User:Tropylium/Pattern - Revision history

GUYTU6J: Red link removal

2021-03-01T03:26:53Z

Red link removal

← Older revision		Revision as of 03:26, 1 March 2021
Line 1:		Line 1:
	A '''pattern''' is any configuration of cells on the plane.		A '''pattern''' is any configuration of cells on the plane.

	Physical limitations preclude the simulation of patterns of infinite complexity. Patterns of finite complexity can be generated in many ways, but the most commonly explored class of patterns involve a static, periodic background (the [[medium]]), and a finite number of perturbations from this. This number may be, in fact, zero, in which case the pattern will be known as an [[agar]].		Physical limitations preclude the simulation of patterns of infinite complexity. Patterns of finite complexity can be generated in many ways, but the most commonly explored class of patterns involve a static, periodic background (the [[User:Tropylium/Mediums\|medium]]), and a finite number of perturbations from this. This number may be, in fact, zero, in which case the pattern will be known as an [[agar]].

	Most other patterns operate using the [[vacuum (empty space)\|vacuum]] as the medium. In this paradigm, the symmetry between different cell types is broken. The cell type that makes up the vacuum are normally called '''dead''', while all other cell types are called '''live'''; any pattern of this sort is, then, characterized by the presence of an infinite number of dead cells, vs. a finite number of live cells. The foregoing discussion will be limited to patterns of this kind.		Most other patterns operate using the [[vacuum (empty space)\|vacuum]] as the medium. In this paradigm, the symmetry between different cell types is broken. The cell type that makes up the vacuum are normally called '''dead''', while all other cell types are called '''live'''; any pattern of this sort is, then, characterized by the presence of an infinite number of dead cells, vs. a finite number of live cells. The foregoing discussion will be limited to patterns of this kind.
Line 8:		Line 8:

	==Area==		==Area==
	An '''area''' is a subset of a pattern that is surrounded by at least two disjoint perimeters of dead cells. For [[rule]]s using the [[Moore neighborhood]], the perimeters will need to be [[rookwise connected]]; for rules using the [[von Neumann neighborhood]], a [[kingwise connected]] perimeter will suffice. The evolution of any area to the next generation will be independent of any other areas, regardless of the exact transition rule.		An '''area''' is a subset of a pattern that is surrounded by at least two disjoint perimeters of dead cells. For [[rule]]s using the [[Moore neighborhood]], the perimeters will need to be rookwise connected; for rules using the [[von Neumann neighborhood]], a kingwise connected perimeter will suffice. The evolution of any area to the next generation will be independent of any other areas, regardless of the exact transition rule.

	Equivalently, given a subset of a pattern, the '''subset neighborhood''' can be defined as:		Equivalently, given a subset of a pattern, the '''subset neighborhood''' can be defined as:
Line 68:		Line 68:
	Equivalently, strict objects are the parts of a partition of pattern into minimal sets of ''live'' cells such that any change of the ''live'' cells to dead within one partition will not affect any of the other partitions.		Equivalently, strict objects are the parts of a partition of pattern into minimal sets of ''live'' cells such that any change of the ''live'' cells to dead within one partition will not affect any of the other partitions.

	At least within Conway's Game of Life, it is '''not''' sufficient to define a strict object merely as a minimal set of live cells such that any change to other live cells will leave the evolution of the pattern unaffected. Consider a 5×5 square of live cells: the removal of any one of the 9 central cells will not affect the evolution of the pattern, as all cells within the central 5×5 square will be dead in the next generation regardless. However, it would seem strange to analyze the "removable" central cells as isolated [[dot spark]]s, since their isolatedness status is not independent. For example, the removal of all nine of them will produce a clearly different object; the 5×5 unfilled square is a large spark instead of a [[honey farm]] predecessor. Therefore, while the dynamics of a 5×5 block can be analyzed on a sub-object level, it cannot be consistently partitioned into smaller objects.		At least within Conway's Game of Life, it is '''not''' sufficient to define a strict object merely as a minimal set of live cells such that any change to other live cells will leave the evolution of the pattern unaffected. Consider a 5×5 square of live cells: the removal of any one of the 9 central cells will not affect the evolution of the pattern, as all cells within the central 5×5 square will be dead in the next generation regardless. However, it would seem strange to analyze the "removable" central cells as isolated [[dot]] [[spark]]s, since their isolatedness status is not independent. For example, the removal of all nine of them will produce a clearly different object; the 5×5 unfilled square is a large spark instead of a [[honey farm]] predecessor. Therefore, while the dynamics of a 5×5 block can be analyzed on a sub-object level, it cannot be consistently partitioned into smaller objects.
	* This issue is perhaps resolvable by adding an intermediate concept. Any one subtraction is an analysis into a dot spark (clearly an object by itself) plus a residual possibly-object; however, different possibly-object analyses will terminate in different minimal possibly-objects. Then, an object is a possibly-object that does not have a unique analysis as minimal possibly-objects (although it may have two or more such minimal analyses).		* This issue is perhaps resolvable by adding an intermediate concept. Any one subtraction is an analysis into a dot spark (clearly an object by itself) plus a residual possibly-object; however, different possibly-object analyses will terminate in different minimal possibly-objects. Then, an object is a possibly-object that does not have a unique analysis as minimal possibly-objects (although it may have two or more such minimal analyses).

Apple Bottom: disambig

2018-01-25T16:59:18Z

disambig

← Older revision		Revision as of 16:59, 25 January 2018
Line 3:		Line 3:
	Physical limitations preclude the simulation of patterns of infinite complexity. Patterns of finite complexity can be generated in many ways, but the most commonly explored class of patterns involve a static, periodic background (the [[medium]]), and a finite number of perturbations from this. This number may be, in fact, zero, in which case the pattern will be known as an [[agar]].		Physical limitations preclude the simulation of patterns of infinite complexity. Patterns of finite complexity can be generated in many ways, but the most commonly explored class of patterns involve a static, periodic background (the [[medium]]), and a finite number of perturbations from this. This number may be, in fact, zero, in which case the pattern will be known as an [[agar]].

	Most other patterns operate using the [[vacuum]] as the medium. In this paradigm, the symmetry between different cell types is broken. The cell type that makes up the vacuum are normally called '''dead''', while all other cell types are called '''live'''; any pattern of this sort is, then, characterized by the presence of an infinite number of dead cells, vs. a finite number of live cells. The foregoing discussion will be limited to patterns of this kind.		Most other patterns operate using the [[vacuum (empty space)\|vacuum]] as the medium. In this paradigm, the symmetry between different cell types is broken. The cell type that makes up the vacuum are normally called '''dead''', while all other cell types are called '''live'''; any pattern of this sort is, then, characterized by the presence of an infinite number of dead cells, vs. a finite number of live cells. The foregoing discussion will be limited to patterns of this kind.

	''(nb: infinite [[wick]]s fit poorly into this scheme)''		''(nb: infinite [[wick]]s fit poorly into this scheme)''

Tropylium: /* Object */

2017-09-13T18:25:23Z

Object

← Older revision		Revision as of 18:25, 13 September 2017
Line 52:		Line 52:

	==Object==		==Object==
			An '''object''', loosely defined, is a part of a minimal partition of a pattern into non-interacting components.

	All [[polyplet]]s (including [[polyomino]]es) are objects.		In the context of general CA, the notion of an object is dependent on the definition of a medium. While the backdrop of cellular automata is a connected space (in the graph-theoretic sense), and no part of a pattern can be truly independent of its surroundings, it is desirable to abstract away from the predictable evolution of empty medium. The evolution of a pattern, then, can be decomposed into the evolution of its medium (infinite but easily predictable) and the evolution of its objects (finite, non-trivial to predict). The following discussion will presume vacuum medium.

			a set of cells in a pattern, finite at least in one dimension, such that no reference to other cells in the pattern is necessary to predict its live cells' immediate development; and that the pattern cannot be divided in a set of smaller objects. An object, therefore, consist of a boundary of dead cells, of one neighborhood thickness (at least some of whose evolution necessarily depends on the cells beyond the boundary), containing one or more live cells.

			''Notes:''
			*''A pattern subset that does not necessarily fulfill the last criterion could be called an "objectoid". Another possibility is "pre-object", but this may be likely to be confused with a [[predecessor]] of an object.''
			*''Infinite but periodic wicks should probably be treated as objects; most agars should probably not, given that they will occupy the entire plane. This leaves "agars with holes" and "agar half-planes" (quarter-planes, etc.) as edge cases.''

			Objects can be further divided in '''strict objects''' and '''pseudo-objects'''. In a strict object, no reference to the ''live'' cells outside the object is necessary to identify cells that remain live, die, or are born. However, cells that remain dead may still be simultaneously within the neighborhood of more than one strict object. An object that can be divided in more than one strict object, then, is a pseudo-object.

			The best-studied pseudo-objects are the [[pseudo still life]]s, but also pseudo-oscillators, pseudo-spaceships, etc. are known (indeed, it is often trivial to extend a strict object into a pseudo-object).

			Equivalently, strict objects are the parts of a partition of pattern into minimal sets of ''live'' cells such that any change of the ''live'' cells to dead within one partition will not affect any of the other partitions.

			At least within Conway's Game of Life, it is '''not''' sufficient to define a strict object merely as a minimal set of live cells such that any change to other live cells will leave the evolution of the pattern unaffected. Consider a 5×5 square of live cells: the removal of any one of the 9 central cells will not affect the evolution of the pattern, as all cells within the central 5×5 square will be dead in the next generation regardless. However, it would seem strange to analyze the "removable" central cells as isolated [[dot spark]]s, since their isolatedness status is not independent. For example, the removal of all nine of them will produce a clearly different object; the 5×5 unfilled square is a large spark instead of a [[honey farm]] predecessor. Therefore, while the dynamics of a 5×5 block can be analyzed on a sub-object level, it cannot be consistently partitioned into smaller objects.
			* This issue is perhaps resolvable by adding an intermediate concept. Any one subtraction is an analysis into a dot spark (clearly an object by itself) plus a residual possibly-object; however, different possibly-object analyses will terminate in different minimal possibly-objects. Then, an object is a possibly-object that does not have a unique analysis as minimal possibly-objects (although it may have two or more such minimal analyses).

			All [[polyplet]]s (including [[polyomino]]es) are objects, though not necessarily elementary objects (a polyplet can still constitute a [[tie]]).

			''It would seem to be desirable to further define a term for a cell subset formed by an object with its boundary removed. It is this level of analysis that is often applied to conventional terms such as [[beehive]]. A [[honeycomb]], for example, does not contain a beehive-as-an-object or a tub-as-an-object, but it does contain sets of cells that consitute a beehive/tub-as-an-object-sans-boundary. Mere shape is not meaningful, though: also identical evolution should be required.''

Tropylium: /* Cluster */

2014-11-25T19:32:16Z

Cluster

← Older revision		Revision as of 19:32, 25 November 2014
Line 30:		Line 30:

	==Cluster==		==Cluster==
	A '''cluster''' is the rule-specific analogue of an area: a subset of cells delineated by less stringent criteria, that regardless ~~shares~~ the property that its evolution can be predicted without reference to other clusters.		A '''cluster''' is the ''rule-specific'' analogue of an area: a subset of cells delineated by less stringent criteria, that regardless has the property that its evolution can be predicted without reference to the other clusters present in the pattern.

	In [[Conway's Game of Life]], ~~a sufficient condition for detecting a clusters is~~ the following criteria:		In [[Conway's Game of Life]], the following criteria are relevant for determining cluster status:
			* The existence of a rookwise connected perimeter of dead cells, where every cell only has one parent within the perimeter

	The same does '''not''' hold in all [[Life-like cellular automata]]; e.g. in the rule [[Seeds]], where two live cells is a sufficient criterion for the birth of a new cell, two Life clusters that occur within the same area can end up interacting.		The same does '''not''' hold in all [[Life-like cellular automata]]; e.g. in the rule [[Seeds]], where two live cells is a sufficient criterion for the birth of a new cell, two Life clusters that occur within the same area can end up interacting.

	The ~~following is an~~ example ~~of a~~ pattern ~~that comprises a single area, yet~~ in Life is composed of ~~two~~ clusters:		The above single-area example pattern is in Life composed of four clusters, marked in the first figure with the letters A, B, C, D; and in the second figure separated by perimeters marked by asterisks:
	.........		........ *******.
	.OO......		.A.B.... OO..**
	...O.....		....BB.. **.OO.
	.....OO..		.....B.. ....O.
	......OO.		..BBB... .OOO..
	.........		......B. .....O
			........ ********
			.C..D... O.O*..
			...D.... **O...
			........ ..****..

	~~Clusters can~~ be ~~furthermore divided~~ in ~~objects~~ and ~~pseudo-objects~~.		A cluster is, however, only determined in the context of the pattern in general. A subset may be a separate cluster in one pattern, but not in another. As an extreme example, consider e.g. clusters A and C above: these comprise only a single live cell, yet in by far most cases determining the evolution of a particular live cell requires reference to other cells.

	==Object==		==Object==

	All [[polyplet]]s (including [[polyomino]]es) are objects.		All [[polyplet]]s (including [[polyomino]]es) are objects.

Tropylium: /* Cluster */

2014-11-25T17:36:12Z

Cluster

← Older revision		Revision as of 17:36, 25 November 2014
Line 34:		Line 34:
	In [[Conway's Game of Life]], a sufficient condition for detecting a clusters is the following criteria:		In [[Conway's Game of Life]], a sufficient condition for detecting a clusters is the following criteria:

	The same does '''not''' hold in all [[Life-like cellular automata]]; e.g. in the rule [[Seeds]], where two, two Life clusters within the same area can ~~and do interact~~.		The same does '''not''' hold in all [[Life-like cellular automata]]; e.g. in the rule [[Seeds]], where two live cells is a sufficient criterion for the birth of a new cell, two Life clusters that occur within the same area can end up interacting.

	The following is an example of a pattern that comprises a single area, yet in Life is composed of two clusters:		The following is an example of a pattern that comprises a single area, yet in Life is composed of two clusters:

Tropylium: thinking out loud

2014-11-25T17:30:55Z

thinking out loud

New page

A '''pattern''' is any configuration of cells on the plane.

Physical limitations preclude the simulation of patterns of infinite complexity. Patterns of finite complexity can be generated in many ways, but the most commonly explored class of patterns involve a static, periodic background (the [[medium]]), and a finite number of perturbations from this. This number may be, in fact, zero, in which case the pattern will be known as an [[agar]].

Most other patterns operate using the [[vacuum]] as the medium. In this paradigm, the symmetry between different cell types is broken. The cell type that makes up the vacuum are normally called '''dead''', while all other cell types are called '''live'''; any pattern of this sort is, then, characterized by the presence of an infinite number of dead cells, vs. a finite number of live cells. The foregoing discussion will be limited to patterns of this kind.

''(nb: infinite [[wick]]s fit poorly into this scheme)''

==Area==
An '''area''' is a subset of a pattern that is surrounded by at least two disjoint perimeters of dead cells. For [[rule]]s using the [[Moore neighborhood]], the perimeters will need to be [[rookwise connected]]; for rules using the [[von Neumann neighborhood]], a [[kingwise connected]] perimeter will suffice. The evolution of any area to the next generation will be independent of any other areas, regardless of the exact transition rule.

Equivalently, given a subset of a pattern, the '''subset neighborhood''' can be defined as:
# All live cells in the subset.
# All dead cells that are neighbors to live cells in the subset.
Two subsets of a pattern then belong in different areas, if their subset neighborhoods are disjoint.

An example of a pattern that forms a single area:
........
.O.O....
....OO..
.....O..
..OOO...
......O.
........
.O..O...
...O....
........

If a rule is however fixed, it will be possible to however establish finer criteria for dividing a pattern in parts.

==Cluster==
A '''cluster''' is the rule-specific analogue of an area: a subset of cells delineated by less stringent criteria, that regardless shares the property that its evolution can be predicted without reference to other clusters.

In [[Conway's Game of Life]], a sufficient condition for detecting a clusters is the following criteria:

The same does '''not''' hold in all [[Life-like cellular automata]]; e.g. in the rule [[Seeds]], where two, two Life clusters within the same area can and do interact.

The following is an example of a pattern that comprises a single area, yet in Life is composed of two clusters:
.........
.OO......
...O.....
.....OO..
......OO.
.........

Clusters can be furthermore divided in objects and pseudo-objects.

==Object==

All [[polyplet]]s (including [[polyomino]]es) are objects.