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Rake (cellular Automaton)
A rake, in the lexicon of cellular automata, is a type of ''puffer train'', which is an automaton that leaves behind a trail of debris. In the case of a rake, however, the debris left behind is a stream of spaceships, which are automata that "travel" by looping through a short series of iterations and end up in a new location after each cycle returns to the original configuration. In Conway's Game of Life, the discovery of rakes was one of the key components needed to form the ''breeder'', the first known pattern in Life in which the number of live cells exhibits quadratic growth. A breeder is formed by arranging several rakes so that the '' gliders''—the smallest possible spaceships—they generate interact to form a sequence of '' glider guns'', patterns which emit gliders. The emitted gliders fill a growing triangle of the plane of the game. More generally, when a rake exists for a cellular automaton rule (a mathematical function defining the next iteration to be derived f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Automata
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessellation structures, and iterative arrays. Cellular automata have found application in various areas, including physics, theoretical biology and microstructure modeling. A cellular automaton consists of a regular grid of ''cells'', each in one of a finite number of '' states'', such as ''on'' and ''off'' (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its ''neighborhood'' is defined relative to the specified cell. An initial state (time ''t'' = 0) is selected by assigning a state for each cell. A new ''generation'' is created (advancing ''t'' by 1), according to some fixed ''rule'' (generally, a mathematical function) that determines the new state of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheels, Life And Other Mathematical Amusements
''Wheels, Life and Other Mathematical Amusements'' is a book by Martin Gardner published in 1983. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Contents ''Wheels, Life and Other Mathematical Amusements'' is a book of 22 mathematical games columns that were revised and extended after being previously published in ''Scientific American''. It is Gardner's 10th collection of columns, and includes material on Conway's Game of Life, supertasks, intransitive dice, braided polyhedra, combinatorial game theory, the Collatz conjecture, mathematical card tricks, and Diophantine equations such as Fermat's Last Theorem. Reception Dave Langford reviewed ''Wheels, Life and Other Mathematical Amusements'' for ''White Dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Day And Night (cellular Automaton)
Day and Night is a cellular automaton rule in the same family as Game of Life. It is defined by rule notation B3678/S34678, meaning that a dead cell becomes live (is born) if it has 3, 6, 7, or 8 live neighbors, and a live cell remains alive (survives) if it has 3, 4, 6, 7, or 8 live neighbors, out of the eight neighbors in the Moore neighborhood. It was invented and named by Nathan Thompson in 1997, and investigated extensively by David I. Bell. The rule is given the name "Day & Night" because its ''on'' and ''off'' states are symmetric: if all the cells in the Universe are inverted, the future states are the inversions of the future states of the original pattern. A pattern in which the entire universe consists of ''off'' cells except for finitely many ''on'' cells can equivalently be represented by a pattern in which the whole universe is covered in ''on'' cells except for finitely many ''off'' cells in congruent locations. Although the detailed evolution of this cellular aut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highlife (cellular Automaton)
Highlife is a cellular automaton similar to Conway's Game of Life. It was devised in 1994 by Nathan Thompson. It is a two-dimensional, two-state cellular automaton in the " Life family" and is described by the rule B36/S23; that is, a cell is born if it has 3 or 6 neighbors and survives if it has 2 or 3 neighbors. Because the rules of HighLife and Conway's Life (rule B3/S23) are similar, many simple patterns in Conway's Life function identically in HighLife. More complicated engineered patterns for one rule, though, typically do not work in the other rule. Replicator The main reason for interest in HighLife comes from the existence of a pattern called the replicator. After running the replicator for twelve generations, the result is two replicators. The replicators will repeatedly reproduce themselves, all on a diagonal line. Whenever two replicators try to expand into each other, the pattern in the middle simply vanishes. The behavior of a row of Replicators interacting with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Speed Of Light (cellular Automaton)
In Conway's Game of Life (and related cellular automata), the speed of light is a propagation rate across the grid of exactly one step (either horizontally, vertically or diagonally) per generation. In a single generation, a cell can only influence its nearest neighbours, and so the speed of light (by analogy with the speed of light in physics) is the maximum rate at which information can propagate. It is therefore an upper bound to the speed at which any pattern can move. Notation As in physics, the speed of light is represented with the letter ''c''. This in turn is used as a reference for describing the average propagation speed of any given type of spaceship. For example, a glider is said to have a speed of ''c''/4, as it takes four generations for a given state to be translated by one cell. Similarly, the "lightweight spaceship" is said to have a speed of ''c''/2, as it takes four generations for a given state to be translated by two cells. Lightspeed propagation While ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was also a leading authority on Lewis Carroll. ''The Annotated Alice'', which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of the Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematicsand by extension, mathematics in generalthroughout the latter half of the 20th century, principally through his "Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Puffer Train
In a cellular automaton, a puffer train, or simply puffer, is a finite pattern that moves itself across the "universe", leaving debris behind. Thus a pattern consisting of only a puffer will grow arbitrarily large over time. While both puffers and spaceships have periods and speeds, unlike puffers, spaceships do not leave debris behind. The period of a puffer can be considered as the combination of ''two'' periods; the first is the period of the puffer itself, while the second is the apparent period of the pattern of debris produced. This is often the same as the period of the puffer, but sometimes is a factor of the period. A puffer for which the apparent period deduced from the debris is smaller than the period of the engine is a pseudoperiod puffer. Such puffers are typically produced by artificial means. A true period puffer is one in which the period of the debris matches that of the puffer. Puffers are divided into two classes, dirty puffers and clean puffers. While there i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gun (cellular Automaton)
In a cellular automaton, a gun is a pattern with a main part that repeats periodically, like an oscillator, and that also periodically emits spaceships. There are then two periods that may be considered: the period of the spaceship output, and the period of the gun itself, which is necessarily a multiple of the spaceship output's period. A gun whose period is larger than the period of the output is a pseudoperiod gun. In the Game of Life, for every ''p'' greater than or equal to 14, it is possible to construct a glider gun in which the gliders are emitted with period ''p''. Since guns continually emit spaceships, the existence of guns in Life means that initial patterns with finite numbers of cells can eventually lead to configurations with limitless numbers of cells, something that John Conway himself originally conjectured to be impossible. However, according to Conway's later testimony, this conjecture was explicitly intended to encourage someone to disprove it – i.e., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glider (Conway's Life)
The glider is a pattern that travels across the board in Conway's Game of Life. It was first discovered by Richard K. Guy in 1969, while John Conway's group was attempting to track the evolution of the R-pentomino. Gliders are the smallest spaceships, and they travel diagonally at a speed of one cell every four generations, or c/4. The glider is often produced from randomly generated starting configurations. The name comes from the fact that, after two steps, the glider pattern repeats its configuration with a glide reflection symmetry. After four steps and two glide reflections, it returns to its original orientation. John Conway remarked that he wished he hadn't called it the glider. The game was developed before the widespread use of interactive computers, and after seeing it animated, he feels the glider looks more like an ant walking across the plane. Importance Gliders are important to the Game of Life because they are easily produced, can be collided with each other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
