Pattern

Encyclopedia

A

These elements repeat in a predictable manner. It can be a template or model which can be used to generate things or parts of a thing, especially if the things that are created have enough in common for the underlying pattern to be inferred, in which case the things are said to

The most basic patterns, called Tessellation

s, are based on repetition and periodicity

. A single template, tile

, or cell, is combined with duplicates without change or modification. For example, simple harmonic oscillators produce repeated patterns of movement.

Other patterns, such as Penrose tiling

and Pongal

or Kolam

patterns from India, use symmetry

which is a form of finite repetition, instead of translation

which can repeat to infinity. Fractal

patterns also use magnification

or scaling

giving an effect known as self-similarity

or scale invariance

. Some plants, like Fern

s, even generate a pattern using an affine transformation

which combines translation

, scaling, rotation

and reflection

.

Pattern matching

is the act of checking for the presence of the constituents of a pattern, whereas the detecting for underlying patterns is referred to as pattern recognition

. The question of how a pattern emerges is accomplished through the work of the scientific field of pattern formation

.

Pattern recognition is more complex when templates are used to generate variants. For example, in English, sentences often follow the "N-VP" (noun - verb phrase) pattern, but some knowledge of the English language

is required to detect the pattern. Computer science

, ethology

, and psychology

are fields which study patterns.

s, and polka dots). Others can be more complicated, however, they may be found anywhere in nature and in art.

The golden ratio

(approximately 1.618) is found frequently in nature. It is defined by two numbers, that form a ratio such that (a+b)/a = a/b (a/b being the golden ratio). This pattern was exploited by Leonardo da Vinci

in his art. The golden ratio can be seen in nature, from the spirals of flowers to the symmetry of the human body (as expressed in Da Vinci's Vitruvian Man

, one of the most referenced and reproduced works of art today. This is still used by many artists).

Patterns of abstraction may not be directly observable - such as patterns in science, drama, maths, english

is commonly described as the "Science of Pattern." Any sequence of numbers that may be modeled by a mathematical function is considered a pattern.

In Pattern theory

, mathematicians attempt to describe the world in terms of patterns. The goal is to lay out the world in a more computationally friendly manner.

Patterns are common in many areas of mathematics. Recurring decimals are one example. These are repeating sequences of digits which repeat infinitely. For example, 1 divided by 81 will result in the answer 0.012345679... the numbers 0-9 (except 8) will repeat forever — 1/81 is a recurring decimal.

Fractal

s are mathematical patterns that are scale invariant. This means that the shape of the pattern does not depend on how closely you look at it. Self-similarity is found in fractals. Examples of natural fractals are coast lines and tree shapes, which repeat their shape regardless of what magnification you view at. While the outer appearance of self-similar patterns can be quite complex, the rules needed to describe or produce their formation

can be extremely simple (e.g. Lindenmayer systems for the description of tree

shapes).

In computer science, complex mathematical models may be designed to create more complex patterns. Patterns may be found in every branch of computer science.

An important use of patterns in computer science is the idea of Design patterns

. Design patterns are general solutions to problems in object-oriented programming. They will not solve a specific problem, but they provide a sort of architectural outline that may be reused in order to speed up the development process of a program. Design patterns have provided the stepping stone for computer science to truly enter the engineering field.

A completely different use of patterns is the JPEG compressed image format. The image is divided into a grid pattern of equal-size tiles. Then each tile is analysed independently to find the dominant patterns in the part of the image it contains. As more compression is applied, the best-match tiles are chosen from a smaller set of available tiles. If excessive compression is applied then both the tiles and the patterns within tiles may be seen.

, a training image is used to provide the spatial model of variability. A pattern-based modeling approach can thus be seen as an image construction algorithm, where the patterns of the training image are used, and tiled next to each other such that a new image with similar characteristics/features is generated.

, a mineral

's crystal structure expresses a recurring pattern. In fact, this is one of the five requirements of a mineral. Minerals must have a fixed chemical composition in a repeating arrangement, such as a crystal matrix. A 2-dimensional crystal structure has 10 different possible planar lattices. Moving up to 3 dimensions, 32 patterns are possible. These are called bravais lattices.

**pattern**, from the FrenchFrench language

French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

*patron*, is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects.These elements repeat in a predictable manner. It can be a template or model which can be used to generate things or parts of a thing, especially if the things that are created have enough in common for the underlying pattern to be inferred, in which case the things are said to

*exhibit*the unique pattern.The most basic patterns, called Tessellation

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

s, are based on repetition and periodicity

Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

. A single template, tile

Tile

A tile is a manufactured piece of hard-wearing material such as ceramic, stone, metal, or even glass. Tiles are generally used for covering roofs, floors, walls, showers, or other objects such as tabletops...

, or cell, is combined with duplicates without change or modification. For example, simple harmonic oscillators produce repeated patterns of movement.

Other patterns, such as Penrose tiling

Penrose tiling

A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original...

and Pongal

Pongal

Thai Ponggal is a harvest festival celebrated by Tamils in the Indian state of Tamil Nadu, Indian Union Territory of Pondicherry and in Sri Lanka. Pongal coincides with the festival Makara Sankranthi celebrated throughout India. Pongal in Tamil means "boiling over" or "spill over". The boiling...

or Kolam

Kolam

Kolam is a form of painting that is drawn using rice powder. A Kolam is a geometrical line drawing composed of curved loops, drawn around a grid pattern of dots. In South India, it is widely practised by female Hindu family members in front of their homes.-Purpose:Kolams are thought to bestow...

patterns from India, use symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

which is a form of finite repetition, instead of translation

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

which can repeat to infinity. Fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

patterns also use magnification

Magnification

Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

or scaling

Scaling (geometry)

In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...

giving an effect known as self-similarity

Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

or scale invariance

Scale invariance

In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...

. Some plants, like Fern

Fern

A fern is any one of a group of about 12,000 species of plants belonging to the botanical group known as Pteridophyta. Unlike mosses, they have xylem and phloem . They have stems, leaves, and roots like other vascular plants...

s, even generate a pattern using an affine transformation

Affine transformation

In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

which combines translation

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

, scaling, rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

and reflection

Reflection (physics)

Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

.

Pattern matching

Pattern matching

In computer science, pattern matching is the act of checking some sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact. The patterns generally have the form of either sequences or tree structures...

is the act of checking for the presence of the constituents of a pattern, whereas the detecting for underlying patterns is referred to as pattern recognition

Pattern recognition

In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...

. The question of how a pattern emerges is accomplished through the work of the scientific field of pattern formation

Pattern formation

The science of pattern formation deals with the visible, orderly outcomes of self-organisation and the common principles behind similar patterns....

.

Pattern recognition is more complex when templates are used to generate variants. For example, in English, sentences often follow the "N-VP" (noun - verb phrase) pattern, but some knowledge of the English language

English language

English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

is required to detect the pattern. Computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, ethology

Ethology

Ethology is the scientific study of animal behavior, and a sub-topic of zoology....

, and psychology

Psychology

Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...

are fields which study patterns.

- "A pattern has an integrity independent of the medium by virtue of which you have received the information that it exists. Each of the chemical elements is a pattern integrity. Each individual is a pattern integrity. The pattern integrity of the human individual is evolutionary and not static."
- R. Buckminster Fuller (1895-1983), U.S.American philosopher and inventor, in
*Synergetics: Explorations in the Geometry of Thinking*(1975), Pattern Integrity 505.201

- R. Buckminster Fuller (1895-1983), U.S.American philosopher and inventor, in

### Visual

Visual patterns are very common such as simple decorative patherns (stripes, zigzagZigzag

A zigzag is a pattern made up of small corners at variable angles, though constant within the zigzag, tracing a path between two parallel lines; it can be described as both jagged and fairly regular....

s, and polka dots). Others can be more complicated, however, they may be found anywhere in nature and in art.

- Penrose tilingPenrose tilingA Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original...

s

#### Art

One recurring pattern in a single piece of art may constitute a motif.The golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

(approximately 1.618) is found frequently in nature. It is defined by two numbers, that form a ratio such that (a+b)/a = a/b (a/b being the golden ratio). This pattern was exploited by Leonardo da Vinci

Leonardo da Vinci

Leonardo di ser Piero da Vinci was an Italian Renaissance polymath: painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer whose genius, perhaps more than that of any other figure, epitomized the Renaissance...

in his art. The golden ratio can be seen in nature, from the spirals of flowers to the symmetry of the human body (as expressed in Da Vinci's Vitruvian Man

Vitruvian Man

The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci circa 1487. It is accompanied by notes based on the work of the famed architect, Vitruvius. The drawing, which is in pen and ink on paper, depicts a male figure in two superimposed positions with his arms and legs apart and...

, one of the most referenced and reproduced works of art today. This is still used by many artists).

- "Art is the imposing of a pattern on experience, and our aesthetic enjoyment is recognition of the pattern."
- Alfred North WhiteheadAlfred North WhiteheadAlfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...

(1861-1947), English philosopher and mathematician.*Dialogues*, June 10, 1943.

- Alfred North Whitehead

Patterns of abstraction may not be directly observable - such as patterns in science, drama, maths, english

## Mathematics

MathematicsMathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

is commonly described as the "Science of Pattern." Any sequence of numbers that may be modeled by a mathematical function is considered a pattern.

In Pattern theory

Pattern theory

Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it...

, mathematicians attempt to describe the world in terms of patterns. The goal is to lay out the world in a more computationally friendly manner.

Patterns are common in many areas of mathematics. Recurring decimals are one example. These are repeating sequences of digits which repeat infinitely. For example, 1 divided by 81 will result in the answer 0.012345679... the numbers 0-9 (except 8) will repeat forever — 1/81 is a recurring decimal.

Fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s are mathematical patterns that are scale invariant. This means that the shape of the pattern does not depend on how closely you look at it. Self-similarity is found in fractals. Examples of natural fractals are coast lines and tree shapes, which repeat their shape regardless of what magnification you view at. While the outer appearance of self-similar patterns can be quite complex, the rules needed to describe or produce their formation

Pattern formation

The science of pattern formation deals with the visible, orderly outcomes of self-organisation and the common principles behind similar patterns....

can be extremely simple (e.g. Lindenmayer systems for the description of tree

Tree

A tree is a perennial woody plant. It is most often defined as a woody plant that has many secondary branches supported clear of the ground on a single main stem or trunk with clear apical dominance. A minimum height specification at maturity is cited by some authors, varying from 3 m to...

shapes).

### Computer science

In computer science, complex mathematical models may be designed to create more complex patterns. Patterns may be found in every branch of computer science.

An important use of patterns in computer science is the idea of Design patterns

Design pattern (computer science)

In software engineering, a design pattern is a general reusable solution to a commonly occurring problem within a given context in software design. A design pattern is not a finished design that can be transformed directly into code. It is a description or template for how to solve a problem that...

. Design patterns are general solutions to problems in object-oriented programming. They will not solve a specific problem, but they provide a sort of architectural outline that may be reused in order to speed up the development process of a program. Design patterns have provided the stepping stone for computer science to truly enter the engineering field.

A completely different use of patterns is the JPEG compressed image format. The image is divided into a grid pattern of equal-size tiles. Then each tile is analysed independently to find the dominant patterns in the part of the image it contains. As more compression is applied, the best-match tiles are chosen from a smaller set of available tiles. If excessive compression is applied then both the tiles and the patterns within tiles may be seen.

### Spatial Statistics

In multiple-point GeostatisticsGeostatistics

Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...

, a training image is used to provide the spatial model of variability. A pattern-based modeling approach can thus be seen as an image construction algorithm, where the patterns of the training image are used, and tiled next to each other such that a new image with similar characteristics/features is generated.

## Science

In geologyGeology

Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

, a mineral

Mineral

A mineral is a naturally occurring solid chemical substance formed through biogeochemical processes, having characteristic chemical composition, highly ordered atomic structure, and specific physical properties. By comparison, a rock is an aggregate of minerals and/or mineraloids and does not...

's crystal structure expresses a recurring pattern. In fact, this is one of the five requirements of a mineral. Minerals must have a fixed chemical composition in a repeating arrangement, such as a crystal matrix. A 2-dimensional crystal structure has 10 different possible planar lattices. Moving up to 3 dimensions, 32 patterns are possible. These are called bravais lattices.

- Cellular Automata
- CrystalsCrystal structureIn mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...

## See also

- Pattern formationPattern formationThe science of pattern formation deals with the visible, orderly outcomes of self-organisation and the common principles behind similar patterns....
- Pattern (sewing)Pattern (sewing)In sewing and fashion design, a pattern is an original garment from which other garments of a similar style are copied, or the paper or cardboard templates from which the parts of a garment are traced onto fabric before cutting out and assembling .Patternmaking, pattern making or pattern cutting is...
- Pattern coinPattern coinA pattern coin is a coin which has not been approved for release, produced for the purpose of evaluating a proposed coin design. They are often off-metal strikes, to proof standard or piedforts...
- Pattern (casting)Pattern (casting)In casting, a pattern is a replica of the object to be cast, used to prepare the cavity into which molten material will be poured during the casting process.Patterns used in sand casting may be made of wood, metal, plastics or other materials...
- Pattern languagePattern languageA pattern language, a term coined by architect Christopher Alexander, is a structured method of describing good design practices within a field of expertise. Advocates of this design approach claim that ordinary people of ordinary intelligence can use it to successfully solve very large, complex...
- Pedagogical patternsPedagogical patternsPedagogical Patterns are high-level patterns that have been recognized in many areas of training and pedagogy such as group work, software design, human computer interaction, education and others. The concept is an extension of pattern languages...
- Pattern (architecture)
- Design patternDesign patternA design pattern in architecture and computer science is a formal way of documenting a solution to a design problem in a particular field of expertise. The idea was introduced by the architect Christopher Alexander in the field of architecture and has been adapted for various other disciplines,...
- Tessellations
- Pattern recognitionPattern recognitionIn machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...
- Design pattern (computer science)Design pattern (computer science)In software engineering, a design pattern is a general reusable solution to a commonly occurring problem within a given context in software design. A design pattern is not a finished design that can be transformed directly into code. It is a description or template for how to solve a problem that...