculate the successor state for the next line. The first buffer is then written to its line and freed to hold
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approach of the inner field regarding its neighbours to a scientific observer's viewpoint: if the sum of all nine fields in a given neighbourhood is three, the inner field state for the next generation will be life; if the all-field sum is four, the inner field retains its current state; and every
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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 may swap roles so that the successor array in the
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in the Game of Life that creates a complete copy of itself, including the instruction tape. In
October 2018, Adam P. Goucher finished his construction of the 0E0P metacell, a metacell capable of self-replication. This differed from previous metacells, such as the OTCA metapixel by Brice Due, which
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based on Conway's Life. In the game, Life continually generates on a group of cells within a "petri dish". The patterns formed are smoothed and rounded to look like a growing amoeba spewing smaller ones (actually gliders). Special "probes" zap the "blob" to keep it from overflowing the dish while
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In principle, the Game of Life field is infinite, but computers have finite memory. This leads to problems when the active area encroaches on the border of the array. Programmers have used several strategies to address these problems. The simplest strategy is to assume that every cell outside the
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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, so a program that keeps
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and other objects within the game. Another similar variation, called QuadLife, involves four different on states. When a new cell is born from three different on neighbours, it takes the fourth value, and otherwise, like
Immigration, it takes the majority value. Except for the variation among on
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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
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Conway originally conjectured that no pattern can grow indefinitely—i.e. that for any initial configuration with a finite number of living cells, the population cannot grow beyond some finite upper limit. In the game's original appearance in "Mathematical Games", Conway offered a prize of fifty
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rulespace, with 2 possible cellular automaton rules (the Game of Life again being one of them). These are rules that use the same square grid as the Life-like rules and the same eight-cell neighbourhood, and are likewise invariant under rotation and reflection. However, in isotropic rules, the
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Since the Game of Life's inception, new, similar cellular automata have been developed. The standard Game of Life is symbolized in rule-string notation as B3/S23. A cell is born if it has exactly three neighbours, survives if it has two or three living neighbours, and dies otherwise. The first
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system for creating a reductionist model of self-replication. Ulam and von
Neumann created a method for calculating liquid motion in the late 1950s. The driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbours'
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The popularity of the Game of Life was helped by its coming into being at the same time as increasingly inexpensive computer access. 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
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Xlife is a cellular-automaton laboratory by Jon
Bennett. The standard UNIX X11 Game of Life simulation application for a long time, it has also been ported to Windows. It can handle cellular automaton rules with the same neighbourhood as the Game of Life, and up to eight possible states per
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Alternatively, programmers may abandon the notion of representing the Game of Life field with a two-dimensional array, and use a different data structure, such as a vector of coordinate pairs representing live cells. This allows the pattern to move about the field unhindered, as long as the
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that any given pattern would do so. Moreover, some "simple initial patterns" should "grow and change for a considerable period of time" before settling into a static configuration or a repeating loop. Conway later wrote that the basic motivation for Life was to create a "universal" cellular
939:), as opposed to moving orthogonally or along a 45° diagonal. This is the first new spaceship movement pattern for an elementary spaceship found in forty-eight years. "Elementary" means that it cannot be decomposed into smaller interacting patterns such as gliders and still lifes.
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cycles. For some, however, the Game of Life had more philosophical connotations. It developed a cult following through the 1970s and beyond; current developments have gone so far as to create theoretic emulations of computer systems within the confines of a Game of Life board.
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Some variations on the Game of Life modify the geometry of the universe as well as the rules. The above variations can be thought of as a two-dimensional square, because the world is two-dimensional and laid out in a square grid. One-dimensional square variations, known as
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population does not exceed the size of the live-coordinate array. The drawback is that counting live neighbours becomes a hash-table lookup or search operation, slowing down simulation speed. With more sophisticated data structures this problem can also be largely solved.
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Mirek's
Cellebration is a freeware one- and two-dimensional cellular automata viewer, explorer, and editor for Windows. It includes powerful facilities for simulating and viewing a wide variety of cellular automaton rules, including the Game of Life, and a scriptable
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can be computed within the Game of Life. Gardner wrote, "Because of Life's analogies with the rise, fall, and alterations of a society of living organisms, it belongs to a growing class of what are called 'simulation games' (games that resemble real-life processes)."
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array is dead. This is easy to program but leads to inaccurate results when the active area crosses the boundary. A more sophisticated trick is to consider the left and right edges of the field to be stitched together, and the top and bottom edges also, yielding a
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Additional Life-like cellular automata exist. The vast majority of these 2 different rules produce universes that are either too chaotic or too desolate to be of interest, but a large subset do display interesting behaviour. A further generalization produces the
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were similar: they represented the patterns as two-dimensional arrays in computer memory. Typically, two arrays are used: one to hold the current generation, and one to calculate its successor. Often 0 and 1 represent dead and live cells, respectively. A nested
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with more states. The name 0E0P is short for "Zero
Encoded by Zero Population", which indicates that instead of a metacell being in an "off" state simulating empty space, the 0E0P metacell removes itself when the cell enters that state, leaving a blank space.
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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.
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of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed, live or dead; births and deaths occur simultaneously, and the discrete moment at which this happens is sometimes called a
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and below it but the new states of the cells to its left and above it, a different cellular automaton results, which is known as NaiveLife because it is a common beginners' mistake among people attempting to program Conway's Game of Life.
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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
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955:, which means that given an initial pattern and a later pattern, no algorithm exists that can tell whether the later pattern is ever going to appear. Given that the Game of Life is Turing-complete, this is a corollary of the
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Frequently occurring examples (in that they emerge frequently from a random starting configuration of cells) of the three aforementioned pattern types are shown below, with live cells shown in black and dead cells in white.
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states, often expressed as two different colours. Whenever a new cell is born, it takes on the on state that is the majority in the three cells that gave it birth. This feature can be used to examine interactions between
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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
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when applied to a single live cell. The
Sierpinski triangle can also be observed in the Game of Life by examining the long-term growth of an infinitely long single-cell-thick line of live cells, as well as in Highlife,
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Computers have been used to follow and simulate the Game of Life since it was first publicized. When John Conway was first investigating how various starting configurations developed, he tracked them by hand using a
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may be useful. There is also a method for implementation of the Game of Life and other cellular automata using arbitrary asynchronous updates while still exactly emulating the behaviour of the synchronous game.
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only worked with already constructed copies near them. The 0E0P metacell works by using construction arms to create copies that simulate the programmed rule. The actual simulation of the Game of Life or other
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began doing experiments in 1968 with a variety of different two-dimensional cellular automaton rules. Conway's initial goal was to define an interesting and unpredictable cellular automaton. According to
155:, meaning that its evolution is determined by its initial state, requiring no further input. One interacts with the Game of Life by creating an initial configuration and observing how it evolves. It is
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editing and simulation, the capability for simulating multiple rules including the Game of Life, and a large library of interesting patterns in the Game of Life and other cellular automaton rules.
1164:, is described by the rule B36/S23, because having six neighbours, in addition to the original game's B3/S23 rule, causes a birth. HighLife is best known for its frequently occurring replicators.
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I'm pretty sure this is because you've accidentally created an implementation of what's sometimes known as NaiveLife (as it's a common mistake made by many people coding CGoL for the first time):
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last iteration becomes the current array in the next iteration, or one may copy the values of the second array into the first array then update the second array from the first array again.
3155:: "A simpler kind of unbounded growth occurs if one starts from an infinite line of black cells. In that case, the evolution is effectively 1D, and turns out to follow elementary rule 22"
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Patterns relating to fractals and fractal systems may also be observed in certain Life-like variations. For example, the automaton B1/S12 generates four very close approximations to the
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that a particular pattern would make endless copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so. This design is known as the
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Early patterns with unknown futures, such as the R-pentomino, led computer programmers to write programs to track the evolution of patterns in the Game of Life. Most of the early
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dollars (equivalent to $ 390 in 2023) to the first person who could prove or disprove the conjecture before the end of 1970. The prize was won in
November by a team from the
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failed to stabilize in a small number of generations. In fact, it takes 1103 generations to stabilize, by which time it has a population of 116 and has generated six escaping
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Brown, Nico; Cheng, Carson; Jacobi, Tanner; Karpovich, Maia; Merzenich, Matthias; Raucci, David; Riley, Mitchell (5 December 2023). "Conway's Game of Life is
Omniperiodic".
1101:, support a choice of the standard infinite field, a field infinite only in one dimension, or a finite field, with a choice of topologies such as a cylinder, a torus, or a
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With the advent of microcomputers and Cromemco's graphics board, Life became a favorite display program for video monitors and led to a revival of interest in the game.
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Until the 2010s, all known spaceships could only move orthogonally or diagonally, whereas the existence of moving patterns that move like knights had been predicted by
1774:
373:, used to convey the somewhat counter-intuitive notion that design and organization can spontaneously emerge in the absence of a designer. For example, philosopher
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positions of neighbour cells relative to each other may be taken into account in determining a cell's future state—not just the total number of those neighbours.
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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
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Since its publication, the Game of Life has attracted much interest because of the surprising ways in which the patterns can evolve. It provides an example of
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315:, Conway experimented with different rules, aiming for rules that would allow for patterns to "apparently" grow without limit, while keeping it difficult to
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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
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board with its black and white stones. This was tedious and prone to errors. The first interactive Game of Life program was written in an early version of
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1430:, a certain pattern that appears throughout the game as a tattoo on the heroine Momo Aizaki has to be entered into the Game of Life to complete the game (
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working within a cellular automaton with a small neighbourhood (only those cells that touch are neighbours; for von Neumann's cellular automata, only
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A sample of a 48-step oscillator along with a 2-step oscillator and a 4-step oscillator from a two-dimensional hexagonal Game of Life (rule H:B2/S34)
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magazine, and Banthorpe followed this with a three-dimensional version in the May 1984 issue. Susan Stepney, Professor of Computer Science at the
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has used the analogy of the Game of Life "universe" extensively to illustrate the possible evolution of complex philosophical constructs, such as
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array is used, a third buffer is needed so that the original state of the first line in the array can be saved until the last line is computed.
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917:. In fact, several different programmable computer architectures have been implemented in the Game of Life, including a pattern that simulates
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Many different types of patterns occur in the Game of Life, which are classified according to their behaviour. Common pattern types include:
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can be built which contains a Turing complete computer, and which can build many types of complex objects, including more copies of itself.
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is a pattern that disappears after a long time. Starting patterns of eight or more cells can be made to die after an arbitrarily long time.
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204:, which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:
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of the Game of Life in 2012. Users who search for the term are shown an implementation of the game in the search results page.
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959:: the problem of determining whether a given program will finish running or continue to run forever from an initial input.
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It is a model and simulation that is interesting to watch and can show that simple things can become complicated problems.
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includes a basic implementation of the Game of Life in it, which is connected to the plot of the novel. Near the end of
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automaton) by Andrew Trevorrow and Tomas Rokicki. It includes the Hashlife algorithm for extremely fast generation, and
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growing patterns. The Game of Life on a finite field is sometimes explicitly studied; some implementations, such as
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1854:, ed., Cerebral Mechanisms in Behavior – The Hixon Symposium, John Wiley & Sons, New York, 1951, pp. 1–31.
1373:, followed this up in 1988 with Life on the Line, a program that generated one-dimensional cellular automata.
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refers to the number of ticks a pattern must iterate through before returning to its initial configuration.
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Motivated by questions in mathematical logic and in part by work on simulation games by Ulam, among others,
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sequencing. A variety of programs exist for creating sound from patterns generated in the Game of Life.
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Glider gun within a toroidal array. The stream of gliders eventually wraps around and destroys the gun.
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since 1982. The spaceships which move neither orthogonally nor diagonally are commonly referred to as
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Two early implementations of the Game of Life on home computers were by Malcolm Banthorpe written in
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examples that implement the basic Game of Life scenario in various programming languages, including
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393:. For many, the Game of Life was simply a programming challenge: a fun way to use otherwise wasted
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The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics
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of the preceding one. The rules continue to be applied repeatedly to create further generations.
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In 2018, the first truly elementary knightship, Sir Robin, was discovered by Adam P. Goucher. A
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Immigration is a variation that is very similar to the Game of Life, except that there are two
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behaviours. Thus was born the first system of cellular automata. Like Ulam's lattice network,
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are two-dimensional, with his self-replicator implemented algorithmically. The result was a
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Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.
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microcomputers, and a display from that program filled the cover of the June 1976 issue of
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To avoid decisions and branches in the counting loop, the rules can be rearranged from an
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is a spaceship that moves two squares left for every one square it moves down (like a
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2614:"A Turing Machine in Conway's Game of Life, extendable to a Universal Turing Machine"
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For exploring large patterns at great time depths, sophisticated algorithms such as
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1747:"Re: Strange spaceship that is supposed to be impossible and infinite cell spread"
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conditions. Patterns which evolve for long periods before stabilizing are called
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Any live cell with more than three live neighbours dies, as if by overpopulation.
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Any live cell with fewer than two live neighbours dies, as if by underpopulation.
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Any live cell with two or three live neighbours lives on to the next generation.
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track of which areas are active can save time by not updating inactive zones.
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takes 5,206 generations to generate 633 cells, including 13 escaped gliders.
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2967:. Alexandria, Virginia, USA: IEEE Computer Society Press. pp. 201–209.
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Various musical composition techniques use the Game of Life, especially in
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357:. A version of Life that incorporates random fluctuations has been used in
80:"Conway game" redirects here. For Conway's surreal number game theory, see
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Note (f) for structures in class 4 systems: Structures in the Game of Life
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Conway's rules may also be generalized such that instead of two states,
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1547:"The fantastic combinations of John Conway's new solitaire game 'life'"
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A color version of the Game of Life was written by Ed Hall in 1976 for
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The game made its first public appearance in the October 1970 issue of
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connected to two counters. This has the same computational power as a
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cells, both of these variations act identically to the Game of Life.
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John von Neumann, "The general and logical theory of automata," in
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1256:
1175:
1091:
1085:
Red glider on the square lattice with periodic boundary conditions
1080:
1072:
1067:
1048:
1042:
905:
using gliders. It is possible to build a pattern that acts like a
88:
2719:
2005:
Conway, private communication to the 'Life list', 14 April 1999.
1246:
266:, Ulam's colleague at Los Alamos, was working on the problem of
3577:
3470:
3243:
Proceedings of the 2004 International Computer Music Conference
2915:
2859:
2390:"Math's 'Game of Life' Reveals Long-Sought Repeating Patterns"
26:
3508:"Type 'Conway's Game of Life' on Google and See What Happens"
3285:"glitchDS – Cellular Automaton Sequencer For The Nintendo DS"
1461: – Two-dimensional Turing machine with emergent behavior
258:
in the 1940s, studied the growth of crystals, using a simple
3423:
459:
2132:
Darwin's Dangerous Idea: Evolution and the Meanings of Life
440:. During this early research, Conway discovered that the R-
3570:, recurrence relation for iterating Conway's Game of Life.
3551:
1337:. The results were published in the October 1970 issue of
120:(green) in its wake, which in turn create gliders (blue) (
2636:"Build a working game of Tetris in Conway's Game of Life"
2415:"Most seen natural occurring ash objects in Game of Life"
1956:
Bialynicki-Birula, Iwo; Bialynicka-Birula, Iwona (2004).
992:
rules is done by simulating an equivalent rule using the
848:, which move along leaving behind a trail of debris; and
291:
cells), and with 29 states per cell. Von Neumann gave an
1906:
Collected works. 4: Continuous geometry and other topics
1694:"NaiveLife Emulated: A reading-order simulation of Life"
3126:"First gliders navigate ever-changing Penrose universe"
412:, which do not change from one generation to the next;
2692:"First replicating creature spawned in life simulator"
177:
an infinite, two-dimensional orthogonal grid of square
2275:. The Online Life-Like CA Soup Search. Archived from
714:, the first-discovered of which was the R-pentomino.
198:, respectively). Every cell interacts with its eight
3233:
Burraston, Dave; Edmonds, Ernest; Livingstone, Dan;
2299:"Spontaneous appeared Spaceships out of Random Dust"
1908:(Repr ed.). Oxford Frankfurt: Pergamon Press.
3849:
3833:
3802:
3765:
3739:
3701:
3615:
3426:. The BBC and Master Computer Public Domain Library
2961:
Self-Reproduction in Asynchronous Cellular Automata
1864:Kemeny, John G. (1955). "Man viewed as a machine".
448:; these were the first spaceships ever discovered.
3563:Cellular Automata FAQ – Conway's Game of Life
3327:"Game Of Life Music Sequencer For iOS, Runxt Life"
2129:
2100:
983:On November 23, 2013, Dave Greene built the first
871:, which worked by leaving behind a trail of guns.
436:, and physical game boards, such as those used in
182:, each of which is in one of two possible states,
3239:"Cellular Automata in MIDI based Computer Music"
2509:, ConwayLife forums, April 28th, 2015, posts by
274:in 1948. Ulam was the one who suggested using a
1827:Cellular Automata: A Discrete View of the World
1558:. Vol. 223, no. 4. pp. 120–123.
1492: – Artificial life program (simulation of
1486: – Wiki dedicated to Conway's Game of Life
2965:2002 NASA/DoD Conference on Evolvable Hardware
2365:"LifeWiki:Game of Life Status page - LifeWiki"
1393:scriptability for both editing and simulation.
424:, which translate themselves across the grid.
3589:
1363:. The first was in the January 1984 issue of
1066:the successor state for the third line. If a
8:
1985:von Neumann, John; Burks, Arthur W. (1966).
1796:. Sterling Publishing Company, Inc. p.
1721:"Re: Thread For Your Accidental Discoveries"
2959:Nehaniv, Chrystopher L. (15–18 July 2002).
2184:"A Turing Machine in Conway's Game of Life"
1958:Modeling Reality: How Computers Mirror Life
1773:: CS1 maint: numeric names: authors list (
3596:
3582:
3574:
2513:("simsim314") and Dongook Lee ("Scorbie").
2046:Alstrøm, Preben; Leão, João (1994-04-01).
2013:
2011:
3250:
3024:HighLife – An Interesting Variant of Life
2980:
2806:"Automate Cellulaire - Passe-science #27"
2348:
2273:"Census Results in Conway's Game of Life"
1669:
1471:Self-organization § Computer science
1062:other sum sets the inner field to death.
65:Learn how and when to remove this message
3060:"Life-like cellular automata - LifeWiki"
2507:The Hunting of the New Herschel Conduits
2206:"Spartan universal computer-constructor"
1819:
1817:
1615:Winning Ways for your Mathematical Plays
788:
776:
748:
738:
728:
619:
536:
463:
369:. The game can also serve as a didactic
107:
2939:"About my Conway's Game of Life Applet"
2741:"Universal Constructor Based Spaceship"
1527:
1507:
3548:, extensive lexicon with many patterns
1932:Cellular Automata: A Discrete Universe
1766:
1160:. Another common Life-like automaton,
2912:"The Online Life-Like CA Soup Search"
2714:
2712:
1596:
1594:
1592:
1590:
1588:
1047:The Game of Life on the surface of a
764:Massachusetts Institute of Technology
7:
1629:
1627:
1625:
1537:
1535:
1533:
1531:
221:The initial pattern constitutes the
175:The universe of the Game of Life is
49:Please help consolidate the article.
3897:Games and sports introduced in 1970
2230:. LifeWiki. 1983. pp. 219, 223
2048:"Self-organized criticality in the
1988:Theory of Self-Reproducing Automata
3489:"Dr. Blob's Organism - It's free!"
2136:. New York: Simon & Schuster.
1564:10.1038/scientificamerican1070-120
1308:Core Duo 2 GHz CPU using Golly in
77:Two-dimensional cellular automaton
25:
2836:"Fully self-directed replication"
2182:Paul Rendell (January 12, 2005).
2019:Paul Chapman (11 November 2002).
1886:10.1038/scientificamerican0455-58
1012:cellular automaton speed of light
836:Later discoveries included other
301:von Neumann universal constructor
3506:Wasserman, Todd (12 July 2012).
2885:"Game of Life Object Statistics"
2680:, LifeWiki. Retrieved 2018-11-21
2457:Koenig, H. (February 21, 2005).
2413:Achim Flammenkamp (2004-09-07).
2301:. Achim Flammenkamp (1995-12-09)
1581:from the original on 2022-10-09.
826:
816:
811:
685:
667:
649:
631:
608:
592:
578:
564:
550:
525:
513:
501:
489:
475:
341:: anything that can be computed
285:universal copier and constructor
262:as his model. At the same time,
239:
31:
3401:from the original on 2022-10-09
3104:"Elementary Cellular Automaton"
2812:from the original on 2021-12-11
1618:(2nd ed.). A K Peters Ltd.
281:von Neumann's cellular automata
3306:"Game Of Life Music Sequencer"
1788:Pickover, Clifford A. (2009).
883:. It is possible to construct
256:Los Alamos National Laboratory
1:
3388:Journal of Cellular Automata
3329:. Synthtopia.com. 2011-01-12
3308:. Synthtopia.com. 2009-04-29
3287:. Synthtopia.com. 2008-05-29
2804:Passe-Science (2019-05-29).
2690:Aron, Jacob (16 June 2010).
2524:"Block-laying switch engine"
1993:University of Illinois Press
1193:variations. A variant using
1187:elementary cellular automata
1151:Life-like cellular automaton
3424:"Acorn User Magazine Scans"
3357:. No. 10. pp. 6–7
3347:Helmers, Carl (June 1976).
2161:. New York: Penguin Books.
1929:Ilachinski, Andrew (2001).
1158:Life-like cellular automata
3918:
3166:"Life Imitates Sierpinski"
2860:"0E0P metacell - LifeWiki"
2840:Complex Projective 4-Space
2388:Stone, Alex (2024-01-18).
2107:. Boston: Back Bay Books.
1904:Von Neumann, John (1976).
1197:grids has also been made.
1148:
879:can be used to simulate a
803:block-laying switch engine
79:
18:Conway's game of life
3537:Conway's Game of Life
3026:by David Bell (.zip file)
2078:10.1103/PhysRevE.49.R2507
2021:"Life Universal Computer"
1830:. Wiley & Sons, Inc.
1671:10.4249/scholarpedia.1816
1404:Dr. Blob's Organism is a
994:von Neumann neighbourhood
466:
3887:Cellular automaton rules
3757:Von Neumann neighborhood
2834:apgoucher (2018-11-12).
2459:"New Methuselah Records"
1824:Schiff, Joel L. (2011).
1191:hexagonal and triangular
911:universal Turing machine
339:universal Turing machine
268:self-replicating systems
44:too many section headers
3450:www-users.cs.york.ac.uk
3168:. ConwayLife.com forums
3007:"Conway's Game of Life"
2973:10.1109/EH.2002.1029886
2658:"Elementary knightship"
2594:. conwaylife.com forums
2592:"Programmable computer"
2103:Consciousness Explained
2099:Dennett, D. C. (1991).
1962:Oxford University Press
1634:Izhikevich, Eugene M.;
1409:destroying its nucleus.
367:nonequilibrium dynamics
299:model, and is called a
254:, while working at the
230:. Each generation is a
144:devised by the British
132:, also known simply as
3082:"Isotropic - LifeWiki"
2157:Dennett, D.C. (2003).
2128:Dennett, D.C. (1995).
1552:. Mathematical Games.
1447: – Field of study
1413:Google implemented an
1313:
1181:
1086:
1078:
1054:
856:asymptotically optimal
794:
782:
754:
744:
734:
125:
105:
3605:Conway's Game of Life
3235:Miranda, Eduardo Reck
3148:A New Kind of Science
2465:on September 10, 2019
1894:1955; 192:6 (errata).
1260:
1179:
1084:
1076:
1046:
926:universal constructor
859:quadratic growth rate
792:
780:
752:
742:
732:
161:universal constructor
134:Conway's Game of Life
111:
92:
3446:"AcornUser articles"
2910:Nathaniel Johnston.
2883:Andrzej Okrasinski.
2784:"Geminoid Challenge"
1935:. World Scientific.
951:The Game of Life is
907:finite-state machine
877:sliding block memory
402:Examples of patterns
3210:Stephen A. Silver.
3187:Stephen A. Silver.
3106:. Wolfram Mathworld
3036:Stephen A. Silver.
2720:"Gemini – LifeWiki"
2568:Stephen A. Silver.
2545:Stephen A. Silver.
2486:"Gosper glider gun"
2484:Stephen A. Silver.
2434:Stephen A. Silver.
2320:Stephen A. Silver.
2249:Stephen A. Silver.
2070:1994PhRvE..49.2507A
2027:on 6 September 2009
1878:1955SciAm.192d..58K
1662:2015SchpJ..10.1816I
1606:Conway, John Horton
1555:Scientific American
1340:Scientific American
1213:Sierpinski triangle
1008:mathematical beauty
990:Moore neighbourhood
389:computer-generated
326:Scientific American
159:and can simulate a
114:puffer-type breeder
3902:John Horton Conway
3747:Moore neighborhood
3726:Life without Death
3471:"Xlife - LifeWiki"
3214:. The Life Lexicon
3191:. The Life Lexicon
3040:. The Life Lexicon
2572:. The Life Lexicon
2549:. The Life Lexicon
2488:. The Life Lexicon
2438:. The Life Lexicon
2324:. The Life Lexicon
2253:. The Life Lexicon
2064:(4): R2507–R2508.
1371:University of York
1314:
1300:) generation of a
1182:
1087:
1079:
1055:
973:oblique spaceships
963:Oblique spaceships
795:
783:
755:
745:
735:
335:Mathematical Games
149:John Horton Conway
142:cellular automaton
126:
116:(red) that leaves
112:A screenshot of a
106:
3892:Self-organization
3874:
3873:
3609:cellular automata
3568:Algebraic formula
3349:"About the Cover"
2547:"Infinite Growth"
2204:Adam P. Goucher.
2168:978-0-14-200384-8
2143:978-0-684-82471-0
2114:978-0-316-18066-5
2057:Physical Review E
1942:978-981-238-183-5
1915:978-0-08-009566-0
1465:Poietic Generator
1420:The visual novel
834:
833:
799:
798:
793:Simkin glider gun
781:Gosper glider gun
759:
758:
700:
699:
693:
692:
616:
615:
605:(period 15)
533:
532:
363:phase transitions
355:self-organization
151:in 1970. It is a
75:
74:
67:
16:(Redirected from
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3497:
3496:
3485:
3479:
3478:
3467:
3461:
3460:
3458:
3457:
3444:Stepney, Susan.
3441:
3435:
3434:
3432:
3431:
3420:
3414:
3413:
3408:
3406:
3400:
3385:
3377:McIntosh, Harold
3373:
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3364:
3362:
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3094:
3093:
3091:
3089:
3084:. Conwaylife.com
3078:
3072:
3071:
3069:
3067:
3062:. Conwaylife.com
3056:
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2984:
2956:
2950:
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2934:
2928:
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2925:
2923:
2914:. Archived from
2907:
2901:
2900:
2898:
2896:
2887:. Archived from
2880:
2874:
2873:
2871:
2870:
2856:
2850:
2849:
2847:
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2786:. Conwaylife.com
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2743:. Conwaylife.com
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2722:. Conwaylife.com
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2514:
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2498:
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2461:. Archived from
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2023:. Archived from
2015:
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1976:
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1602:Berlekamp, E. R.
1598:
1583:
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1551:
1545:(October 1970).
1539:
1516:
1512:
1477:Of Man and Manta
1299:
1297:
1291:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
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1266:
1253:Notable programs
1220:, and Wolfram's
1195:aperiodic tiling
1137:can be found at
979:Self-replication
830:
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815:
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773:
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264:John von Neumann
243:
153:zero-player game
70:
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35:
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3766:Implementations
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3493:digital-eel.com
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3143:Stephen Wolfram
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3013:. June 7, 2024.
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2638:. StackExchange
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2159:Freedom Evolves
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1719:Goucher, Adam.
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1636:Conway, John H.
1633:
1632:
1623:
1600:
1599:
1586:
1578:
1549:
1543:Gardner, Martin
1541:
1540:
1529:
1525:
1520:
1519:
1513:
1509:
1504:
1499:
1445:Artificial life
1440:
1295:
1293:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1262:
1255:
1243:
1153:
1147:
1024:
1003:
981:
969:Elwyn Berlekamp
965:
957:halting problem
945:
937:knight in chess
915:Turing complete
825:
733:The R-pentomino
701:
681:
679:
677:
663:
661:
659:
645:
643:
641:
604:
602:
588:
574:
560:
546:
485:
404:
343:algorithmically
293:existence proof
272:Hixon Symposium
260:lattice network
249:
244:
173:
157:Turing complete
85:
78:
71:
60:
54:
51:
48:
36:
32:
23:
22:
15:
12:
11:
5:
3915:
3913:
3905:
3904:
3899:
3894:
3889:
3879:
3878:
3872:
3871:
3869:
3868:
3861:
3853:
3851:
3847:
3846:
3844:
3843:
3837:
3835:
3831:
3830:
3828:
3827:
3822:
3817:
3815:Martin Gardner
3812:
3806:
3804:
3800:
3799:
3797:
3796:
3793:Anonymous;Code
3789:
3782:
3775:
3769:
3767:
3763:
3762:
3760:
3759:
3754:
3752:Speed of light
3749:
3743:
3741:
3737:
3736:
3734:
3733:
3728:
3723:
3718:
3713:
3707:
3705:
3699:
3698:
3696:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3645:
3640:
3635:
3630:
3628:Garden of Eden
3625:
3619:
3617:
3613:
3612:
3603:
3601:
3600:
3593:
3586:
3578:
3572:
3571:
3565:
3560:
3554:
3549:
3543:
3532:
3531:External links
3529:
3527:
3526:
3498:
3480:
3475:conwaylife.com
3462:
3436:
3415:
3381:"Introduction"
3368:
3339:
3318:
3297:
3276:
3269:
3225:
3202:
3179:
3157:
3135:
3117:
3095:
3073:
3051:
3028:
3016:
2998:
2991:
2951:
2929:
2902:
2875:
2864:conwaylife.com
2851:
2826:
2796:
2775:
2753:
2732:
2708:
2682:
2670:
2649:
2627:
2616:. Paul Rendell
2605:
2583:
2560:
2537:
2515:
2511:Michael Simkin
2499:
2476:
2449:
2426:
2405:
2380:
2369:conwaylife.com
2356:
2335:
2312:
2290:
2264:
2241:
2219:
2196:
2174:
2167:
2149:
2142:
2120:
2113:
2091:
2038:
2007:
1998:
1977:
1971:978-0198531005
1970:
1948:
1941:
1921:
1914:
1896:
1856:
1843:
1836:
1813:
1807:978-1402757969
1806:
1780:
1751:ConwayLife.com
1737:
1725:ConwayLife.com
1711:
1698:ConwayLife.com
1685:
1644:"Game of Life"
1642:(2015-06-21).
1621:
1584:
1526:
1524:
1521:
1518:
1517:
1506:
1505:
1503:
1500:
1498:
1497:
1487:
1481:
1473:
1468:
1462:
1456:
1448:
1441:
1439:
1436:
1428:Anonymous;Code
1423:Anonymous;Code
1411:
1410:
1402:
1398:
1394:
1302:Turing machine
1254:
1251:
1242:
1239:
1149:Main article:
1146:
1143:
1023:
1020:
1002:
999:
980:
977:
964:
961:
944:
943:Undecidability
941:
832:
831:
822:
821:
797:
796:
785:
784:
757:
756:
746:
736:
698:
697:
691:
690:
683:
673:
672:
665:
655:
654:
647:
637:
636:
629:
625:
624:
617:
614:
613:
606:
598:
597:
590:
584:
583:
576:
570:
569:
562:
556:
555:
548:
542:
541:
534:
531:
530:
523:
519:
518:
511:
507:
506:
499:
495:
494:
487:
481:
480:
473:
469:
468:
458:
403:
400:
375:Daniel Dennett
331:Martin Gardner
313:Martin Gardner
252:Stanislaw Ulam
248:
245:
238:
219:
218:
215:
212:
209:
172:
169:
165:Turing machine
82:Surreal number
76:
73:
72:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3914:
3903:
3900:
3898:
3895:
3893:
3890:
3888:
3885:
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3859:
3855:
3854:
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3848:
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3838:
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3832:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3807:
3805:
3801:
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3790:
3788:
3787:
3783:
3781:
3780:
3776:
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3764:
3758:
3755:
3753:
3750:
3748:
3745:
3744:
3742:
3738:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3711:Day and Night
3709:
3708:
3706:
3704:
3703:Life variants
3700:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
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3328:
3322:
3319:
3307:
3301:
3298:
3286:
3280:
3277:
3272:
3270:9780971319226
3266:
3262:
3258:
3253:
3252:10.1.1.6.3882
3248:
3244:
3240:
3236:
3229:
3226:
3213:
3206:
3203:
3190:
3189:"Immigration"
3183:
3180:
3167:
3161:
3158:
3154:
3150:
3149:
3144:
3139:
3136:
3131:
3130:New Scientist
3127:
3121:
3118:
3105:
3099:
3096:
3083:
3077:
3074:
3061:
3055:
3052:
3039:
3032:
3029:
3025:
3020:
3017:
3012:
3008:
3002:
2999:
2994:
2992:0-7695-1718-8
2988:
2983:
2978:
2974:
2970:
2966:
2962:
2955:
2952:
2940:
2937:Alan Hensel.
2933:
2930:
2918:on 2009-09-10
2917:
2913:
2906:
2903:
2891:on 2009-07-27
2890:
2886:
2879:
2876:
2865:
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2763:
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2713:
2709:
2697:
2696:New Scientist
2693:
2686:
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2637:
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2316:
2313:
2300:
2294:
2291:
2279:on 2009-09-10
2278:
2274:
2268:
2265:
2252:
2245:
2242:
2229:
2228:"R-pentomino"
2223:
2220:
2207:
2200:
2197:
2185:
2178:
2175:
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2164:
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1967:
1963:
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1879:
1875:
1871:
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1853:
1852:L.A. Jeffress
1847:
1844:
1839:
1837:9781118030639
1833:
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1712:
1700:. 24 May 2020
1699:
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1616:
1612:(2001–2004).
1611:
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1459:Langton's ant
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891:
886:
882:
878:
872:
870:
866:
865:
860:
857:
853:
852:
847:
846:
845:puffer trains
841:
840:
829:
824:
823:
819:
814:
810:
809:
806:
804:
791:
787:
786:
779:
775:
774:
771:
769:
765:
751:
747:
741:
737:
731:
727:
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721:
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713:
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425:
423:
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417:
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410:
401:
399:
396:
392:
386:
384:
380:
379:consciousness
376:
372:
368:
364:
360:
356:
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344:
340:
336:
332:
328:
327:
321:
318:
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290:
286:
282:
277:
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269:
265:
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237:
235:
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233:pure function
229:
224:
216:
213:
210:
207:
206:
205:
203:
202:
197:
193:
189:
185:
181:
178:
170:
168:
166:
163:or any other
162:
158:
154:
150:
147:
146:mathematician
143:
139:
135:
131:
123:
119:
115:
110:
104:
100:
96:
91:
87:
83:
69:
66:
58:
46:
45:
40:This article
38:
29:
28:
19:
3863:
3856:
3791:
3784:
3779:Life Genesis
3777:
3653:Puffer train
3607:and related
3604:
3546:Life Lexicon
3517:. Retrieved
3511:
3501:
3492:
3483:
3474:
3465:
3454:. Retrieved
3449:
3439:
3428:. Retrieved
3418:
3410:
3403:. Retrieved
3391:
3387:
3371:
3361:February 18,
3359:. Retrieved
3352:
3342:
3331:. Retrieved
3321:
3310:. Retrieved
3300:
3289:. Retrieved
3279:
3242:
3228:
3216:. Retrieved
3205:
3193:. Retrieved
3182:
3170:. Retrieved
3160:
3146:
3138:
3129:
3120:
3108:. Retrieved
3098:
3086:. Retrieved
3076:
3064:. Retrieved
3054:
3042:. Retrieved
3038:"Replicator"
3031:
3019:
3011:Rosetta Code
3010:
3001:
2960:
2954:
2942:. Retrieved
2932:
2920:. Retrieved
2916:the original
2905:
2893:. Retrieved
2889:the original
2878:
2867:. Retrieved
2863:
2854:
2843:. Retrieved
2839:
2829:
2819:– via
2814:. Retrieved
2799:
2788:. Retrieved
2778:
2766:. Retrieved
2756:
2745:. Retrieved
2735:
2724:. Retrieved
2699:. Retrieved
2695:
2685:
2678:"Elementary"
2673:
2661:. Retrieved
2652:
2640:. Retrieved
2630:
2618:. Retrieved
2608:
2596:. Retrieved
2586:
2574:. Retrieved
2563:
2551:. Retrieved
2540:
2528:. Retrieved
2518:
2502:
2490:. Retrieved
2479:
2467:. Retrieved
2463:the original
2452:
2440:. Retrieved
2429:
2418:. Retrieved
2408:
2397:. Retrieved
2393:
2383:
2372:. Retrieved
2368:
2359:
2338:
2326:. Retrieved
2315:
2303:. Retrieved
2293:
2281:. Retrieved
2277:the original
2267:
2255:. Retrieved
2244:
2232:. Retrieved
2222:
2210:. Retrieved
2199:
2187:. Retrieved
2177:
2158:
2152:
2131:
2123:
2102:
2094:
2061:
2055:
2050:game of Life
2049:
2041:
2029:. Retrieved
2025:the original
2001:
1987:
1980:
1957:
1951:
1931:
1924:
1905:
1899:
1891:
1872:(4): 58–67.
1869:
1865:
1859:
1846:
1826:
1791:
1783:
1761:
1754:. Retrieved
1750:
1740:
1728:. Retrieved
1724:
1714:
1702:. Retrieved
1697:
1688:
1653:
1649:Scholarpedia
1647:
1614:
1553:
1510:
1476:
1452:Glory Season
1451:
1432:Kok's galaxy
1427:
1421:
1419:
1412:
1406:Shoot 'em up
1375:
1364:
1358:
1351:
1345:
1338:
1335:S. R. Bourne
1331:M. J. T. Guy
1315:
1244:
1228:
1226:
1218:Seeds (B2/S)
1210:
1205:
1201:
1199:
1183:
1169:
1166:
1154:
1139:Rosetta Code
1111:
1107:
1103:Möbius strip
1096:
1088:
1064:
1056:
1052:trefoil knot
1038:
1025:
1015:
1014:and denoted
1004:
982:
972:
966:
950:
946:
932:
930:
925:
923:
900:
894:
888:
876:
873:
868:
862:
849:
843:
837:
835:
802:
800:
760:
719:
715:
709:
704:
702:
694:
540:Oscillators
467:Still lifes
453:
450:
426:
419:
413:
407:
405:
387:
348:
324:
322:
316:
305:
297:tessellation
275:
250:
231:
227:
222:
220:
199:
195:
191:
187:
183:
179:
174:
137:
133:
130:Game of Life
129:
127:
86:
61:
52:
41:
3825:Richard Guy
3820:Bill Gosper
3810:John Conway
3678:Spacefiller
3452:. AcornUser
3394:: 181–186.
2530:December 5,
2469:January 24,
2234:December 5,
2212:December 5,
1756:29 November
1730:29 November
1704:29 November
1656:(6): 1816.
1119:Source code
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3648:Oscillator
3643:Methuselah
3616:Structures
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3430:2018-12-29
3405:3 November
3333:2012-06-24
3312:2012-06-24
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2845:2019-06-25
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2701:12 October
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3237:(2004).
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3384:(PDF)
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