Summary Conways Game of Life Omniperiodic Cellular Automata arxiv.org
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Conway's Game of Life is a cellular automaton known for its complex behavior and oscillators, with specific periods of 19 and 41.
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Key Points
- Conway's Game of Life is a cellular automaton that has complex behavior and has been extensively studied.
- An omniperiodic cellular automaton has oscillators of all periods.
- The search for oscillators in Conway's Game of Life has ended with the discovery of oscillators with the final two periods, 19 and 41.
- Patterns in Life can be categorized by their period, and it has long been conjectured that oscillators of every period exist in Life.
- Various techniques, including brute force computer searches, have been used to find oscillators in Life.
- The authors provide an overview of Conway's Game of Life, its history, and its significance in various fields.
- They present a comprehensive list of known omniperiodic patterns in the Game of Life.
- Various software tools and online resources are mentioned for studying and analyzing the Game of Life.
Summaries
19 word summary
Conway's Game of Life is a well-studied cellular automaton with complex behavior and oscillators, including periods 19 and 41.
68 word summary
Conway's Game of Life is a well-studied cellular automaton with complex behavior and oscillators. David Buckingham determined the maximum period of oscillators in the game, leading to the discovery of oscillators with periods 19 and 41. The game takes place on an infinite grid, with patterns categorized by their period. The document provides an overview of the game, its applications, known patterns, and collaborative nature of the community.
205 word summary
Conway's Game of Life, a well-studied cellular automaton, is known for its complex behavior and oscillators. David Buckingham determined the maximum period of oscillators in the game, and subsequent advancements in computer speed and search techniques led to the discovery of oscillators with periods 19 and 41. A comprehensive table of oscillators of all periods confirms the omniperiodicity of Life.
The game takes place on an infinite grid, where each cell is alive or dead. Simple rules dictate changes in the grid's state over time. Chaotic activity settles into small patterns, including basic arrangements like blocks and blinkers, as well as more complex patterns like universal constructors and computers.
Patterns in Life can be categorized by their period. The existence of oscillators with every period has been conjectured and various techniques have been used to find low-period oscillators. Collaboration among individuals has been crucial to these discoveries.
The document provides an overview of Conway's Game of Life, its popularity, and its applications in mathematics, computer science, and artificial life. It presents a comprehensive list of known omniperiodic patterns and mentions relevant software tools, online resources, and notable contributors. The collaborative nature of the Game of Life community is emphasized, with links provided for further engagement.
417 word summary
Conway's Game of Life is a well-studied cellular automaton known for its complex behavior and oscillators. David Buckingham established a finite bound for the maximum period of oscillators in Conway's Game of Life. Over the years, computer speed and search techniques improved, leading to the discovery of oscillators with the final two periods, 19 and 41. A table with oscillators of all periods proves that Life is omniperiodic.
Conway's Game of Life takes place on an infinite plane of square grid cells, where each cell is either alive or dead. The state of the entire plane changes according to simple rules at each time step. Starting with a random initial state, Life exhibits chaotic activity that settles into small patterns of alive cells. There are basic arrangements like the block and the blinker, as well as more complex patterns including universal constructors and computers.
Patterns in Life can be categorized by their period, which is the number of generations after which the state of the pattern repeats. Still lifes have a period of 1, oscillators have a period of 2 or more, and spaceships translate in the plane while repeating their state. It has been conjectured that oscillators of every period exist in Life, and this problem has been discussed among hobbyists.
Low-period oscillators can be found through manual exploration or brute force computer searches. From 1996 to July 2023, various techniques were used to fill in the remaining gaps and find oscillators with periods below 43. These techniques include stabilizing repeating patterns, hassling common active objects, finding oscillating perturbations, multiplying the period of low-period oscillators, and finding dependent glider loops. The collaboration of many individuals has been instrumental in making these discoveries possible.
The authors provide an overview of Conway's Game of Life, emphasizing its popularity and widespread use in mathematics, computer science, and artificial life. They discuss the concept of omniperiodic cellular automata and present a comprehensive list of known omniperiodic patterns in the Game of Life. Various software tools and online resources are mentioned, along with the contributions of researchers such as David Buckingham and Bill Gosper.
Throughout the document, the authors highlight the collaborative nature of the Game of Life community and encourage readers to engage with others in forums and online communities. They provide links to relevant websites, forum discussions, and software repositories for further exploration. The document serves as a clear and concise overview of Conway's Game of Life, providing historical context, patterns, software tools, and notable contributors to the field.
968 word summary
Conway's Game of Life is a cellular automaton that has been extensively studied for its complex behavior. In the theory of cellular automata, an oscillator is a pattern that repeats itself after a fixed number of generations, known as its period. An omniperiodic cellular automaton is one that has oscillators of all periods. Conway's Game of Life is the most famous cellular automaton and has been studied for its oscillators.
In the early days of studying cellular automata, David Buckingham established a finite bound above which oscillators of every period could be built. For periods below this cutoff, oscillators needed to be found individually. At the turn of the millennium, there were only twelve periods remaining to be found in Conway's Game of Life: 19, 23, 27, 31, 34, 37, 38, 39, 41, 43, 51, and 53. Over the last couple of decades, these periods were gradually filled in as computer speed and search techniques improved. The search for oscillators in Conway's Game of Life has finally ended with the discovery of oscillators with the final two periods, 19 and 41. A table with oscillators of all periods is provided as proof that Life is omniperiodic.
Conway's Game of Life occurs on an infinite plane of square grid cells, each of which is either alive or dead. At each time step, or generation, the entire plane changes state according to simple rules. If a dead cell has exactly three live neighbors, it becomes alive. If an alive cell has exactly two or three live neighbors, it stays alive. Otherwise, it becomes dead. These rules give rise to complex behavior that has been studied for over 50 years.
Starting with a random initial state, running Life forwards in time typically causes an initial burst of chaotic activity that settles down into small patterns of alive cells. There are basic arrangements that occur naturally in Life, such as the block, which is a 2x2 grid of alive cells that does not change, and the blinker, which is a 1x3 line of alive cells that alternates between horizontal and vertical at each time step. There are also more complex patterns, both naturally occurring and intentionally engineered. These include working universal constructors and computers.
Patterns in Life can be categorized by their period, which is the number of generations after which the state of the pattern repeats. Still lifes are patterns with a period of 1, oscillators have a period of 2 or more, and spaceships repeat their state but translate in the plane. It has long been conjectured that oscillators of every period exist in Life, meaning that Life is omniperiodic. This problem has been open for discussion among hobbyists for many years.
Low-period oscillators in Life can be found by playing with patterns by hand or using brute force computer searches. Some periods, particularly those that are prime, proved more difficult to find. From 1996 to July 2023, a sequence of discoveries using new mechanisms and improved search algorithms filled in the remaining gaps. These techniques include stabilizing infinitely repeating patterns, hassling common active objects, finding oscillating perturbations of stable background patterns, multiplying the period of low-period oscillators, and finding dependent glider loops. Together, these techniques cover all periods below 43, proving that Life is omniperiodic.
The search for oscillators in Life has been a collaborative effort involving many individuals who have made new discoveries, proposed and implemented new approaches, and run the searches that make these discoveries possible. The first oscillator of each period is listed in the appendix.
There have been various techniques used to search for oscillators in Life. One approach is to perform soup searches, which involve starting with a random configuration and looking for oscillators that stabilize. Another approach is to use direct searches, where each cell in a finite grid is tested for its oscillatory
In this document, the authors provide an overview of Conway's Game of Life, a cellular automaton invented by mathematician John H. Conway in 1970. They discuss the history and significance of the Game of Life, highlighting its popularity and widespread use in various fields such as mathematics, computer science, and artificial life.
The authors also explore the concept of omniperiodic cellular automata, which are patterns that repeat themselves on an infinite grid with any desired period. They present a comprehensive list of known omniperiodic patterns in the Game of Life, ranging from period 48 to period 2041. The list includes various types of oscillators, glider shuttles, and other complex structures.
The authors mention the use of various software tools and programs for studying and analyzing the Game of Life, such as Golly, LifeViewer, and Bellman. They also reference online resources like LifeWiki and Catagolue, which provide extensive collections of patterns and information about the Game of Life.
Throughout the document, the authors cite numerous contributors and researchers who have made significant contributions to the study of the Game of Life. They mention the work of David Buckingham, Dean Hickerson, Bill Gosper, and many others who have discovered and studied various patterns and phenomena in the Game of Life.
The authors provide links to relevant websites, forum discussions, and software repositories where readers can find more information and explore the patterns and tools discussed in the document. They emphasize the collaborative nature of the Game of Life community and encourage readers to engage with others in forums and online communities to further their understanding and contribute to the field.
Overall, this document serves as a comprehensive overview of Conway's Game of Life and its various aspects. It provides a wealth of information about the history, patterns, software tools, and notable contributors to the field. The authors present the material in a clear and concise manner, making it accessible to readers with varying levels of familiarity with the Game of Life.