About

John Conway’s Game of Life

Description taken from Wikipedia, the free encyclopedia.

The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is the best-known example of a cellular automaton. The "game" is actually a zero-player game, meaning that its evolution is determined by its initial state, needing no input from human players. One interacts with the Game of Life by creating an initial configuration and observing how it evolves. A variant exists where two players compete. Rules: 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, live or dead. Every cell interacts with its eight neighbours, which are the cells that are directly 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 loneliness.
• Any live cell with more than three live neighbors dies, as if by overcrowding.
• Any live cell with two or three live neighbors lives, unchanged, to the next generation.
• Any dead cell with exactly three live neighbors comes to life.

The initial pattern constitutes the 'seed' of the system. The next generation is created by applying the above rules simultaneously to every cell in the seed - births and deaths happen simultaneously, and the discrete moment at which this happens is sometimes called a tick. (In other words, each generation is a pure function of the one before.) The rules continue to be applied repeatedly to create further generations.  

Translations

Code	Language		Translator	Run

Credits

Francisco Esquembre; Wolfgang Christian; Tan Wei Chiong; lookang

Summary:

This document describes an interactive JavaScript simulation of John Conway's Game of Life, a classic example of a cellular automaton. The resource is part of the Open Educational Resources / Open Source Physics project at Singapore. The game/simulation is presented as a zero-player game where the initial state determines the evolution of the simulation. The user interacts by setting up the initial conditions and observing the resulting patterns.

Key Concepts and Ideas:

• John Conway's Game of Life: A cellular automaton created by John Horton Conway in 1970. It's a zero-player game where the initial state dictates how it evolves.
• "The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is the best-known example of a cellular automaton. The "game" is actually a zero-player game, meaning that its evolution is determined by its initial state, needing no input from human players."
• Cellular Automaton: A discrete model studied in computer science, mathematics, and theoretical biology. It consists of a regular grid of cells, each in a finite number of states. The state of each cell updates simultaneously based on a set of rules depending on the states of its neighbors.
• Grid and Cells: The game takes place on an "infinite two-dimensional orthogonal grid of square cells." Each cell can be either "live or dead."
• "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, live or dead."
• Neighborhood: Each cell interacts with its eight neighbors (horizontally, vertically, and diagonally adjacent cells).
• "Every cell interacts with its eight neighbours, which are the cells that are directly horizontally, vertically, or diagonally adjacent."
• Rules of Evolution: The core of the Game of Life lies in its simple rules that determine how cells transition between live and dead states in each generation:
• Underpopulation: "Any live cell with fewer than two live neighbors dies, as if by loneliness."
• Survival: "Any live cell with two or three live neighbors lives, unchanged, to the next generation."
• Overpopulation: "Any live cell with more than three live neighbors dies, as if by overcrowding."
• Reproduction: "Any dead cell with exactly three live neighbors comes to life."
• Seed and Generation: The initial configuration of live and dead cells is the "seed." Applying the rules simultaneously to all cells creates the next "generation."
• "The initial pattern constitutes the 'seed' of the system. The next generation is created by applying the above rules simultaneously to every cell in the seed - births and deaths happen simultaneously, and the discrete moment at which this happens is sometimes called a tick."
• Simulation: The resource offers an interactive JavaScript simulation, allowing users to create initial patterns and observe their evolution. The simulation provides preset configurations like "gliders, diehards and acorns" to demonstrate different behaviors.
• Educational Resource: This is presented as an open educational resource suitable for primary, secondary and junior college students.

Technical Details:

• The simulation is implemented using JavaScript and HTML5.
• The page provides an embed code to include the simulation in other webpages.
• Credits are given to Francisco Esquembre, Wolfgang Christian, Tan Wei Chiong, and lookang for the development of the simulation.

Potential Uses:

• Demonstrating complex behavior arising from simple rules.
• Exploring concepts in cellular automata and computational science.
• Engaging students in interactive learning about mathematical models.
• Illustrating emergent phenomena.

Key Quotes:

• "The 'game' is actually a zero-player game, meaning that its evolution is determined by its initial state, needing no input from human players."
• "One interacts with the Game of Life by creating an initial configuration and observing how it evolves."
• "Interestingly, certain cellular constructs can be formed based on these rules. Three of them (gliders, diehards and acorns) are already preset in the simulation for your convenience."

John Conway's Game of Life: A Study Guide

Quiz

Answer the following questions in 2-3 sentences each.

1. Who created the Game of Life, and when?
2. What type of game is the Game of Life considered to be? Why?
3. Describe the "universe" in which the Game of Life is played.
4. How many neighbors does each cell in the Game of Life interact with?
5. Explain the rule regarding "under-population" in the Game of Life.
6. Explain the rule regarding "over-population" in the Game of Life.
7. What happens to a live cell with exactly two or three live neighbors?
8. What condition must be met for a dead cell to come to life in the Game of Life?
9. What is the "seed" of the Game of Life?
10. Explain the concept of a "tick" in the Game of Life.

Quiz Answer Key

1. The Game of Life was created by the British mathematician John Horton Conway in 1970. It is a cellular automaton and a popular example of such games.
2. The Game of Life is considered a zero-player game because its evolution is entirely determined by the initial state and does not require any input from human players after the initial setup. The evolution of the game is dependent on the original configuration.
3. The universe in the Game of Life is an infinite two-dimensional orthogonal grid of square cells. Each cell can exist in one of two states: live or dead.
4. Each cell in the Game of Life interacts with its eight neighbors. These neighbors are the cells that are directly horizontally, vertically, or diagonally adjacent to the cell.
5. The "under-population" rule states that any live cell with fewer than two live neighbors dies. This simulates the cell's inability to survive in isolation or with very few surrounding cells.
6. The "over-population" rule states that any live cell with more than three live neighbors dies. This simulates the cell's demise due to limited resources or excessive competition in a crowded environment.
7. A live cell with exactly two or three live neighbors lives on, unchanged, to the next generation. This represents a stable condition where the cell has enough, but not too many, neighbors to sustain its life.
8. A dead cell comes to life if it has exactly three live neighbors. This simulates reproduction, where a dead cell becomes alive when surrounded by a specific number of living cells.
9. The "seed" of the Game of Life refers to the initial pattern or configuration of live and dead cells. This initial pattern determines the entire evolution of the game.
10. A "tick" in the Game of Life refers to a discrete moment in time when the rules are applied simultaneously to every cell, creating the next generation. Births and deaths occur at the same time, and the next generation is always a product of the previous one.

Essay Questions

1. Discuss the significance of the Game of Life as an example of a cellular automaton. What makes it compelling from a mathematical and computational perspective?
2. Explain how the simple rules of the Game of Life can lead to complex and unpredictable patterns. Provide examples of emergent behaviors that can be observed in the simulation.
3. Analyze the impact of the initial configuration on the evolution of the Game of Life. How do different "seeds" lead to different outcomes?
4. Compare and contrast the Game of Life with real-world phenomena. Can the principles observed in the simulation be applied to other complex systems?
5. Critically evaluate the Game of Life as a model for understanding self-organization and emergent behavior. What are its strengths and limitations?

Glossary of Key Terms

• Cellular Automaton: A discrete, abstract computational system composed of a grid of cells, each with a finite number of states, that evolves through discrete time steps according to a set of rules based on the states of neighboring cells.
• Game of Life: A specific cellular automaton devised by John Horton Conway, characterized by simple rules that govern the evolution of cells on a grid based on the states of their neighbors.
• Live Cell: A cell in the Game of Life that is in the "alive" state, often represented visually as filled or colored.
• Dead Cell: A cell in the Game of Life that is in the "dead" state, often represented visually as empty or uncolored.
• Neighbors: The eight cells surrounding a given cell in the Game of Life's grid, including those horizontally, vertically, and diagonally adjacent.
• Under-population: The condition where a live cell has fewer than two live neighbors, causing it to die in the next generation.
• Over-population: The condition where a live cell has more than three live neighbors, causing it to die in the next generation.
• Seed: The initial configuration of live and dead cells in the Game of Life, which determines the system's evolution.
• Tick: A discrete time step or generation in the Game of Life, during which the rules are applied simultaneously to all cells.
• Emergent Behavior: Complex patterns or behaviors that arise from the interaction of simple rules within a system, such as the Game of Life, which were not explicitly programmed or intended.

Sample Learning Goals

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For Teachers

This is a simulation of John Conway's Game of Life, a cellular automation in which cells are placed in a grid, and individual cells either live or die according to these rules:

1) Any live cell with fewer than two live neighbours dies, as if caused by under-population.
2) Any live cell with two or three live neighbours lives on to the next generation.
3) Any live cell with more than three live neighbours dies, as if by over-population.
4) Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

This simulation only requires user input for the initial state, as every other state is determined by the iteration before it.

Interestingly, certain cellular constructs can be formed based on these rules. Three of them (gliders, diehards and acorns) are already preset in the simulation for your convenience.

Research

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Video

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 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

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Game of Life FAQ

What is the Game of Life?

The Game of Life is a cellular automaton, a zero-player game created by mathematician John Horton Conway in 1970. It takes place on an infinite two-dimensional grid of cells, each of which is either "alive" or "dead". The game's evolution is determined solely by its initial state; no further input is needed from a user.

What are the rules that govern the Game of Life?

The Game of Life has four simple rules that determine the state of each cell in the next generation:

1. A live cell with fewer than two live neighbors dies (underpopulation).
2. A live cell with two or three live neighbors lives on to the next generation.
3. A live cell with more than three live neighbors dies (overpopulation).
4. A dead cell with exactly three live neighbors becomes a live cell (reproduction).

How does a "generation" progress in the Game of Life?

A new generation is created by applying the rules to every cell in the grid simultaneously. Births and deaths occur at the same time. The next state is determined by the previous state before any changes are applied to the current state.

What is the role of the initial pattern, or "seed"?

The initial pattern, or "seed," is crucial, as it determines the evolution of the Game of Life. By creating different starting configurations, you can observe a wide variety of behaviors, from simple still lifes to complex oscillating patterns and even seemingly random growth.

What are some of the interesting patterns that can emerge in the Game of Life?

The simulation allows for the formations of gliders, diehards, and acorns, among other cellular constructs.

What does it mean that the Game of Life is a "zero-player" game?

The Game of Life is considered a zero-player game because, after the initial state is set, the game evolves automatically according to its rules, without any further human interaction. The user only needs to set up the starting conditions and then observe how the simulation unfolds.

What is the educational value of using a Game of Life simulation?

The Game of Life provides a visual and intuitive way to explore concepts such as emergence, self-organization, and complex behavior arising from simple rules. It can be used to teach about computation, mathematics, and even biological concepts like population dynamics.

Where can I find an interactive Game of Life simulation?

The document refers to a JavaScript simulation applet of the Game of Life, and includes an embed code. The simulation requires no user input after initialization, since iterations are determined by the previous state.