# Pattern

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A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition.

Wallpaper is a material used in interior decoration to decorate the interior walls of domestic and public buildings. It is usually sold in rolls and is applied onto a wall using wallpaper paste. Wallpapers can come plain as "lining paper", textured, with a regular repeating pattern design, or, much less commonly today, with a single non-repeating large design carried over a set of sheets. The smallest rectangle that can be tiled to form the whole pattern is known as the pattern repeat.

## Contents

Any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, never exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; [1] indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world.

A sense is a physiological capacity of organisms that provides data for perception. The senses and their operation, classification, and theory are overlapping topics studied by a variety of fields, most notably neuroscience, cognitive psychology, and philosophy of perception. The nervous system has a specific sensory nervous system, and a sense organ, or sensor, dedicated to each sense.

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

Mathematics includes the study of such topics as quantity, structure, space, and change.

In art and architecture, decorations or visual motifs may be combined and repeated to form patterns designed to have a chosen effect on the viewer. In computer science, a software design pattern is a known solution to a class of problems in programming. In fashion, the pattern is a template used to create any number of similar garments.

In art and iconography, a motif(pronunciation)  is an element of an image. A motif may be repeated in a pattern or design, often many times, or may just occur once in a work.

In software engineering, a software design pattern is a general, reusable solution to a commonly occurring problem within a given context in software design. It is not a finished design that can be transformed directly into source or machine code. It is a description or template for how to solve a problem that can be used in many different situations. Design patterns are formalized best practices that the programmer can use to solve common problems when designing an application or system.

A die is a specialized tool used in manufacturing industries to cut or shape material mostly using a press. Like molds, dies are generally customized to the item they are used to create. Products made with dies range from simple paper clips to complex pieces used in advanced technology.

## Nature

Nature provides examples of many kinds of pattern, including symmetries, trees and other structures with a fractal dimension, spirals, meanders, waves, foams, tilings, cracks and stripes. [2]

Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.

In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals tend to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set; Because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales called self similarity, also known as expanding symmetry or unfolding symmetry; If this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar.

A meander is one of a series of regular sinuous curves, bends, loops, turns, or windings in the channel of a river, stream, or other watercourse. It is produced by a stream or river swinging from side to side as it flows across its floodplain or shifts its channel within a valley. A meander is produced by a stream or river as it erodes the sediments comprising an outer, concave bank and deposits this and other sediment downstream on an inner, convex bank which is typically a point bar. The result of sediments being eroded from the outside concave bank and their deposition on an inside convex bank is the formation of a sinuous course as a channel migrates back and forth across the down-valley axis of a floodplain. The zone within which a meandering stream shifts its channel across either its floodplain or valley floor from time to time is known as a meander belt. It typically ranges from 15 to 18 times the width of the channel. Over time, meanders migrate downstream, sometimes in such a short time as to create civil engineering problems for local municipalities attempting to maintain stable roads and bridges.

### Symmetry

Symmetry is widespread in living things. Animals that move usually have bilateral or mirror symmetry as this favours movement. [3] Plants often have radial or rotational symmetry, as do many flowers, as well as animals which are largely static as adults, such as sea anemones. Fivefold symmetry is found in the echinoderms, including starfish, sea urchins, and sea lilies. [4]

Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

Sea anemones are a group of marine, predatory animals of the order Actiniaria. They are named after the anemone, a terrestrial flowering plant, because of the colourful appearance of many. Sea anemones are classified in the phylum Cnidaria, class Anthozoa, subclass Hexacorallia. As cnidarians, sea anemones are related to corals, jellyfish, tube-dwelling anemones, and Hydra. Unlike jellyfish, sea anemones do not have a medusa stage in their life cycle.

Among non-living things, snowflakes have striking sixfold symmetry: each flake is unique, its structure recording the varying conditions during its crystallisation similarly on each of its six arms. [5] Crystals have a highly specific set of possible crystal symmetries; they can be cubic or octahedral, but cannot have fivefold symmetry (unlike quasicrystals). [6]

A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, then falls through the Earth's atmosphere as snow. Each flake nucleates around a dust particle in supersaturated air masses by attracting supercooled cloud water droplets, which freeze and accrete in crystal form. Complex shapes emerge as the flake moves through differing temperature and humidity zones in the atmosphere, such that individual snowflakes differ in detail from one another, but may be categorized in eight broad classifications and at least 80 individual variants. The main constituent shapes for ice crystals, from which combinations may occur, are needle, column, plate, and rime. Snow appears white in color despite being made of clear ice. This is due to diffuse reflection of the whole spectrum of light by the small crystal facets of the snowflakes.

A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification.

In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. A single crystal's habit is a description of its general shape and its crystallographic forms, plus how well developed each form is.

### Spirals

Spiral patterns are found in the body plans of animals including molluscs such as the nautilus, and in the phyllotaxis of many plants, both of leaves spiralling around stems, and in the multiple spirals found in flowerheads such as the sunflower and fruit structures like the pineapple. [7]

### Chaos, turbulence, meanders and complexity

Chaos theory predicts that while the laws of physics are deterministic, there are events and patterns in nature that never exactly repeat because extremely small differences in starting conditions can lead to widely differing outcomes. [8] . The patterns in nature tend to be static due to dissipation on the emergence process, but when there is interplay between injection of energy and dissipation there can arise a complex dynamic [9] . Many natural patterns are shaped by this complexity, including vortex streets [10] , other effects of turbulent flow such as meanders in rivers. [11] or nonlinear interaction of the system [12]

### Waves, dunes

Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it oscillate as they pass by. [13] Wind waves are surface waves that create the chaotic patterns of the sea. As they pass over sand, such waves create patterns of ripples; similarly, as the wind passes over sand, it creates patterns of dunes. [14]

### Bubbles, foam

Foams obey Plateau's laws, which require films to be smooth and continuous, and to have a constant average curvature. Foam and bubble patterns occur widely in nature, for example in radiolarians, sponge spicules, and the skeletons of silicoflagellates and sea urchins. [15] [16]

### Cracks

Cracks form in materials to relieve stress: with 120 degree joints in elastic materials, but at 90 degrees in inelastic materials. Thus the pattern of cracks indicates whether the material is elastic or not. Cracking patterns are widespread in nature, for example in rocks, mud, tree bark and the glazes of old paintings and ceramics. [17]

### Spots, stripes

Alan Turing, [18] and later the mathematical biologist James D. Murray [19] and other scientists, described a mechanism that spontaneously creates spotted or striped patterns, for example in the skin of mammals or the plumage of birds: a reaction-diffusion system involving two counter-acting chemical mechanisms, one that activates and one that inhibits a development, such as of dark pigment in the skin. [20] These spatiotemporal patterns slowly drift, the animals' appearance changing imperceptibly as Turing predicted.

## Art and architecture

### Tilings

In visual art, pattern consists in regularity which in some way "organizes surfaces or structures in a consistent, regular manner." At its simplest, a pattern in art may be a geometric or other repeating shape in a painting, drawing, tapestry, ceramic tiling or carpet, but a pattern need not necessarily repeat exactly as long as it provides some form or organizing "skeleton" in the artwork. [21] In mathematics, a tessellation is the tiling of a plane using one or more geometric shapes (which mathematicians call tiles), with no overlaps and no gaps. [22]

### In architecture

In architecture, motifs are repeated in various ways to form patterns. Most simply, structures such as windows can be repeated horizontally and vertically (see leading picture). Architects can use and repeat decorative and structural elements such as columns, pediments, and lintels. [23] Repetitions need not be identical; for example, temples in South India have a roughly pyramidal form, where elements of the pattern repeat in a fractal-like way at different sizes. [24]

## Science and mathematics

Mathematics is sometimes called the "Science of Pattern", in the sense of rules that can be applied wherever needed. [25] For example, any sequence of numbers that may be modeled by a mathematical function can be considered a pattern. Mathematics can be taught as a collection of patterns. [26]

### Fractals

Some mathematical rule-patterns can be visualised, and among these are those that explain patterns in nature including the mathematics of symmetry, waves, meanders, and fractals. Fractals 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 self-similar patterns can appear indefinitely complex, the rules needed to describe or produce their formation can be simple (e.g. Lindenmayer systems describing tree shapes). [27]

In pattern theory, devised by Ulf Grenander, mathematicians attempt to describe the world in terms of patterns. The goal is to lay out the world in a more computationally friendly manner. [28]

In the broadest sense, any regularity that can be explained by a scientific theory is a pattern. As in mathematics, science can be taught as a set of patterns. [29]

## Computer science

In computer science, a software design pattern, in the sense of a template, is a general solution to a problem in programming. A design pattern provides a reusable architectural outline that may speed the development of many computer programs. [30]

## Fashion

In fashion, the pattern is a template, a technical two-dimensional tool used to create any number of identical garments. It can be considered as a means of translating from the drawing to the real garment. [31]

## Related Research Articles

Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions. It is associated with the belief that a god is the geometer of the world. The geometry used in the design and construction of religious structures such as churches, temples, mosques, religious monuments, altars, and tabernacles has sometimes been considered sacred. The concept applies also to sacred spaces such as temenoi, sacred groves, village greens, and holy wells, and the creation of religious art.

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behaviour that mimics the appearance of natural terrain. In other words, the result of the procedure is not a deterministic fractal surface, but rather a random surface that exhibits fractal behaviour.

Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

A meander or meandros is a decorative border constructed from a continuous line, shaped into a repeated motif. Such a design is also called the Greek fret or Greek key design, although these are modern designations.

In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is given its name because several streets in Cairo are paved in this design. It is one of 15 known monohedral pentagon tilings. It is also called MacMahon's net after Percy Alexander MacMahon and his 1921 publication New Mathematical Pastimes. Conway calls it a 4-fold pentille.

A form constant is one of several geometric patterns which are recurringly observed during hallucinations and altered states of consciousness.

Girih tiles are a set of five tiles that were used in the creation of Islamic geometric patterns using strapwork (girih) for decoration of buildings in Islamic architecture. They have been used since about the year 1200 and their arrangements found significant improvement starting with the Darb-i Imam shrine in Isfahan in Iran built in 1453.

Girih is a decorative Islamic geometric artform used in architecture and handicraft objects, consisting of angled lines that form an interlaced strapwork pattern.

Mathematical beauty describes the notion that some mathematicians may derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Mathematicians describe mathematics as an art form or, at a minimum, as a creative activity. Comparisons are often made with music and poetry.

Islamic decoration, which tends to avoid using figurative images, makes frequent use of geometric patterns which have developed over the centuries.

Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.

Jay Kappraff is an American professor of mathematics at the New Jersey Institute of Technology and author.

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by the American mathematician Solomon W. Golomb, who used it to describe self-replicating tilings. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

A geometric object has symmetry if there is an "operation" or "transformation" that maps the figure/object onto itself; i.e., it is said that the object has an invariance under the transform. For instance, a circle rotated about its center will have the same shape and size as the original circle—all points before and after the transform would be indistinguishable. A circle is said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure, the figure is said to have reflectional symmetry or line symmetry; moreover, it is possible for a figure/object to have more than one line of symmetry.

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## Bibliography

### In art and architecture

• Alexander, C. A Pattern Language: Towns, Buildings, Construction. Oxford, 1977.
• de Baeck, P. Patterns. Booqs, 2009.
• Garcia, M. The Patterns of Architecture. Wiley, 2009.
• Kiely, O. Pattern. Conran Octopus, 2010.
• Pritchard, S. V&A Pattern: The Fifties. V&A Publishing, 2009.

### In science and mathematics

• Adam, J. A. Mathematics in Nature: Modeling Patterns in the Natural World. Princeton, 2006.
• Resnik, M. D. Mathematics as a Science of Patterns. Oxford, 1999.

### In computing

• Gamma, E., Helm, R., Johnson, R., Vlissides, J. Design Patterns . Addison-Wesley, 1994.
• Bishop, C. M. Pattern Recognition and Machine Learning. Springer, 2007.