# Camille Zajac

Art, Economics, Physics and Mathematics.

## 21.11.18

### Conway's Game of Life

The

*, also known simply as***Game of Life***, is a cellular automaton devised by the British mathematician John Horton Conway in 1970.***Life**^{[1]}
The game is a zero-player game, 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, or, for advanced players, by creating patterns with particular properties.

## Rules[edit]

The universe of the

*Game of Life*is an infinite, two-dimensional orthogonalgrid 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:- Any live cell with fewer than two live neighbors dies, as if by underpopulation.
- Any live cell with two or three live neighbors lives on to the next generation.
- Any live cell with more than three live neighbors dies, as if by overpopulation.
- Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

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.## Origins[edit]

In late 1940, 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.^{[2]}This turned out not to be realistic with the technology available at the time. Thus, ingeniously, Stanisław Ulam invented cellular automata, which were intended to simulate von Neumann's theoretical electromagnetic constructions. 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.^{[citation needed]}
Motivated by questions in mathematical logic and in part by work on simulation games by Ulam, among others, John Conway began doing experiments in 1968 with a variety of different 2D cellular automaton rules.

^{[3]}Conway's initial goal was to define an interesting and unpredictable cell automaton. 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.
Conway chose his rules carefully, after considerable experimentation, to meet these criteria:

- There should be no explosive growth.
- There should exist small initial patterns with chaotic, unpredictable outcomes.
- There should be potential for von Neumann universal constructors.
- The rules should be as simple as possible, whilst adhering to the above constraints.
^{[4]}

The game made its first public appearance in the October 1970 issue of

*Scientific American*, in Martin Gardner's "Mathematical Games" column. Theoretically,*Conway's Life*has the power of a universal Turing machine: anything that can be computed algorithmically can be computed within*Life*.^{[5]}^{[6]}^{[7]}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)."^{[8]}
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.^{[citation needed]}The game can also serve as a didactic analogy, used to convey the somewhat counter-intuitive notion that design and organization can spontaneously emerge in the absence of a designer. 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.^{[9]}^{[10]}^{[11]}
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: a fun way to use otherwise wasted CPU cycles. For some, however,*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 Life board.^{[12]}^{[13]}## Examples of patterns[edit]

Many different types of patterns occur in the

*Game of Life*, which are classified according to their behaviour. Common pattern types include:*still lifes*, which do not change from one generation to the next;*oscillators*, which return to their initial state after a finite number of generations; and*spaceships*, which translate themselves across the grid.
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 1103 generations to stabilize, by which time it has a population of 116 and has generated six escaping gliders;^{[14]}these were the first spaceships ever discovered.^{[15]}
Some frequently occurring

^{[16]}^{[17]}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*^{[18]}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,^{[19]}and oscillators of periods 4, 8, 14, 15, 30 and a few others have been seen to arise from random initial conditions.^{[20]}Patterns which evolve for long periods before stabilizing are called*Methuselahs*, the first-discovered of which was the R-pentomino.*Diehard*is a pattern that eventually disappears, rather than stabilizing, after 130 generations, which is conjectured to be maximal for patterns with seven or fewer cells.^{[21]}*Acorn*takes 5206 generations to generate 633 cells, including 13 escaped gliders.^{[22]}
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 dollars 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 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.^{[23]}In 2015 a period-120 gun called the Simkin glider gun was discovered that has fewer live cells but which is spread out across a larger bounding box at its extremities.^{[24]}
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*: a configuration that leaves behind two-by-two still life blocks as its translates itself across the game's universe.^{[25]}The third configuration creates two such patterns. The first has only ten live cells, which has been proven to be minimal.^{[26]}The second fits in a five-by-five square, and the third is only one cell high.
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.^{[27]}Gosper also constructed the first pattern with an asymptotically optimal quadratic growth rate, called a*breeder*or*lobster*, which worked by leaving behind a trail of guns.
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 construct logic gates such as*AND*,*OR*and*NOT*using gliders. 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.^{[5]}^{[6]}In fact, several different programmable computer architectures^{[28]}^{[29]}have been implemented in Conway's Life, including a pattern that simulates Tetris.^{[30]}
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.^{[6]}
On March 6, 2018, the first truly elementary knightship, Sir Robin, was discovered by Adam P. Goucher.

^{[31]}A*knightship*is a spaceship that moves two squares left for every one square it moves down, as opposed to normal spaceships which move side to side, or a glider which moves exactly diagonally. This is the first new spaceship movement pattern for an elementary spaceship found in forty-eight years.## Undecidability[edit]

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: the problem of determining whether a given program will finish running or continue to run forever from an initial input.

^{[32]}

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.## Self-replication[edit]

On May 18, 2010, Andrew J. Wade announced a self-constructing pattern dubbed Gemini that creates a copy of itself while destroying its parent.

^{[33]}^{[34]}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. 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.^{[35]}^{[36]}
On November 23, 2013, Dave Greene built the first replicator in

*Conway's Game of Life*that creates a complete copy of itself, including the instruction tape.^{[37]}In December 2015, diagonal versions of the Gemini were built.^{[38]}## Iteration[edit]

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;

^{[39]}^{[40]}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*.## Algorithms[edit]

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: they represented

*Life*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 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. So, a program that keeps track of which areas are active can save time by not updating inactive zones.

^{[41]}
To avoid decisions and branches in the counting loop, the rules can be rearranged from an egocentric approach of the inner field regarding its neighbours to a scientific observer's viewpoint: if the sum of all nine fields in a given neighborhood 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 other sum sets the inner field to death.

If it is desired 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 the successor state for the third line. If a toroidal 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.

In principle, the

*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 simply to assume that every cell outside the 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 toroidal 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 growing patterns.*Life*on a finite field is sometimes explicitly studied; some implementations, such as*Golly*, 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 Möbius strip.
Alternatively, the programmer may abandon the notion of representing the

*Life*field with a 2-dimensional array, and use a different data structure, such as a vector of coordinate pairs representing live cells. This approach allows the pattern to move about the field unhindered, as long as the 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.
For exploring large patterns at great time depths, sophisticated algorithms such as Hashlife may be useful. There is also a method, applicable to other cellular automata too, for implementation of the

*Game of Life*using arbitrary asynchronous updates whilst still exactly emulating the behaviour of the synchronous game.^{[42]}Source code examples that implement the basic*Game of Life*scenario in various programming languages, including C, C++, Java and Python can be found at Rosetta Code.^{[43]}## Variations[edit]

Main article: Life-like cellular automaton

Since

*Life*'s inception, new, similar cellular automata have been developed. The standard*Game of Life*is symbolized as B3/S23. A cell is*if it has exactly three neighbours,***B**orn*if it has two or three living neighbours, and dies otherwise. The first 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 neighbours, and lives on if there are either one or six neighbours". Cellular automata on a two-dimensional grid that can be described in this way are known as***S**urvives*Life*-like cellular automata. Another common*Life*-like automaton,*Highlife*, 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.^{[44]}^{[45]}Additional*Life*-like cellular automata exist, although the vast majority of them produce universes that are either too chaotic or too desolate to be of interest.
Some variations on

*Life*modify the geometry of the universe as well as the rule. The above variations can be thought of as 2-D square, because the world is two-dimensional and laid out in a square grid. 1-D square variations, known as elementary cellular automata,^{[46]}and 3-D square variations have been developed, as have 2-D hexagonal and 2-D triangular variations. A variant using non-periodic tile grids has also been made.^{[47]}
Conway's rules may also be generalized such that instead of two states,

*live*and*dead*, 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*Conway's Life*.
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 Sierpiński trianglewhen applied to a single live cell. The Sierpiński triangle can also be observed in Conway's Game of Life by examining the long-term growth of a long single-cell-thick line of live cells,^{[48]}as well as in*Highlife*,*Seeds*(B2/S), and Wolfram's Rule 90.^{[49]}*Immigration*is a variation that is very similar to

*Conway's Game of Life*, except that there are two

*on*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 spaceships and other objects within the game.

^{[50]}Another similar variation, called

*QuadLife*, involves four different on states. When a new cell is born from three different on neighbours, it takes on the fourth value, and otherwise, like

*Immigration*, it takes the majority value.

^{[51]}Except for the variation among on cells, both of these variations act identically to Life.

## 20.11.18

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