Pattern

Last updated
Tilings, such as these from Igreja de Campanha, Porto, Portugal, are visual patterns used for decoration. Igreja de Campanha Azulejo 4.jpg
Tilings, such as these from Igreja de Campanhã, Porto, Portugal, are visual patterns used for decoration.

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 branch of mathematics that measures the shape, size and position of objects

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.

Shape form of an object or its external boundary

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 material used to cover and decorate interior walls

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.

Sense Physiological capacity of organisms that provides data for perception

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 systematic enterprise that builds and organizes knowledge

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

Mathematics Field of study concerning quantity, patterns and change

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.

Motif (visual arts) in the visual arts, individual design element, alone or combined to produce a pattern

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 state; balance of object

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.

Fractal A self-similar pattern or set

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.

Meander A sinuous bend in a series in the channel of a river

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

Snowflake sixfold symmetry Schnee1.jpg
Snowflake sixfold 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 symmetry with respect to a plane, when the shape does not change by reflecting all of its part from a mirror plane

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 Symmetry (something looking the same) under rotation

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 anemone Order of cnidarians

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]

Snowflake single ice crystal or an aggregation of ice crystals which falls through the Earths atmosphere

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.

Crystal solid material whose constituent atoms, molecules, or ions are arranged in an ordered pattern extending in all three spatial dimensions

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.

Crystal habit mineralogical term for the visible shape of a mineral

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

Aloe polyphylla phyllotaxis Aloe polyphylla spiral.jpg
Aloe polyphylla phyllotaxis

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

Vortex street turbulence Vortex-street-1-90deg.jpg
Vortex street turbulence

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

Dune ripple Sand dune ripples.jpg
Dune ripple

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

Foam of soap bubbles Foam - big.jpg
Foam of soap bubbles

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

Shrinkage Cracks Cracked earth in the Rann of Kutch.jpg
Shrinkage 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

Giant pufferfish skin Giant Pufferfish skin pattern detail.jpg
Giant pufferfish skin

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

Elaborate ceramic tiles at Topkapi Palace Enderun library Topkapi 42.JPG
Elaborate ceramic tiles at Topkapi Palace

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

Patterns in architecture: the Virupaksha temple at Hampi has a fractal-like structure where the parts resemble the whole. Hampi1.jpg
Patterns in architecture: the Virupaksha temple at Hampi has a fractal-like structure where the parts resemble the whole.

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

Fractal model of a fern illustrating self-similarity Fractal fern explained.png
Fractal model of a fern illustrating self-similarity

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]

See also

Related Research Articles

Sacred geometry

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.

Spiral curve which emanates from a point, moving farther away as it revolves around the point

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

Fractal landscape

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

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.

Tessellation tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps

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.

Meander (art) motif

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.

Cairo pentagonal tiling a tiling of the plane by pentagons, in which groups of four tiles form two interlaced hexagonal patterns

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.

Form constant

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

Girih tiles five tiles used in Islamic decorative art

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 geometric lines used in Islamic decorative art

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 notion that some mathematicians may derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful

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 geometric patterns geometric pattern characteristic of Muslim art

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

Mathematics and art

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.

Penrose tiling non-periodic tiling of the plane

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.

Rep-tile a tiling of the plane in which a prototile is recursively subdivided into copies of itself

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 visible regularity of form found in the natural world

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.

Symmetry (geometry) geometrical property and transformation

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.

References

  1. Stewart, 2001. Page 6.
  2. Stevens, Peter. Patterns in Nature, 1974. Page 3.
  3. Stewart, Ian. 2001. Pages 48-49.
  4. Stewart, Ian. 2001. Pages 64-65.
  5. Stewart, Ian. 2001. Page 52.
  6. Stewart, Ian. 2001. Pages 82-84.
  7. Kappraff, Jay (2004). "Growth in Plants: A Study in Number" (PDF). Forma. 19: 335–354.
  8. Crutchfield, James P; Farmer, J Doyne; Packard, Norman H; Shaw, Robert S (December 1986). "Chaos". Scientific American. 254 (12): 46–57.
  9. Clerc, Marcel G.; González-Cortés, Gregorio; Odent, Vincent; Wilson, Mario (29 June 2016). "Optical textures: characterizing spatiotemporal chaos". Optics Express. 24 (14): 15478. doi:10.1364/OE.24.015478.
  10. von Kármán, Theodore. Aerodynamics. McGraw-Hill (1963): ISBN   978-0070676022. Dover (1994): ISBN   978-0486434858.
  11. Lewalle, Jacques (2006). "Flow Separation and Secondary Flow: Section 9.1" (PDF). Lecture Notes in Incompressible Fluid Dynamics: Phenomenology, Concepts and Analytical Tools. Syracuse, NY: Syracuse University. Archived from the original (PDF) on 2011-09-29.
  12. Scroggie, A.J; Firth, W.J; McDonald, G.S; Tlidi, M; Lefever, R; Lugiato, L.A (August 1994). "Pattern formation in a passive Kerr cavity". Chaos, Solitons & Fractals. 4 (8–9): 1323–1354. doi:10.1016/0960-0779(94)90084-1.
  13. French, A.P. Vibrations and Waves. Nelson Thornes, 1971.
  14. Tolman, H.L. (2008), "Practical wind wave modeling", in Mahmood, M.F. (ed.), CBMS Conference Proceedings on Water Waves: Theory and Experiment (PDF), Howard University, USA, 13–18 May 2008: World Scientific Publ.
  15. Philip Ball. Shapes, 2009. pp 68, 96-101.
  16. Frederick J. Almgren, Jr. and Jean E. Taylor, The geometry of soap films and soap bubbles, Scientific American, vol. 235, pp. 82–93, July 1976.
  17. Stevens, Peter. 1974. Page 207.
  18. Turing, A. M. (1952). "The Chemical Basis of Morphogenesis". Philosophical Transactions of the Royal Society B . 237 (641): 37–72. Bibcode:1952RSPTB.237...37T. doi:10.1098/rstb.1952.0012.
  19. Murray, James D. (9 March 2013). Mathematical Biology. Springer Science & Business Media. pp. 436–450. ISBN   978-3-662-08539-4.
  20. Ball, Philip. Shapes. 2009. Pages 159–167.
  21. Jirousek, Charlotte (1995). "Art, Design, and Visual Thinking". Pattern. Cornell University. Retrieved 12 December 2012.
  22. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman.
  23. Adams, Laurie (2001). A History of Western Art. McGraw Hill. p. 99.
  24. Jackson, William Joseph (2004). Heaven's Fractal Net: Retrieving Lost Visions in the Humanities. Indiana University Press. p. 2.
  25. Resnik, Michael D. (November 1981). "Mathematics as a Science of Patterns: Ontology and Reference". Noûs. 15 (4): 529–550. doi:10.2307/2214851. JSTOR   2214851.
  26. Bayne, Richard E (2012). "MATH 012 Patterns in Mathematics - spring 2012" . Retrieved 16 January 2013.
  27. Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN   978-0-7167-1186-5.
  28. Grenander, Ulf; Miller, Michael (2007). Pattern Theory: From Representation to Inference. Oxford University Press.
  29. "Causal Patterns in Science". Harvard Graduate School of Education. 2008. Retrieved 16 January 2013.
  30. Gamma et al, 1994.
  31. "An Artist Centric Marketplace for Fashion Sketch Templates, Croquis & More". Illustrator Stuff. Retrieved 7 January 2018.

Bibliography

In nature

In art and architecture

In science and mathematics

In computing