Pattern

Last updated
Ionic frieze from the Erechtheum, dimensions 130 x 50 cm, in the Glyptothek.jpg
Kleophrades Painter ARV 189 78bis mission to Achilles.jpg
William Morris - "Pimpernel" - Google Art Project.jpg
Strawberrythief.jpg
Settee MET SF2007 368 img4.jpg
Nikoxenos Painter ARV 221 14 athletes with trainer and flute player - satyrs as athletes (05).jpg
D.A. Sturdza House, Bucharest (Romania) 32.jpg
Various examples of patterns

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.

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.

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.

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

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]

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]

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.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]

See also: pattern book.

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

Fractal Self similar mathematical structures

In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive 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. Fractal geometry lies within the mathematical branch of measure theory.

M. C. Escher Dutch graphic artist known for his mathematically-inspired works

Maurits Cornelis Escher was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for long somewhat neglected in the art world, even in his native Netherlands. He was 70 before a retrospective exhibition was held. In the twenty-first century, he became more widely appreciated, with exhibitions across the world.

Sacred geometry Symbolic and sacred meanings ascibed to certain geometric shapes

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.

Symmetry Mathematical invariance under transformations

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

Fractal landscape

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behavior 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 behavior.

Mathematics and architecture overview about the correlation of 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 tiling or tessellation of a flat surface is the covering 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.

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 One of several geometric patterns which are recurringly observed during hypnagogia, hallucinations and altered states of consciousness.

A form constant is one of several geometric patterns which are recurringly observed during hypnagogia, 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.

Geometry Branch of mathematics

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.

Mathematical beauty Notion that some mathematicians may derive aesthetic pleasure from mathematics

Mathematical beauty is the aesthetic pleasure typically derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians often express this pleasure by describing mathematics as beautiful. They might also 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 Relationship between 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 an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.

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 recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. 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

In geometry, an object has symmetry if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also 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. doi:10.1038/scientificamerican1286-46.
  9. Clerc, Marcel G.; González-Cortés, Gregorio; Odent, Vincent; Wilson, Mario (29 June 2016). "Optical textures: characterizing spatiotemporal chaos". Optics Express. 24 (14): 15478–85. arXiv: 1601.00844 . doi:10.1364/OE.24.015478. PMID   27410822.
  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" (PDF). 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.CS1 maint: location (link)
  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 .CS1 maint: ref=harv (link)
  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, Benoit 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