A thousand cells propagating across an endless grid, evolving and devolving into a seemingly random pattern. But it’s not random, these “organisms” are governed by a simple set of laws that work together to produce a beautifully enigmatic and astonishingly diverse simulation. This is the legendary Game of Life (abbreviated by GoL, or simply, Life), the awkward son of English mathematician John Horton Conway. Created over 50 years ago in 1970, Conway published his game in the Mathematical Games section of popular Scientific American columnist, Martin Gardner (Roberts, 2020). From there Life has drawn a cult following, devoted to the simplistic beauty of Conway’s Game, which demonstrates the principles of Deterministic Chaos and the Hierarchy of Life.
Elementary Cellular Automata from Wolfram’s “A New Kind of Science” (Weisstein)
What is a Cellular Automaton?
First theorized by the mathematicians John von Neumann and Stanislaw Ulam, whom Conway mentions as an inspiration for his game (Emmite, Reggia, Sipper, 2008), cellular automaton (plural, automata) is a computational model of a set of colored cells on a grid. Over iterative generations, the states of the cells are updated based on the surrounding states of the neighboring cells (Weisstein). Specifications such as grid shape, the number of cell states (essentially colors; often expressed as k = 2), and the “neighborhood” which is the size of the surrounding cells that can affect the central cell - that is generally either a Moore neighborhood (square; 8 cells) or the von Neumann neighborhood (diamond; 4 cells). Based on these factors the composition of the grid evolves into an intricate pattern as the simulation progresses.
Since then, cellular automata and its properties have been used in a variety of research, applied to physical, biological and other such models, to study fields ranging from weather forecasts to simulating galaxies. One such model is the Lattice Gas Cellular Automaton, which is used to simulate the flow of fluids (Wolfram, 2002). Later, in 2002, Stephen Wolfram published A New Kind of Science, giving insight into his comprehensive studies of cellular automata.
Game of Life - A How to Play
Conway’s Game of Life is the most well known model of cellular automata. GoL is set on an endless 2d grid, with k = 2, and a Moore neighborhood at a range of 1. The player turns on any configuration of cells, and through each successive generation, that cell and its surrounding cells are turned off and on (this is known as being “dead” or “live''). There are only 3 basic rules the game follows, and from iteration to iteration, as the game advances, these 3 rules dictate how the system plays out (Gardner, 1970):
Death: A live cell with less than 2 or greater than 3 living neighbors dies
Survival: A live cell with 2 or 3 neighbors remains live
Birth: A dead cell with 3 living neighbors becomes live
Life Forms and Discoveries
When Conway originally experimented with GoL it was a long time before the invention of the computer. Thus, hunting for patterns and “life forms” - as you can imagine - took a long time. With the development and popularization of computers, GoL attracted many followers who sought to find new undiscovered life (which are recurring patterns that interact with the environment).
Three life form classifications with examples (Lefebvre, 2022)
Conway categorized the discoveries into 3 distinct categories (though there are some life forms that fit into multiple classifications): Still Lifes, Oscillators, and Spaceships.
Still Lifes: (e.g. bee-hive) Stable patterns that once established, don’t change over generations.
Oscillators: (e.g. blinker) Repeating patterns over successive generations.
Spaceships: (e.g. glider) Patterns that form in a different location after generation(s).
To Conway, spaceships were the most interesting class of life. In larger, more elaborate simulations, these spaceships are key to transmitting information across the screen. The Glider Gun was one of the first (and smallest) spaceships to be discovered, and today some are over a hundred cells large.
Turing Machine in Game of Life (Johnston, 2009)
Though there are only 3 rules, GoL is deceiving; It is anything but a simple game. Vast megastructures have been constructed with thousands of cells, with complex interactions and exchanges of information. GoL has been proven to be a universal cellular automaton, meaning that it is capable of simulating a Turing Machine, so theoretically GoL is able to do anything a computer is capable of (De Mol, 2018). Experimentation with GoL has gone so far to construct a digital clock and even GoL inside GoL (search it up, it’s pretty cool).
Game of Life and Deterministic Chaos
The patterns and cell formations produced in GoL appear to be erratic, but they’re just a product of the initial structure in cooperation with the rules. This is exactly what the field of Chaos Theory is concerned with: The study of apparently random or unpredictable behaviors in a system governed by deterministic laws (Bishop, 2017). Deterministic chaos suggests a connection between the familiar and the chaotic that is often regarded as incompatible. Of course, this is a rather crude simplification, but there is an undeniable correlation in concept.
It’s important to understand that Chaos Theory deals with the nonlinear, the effectively impossible to predict, such as turbulence, weather, etc. In GoL a small arrangement can generate an seemingly unpredictable system with chaotic behavior. Thus, it makes sense that cellular automata have likewise been used to simulate real-world processes. GoL demonstrates one of the core Chaos Theory principles: The Butterfly Effect. Small changes in the initial conditions lead to drastic changes in the results (Wolfram, 2002).
Game of Life and Hierarchy of Life
GoL is intrinsically alike to the real world, reflecting the processes of biology. After all, it is a game of life bringing out “analogies with the rise, fall and alternations of a society of living organisms” (Gardner, 1970). GoL mirrors the hierarchy of life in a computer simulation, with cells, organisms, and communities working together. The operation of the 3 rules produces emergent new forms and properties, at first a rather humble model, that matures into a labyrinthine structure. In GoL, cells conjoin to form life forms, in some intricate simulations those life forms transfer information and work towards a common function, in biology this is comparable to an organ system.
The interactions between the cells have similar biological counterparts: Cell division, cell movement, cell adhesion, differentiation, induction, and cell death (Caballero, Hodge, Hernandez, 2016). Cell death is exhibited by the death rule, wherein, the ecological concepts of overpopulation and isolation are at play.
Conclusion
Conway died in April of 2020, the 50th anniversary of the publication of GoL of which he called “a fantastic solitaire pastime” (Gardner, 1970). His game has become somewhat of a cultural wonder in the scientific community, introducing millions of people into the world of cellular automata. Since then, GoL has taught the importance of sensitivity to initial conditions and complexity arises from simplicity.
This article was written for the high school club, Staples STEM Journal.
It was originally meant to be published in the winter, but was unfortunately delayed due to complications with the printer.
Be sure to see the following link to digitally view the published issue containing 10 amazing articles written by different students... https://issuu.com/staplesstemjournal/docs/staples_2023_stem_journal_winter_issue
References are in the following pdf document...
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