Pattern

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Various examples of patterns

A pattern is a regularity in the world, in human-made design, [1] 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.

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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, rarely 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; [2] : 6  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 many areas of the decorative arts, from ceramics and textiles to wallpaper, "pattern" is used for an ornamental design that is manufactured, perhaps for many different shapes of object. 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.

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. [3]

Symmetry

Snowflake sixfold symmetry First Snowfall (38115232341).jpg
Snowflake sixfold symmetry

Symmetry is widespread in living things. Animals that move usually have bilateral or mirror symmetry as this favours movement. [2] : 48–49  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. [2] : 64–65 

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. [2] : 52  Crystals have a highly specific set of possible crystal symmetries; they can be cubic or octahedral, but cannot have fivefold symmetry (unlike quasicrystals). [2] : 82–84 

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. [4]

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. [5] 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. [6] Many natural patterns are shaped by this complexity, including vortex streets, [7] other effects of turbulent flow such as meanders in rivers. [8] or nonlinear interaction of the system [9]

Waves, dunes

Dune ripple Sand dune ripples.jpg
Dune ripple
Dune ripples and boards form a symmetrical pattern. Monster - Sand - Brador - 2021.jpg
Dune ripples and boards form a symmetrical pattern.

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. [10] 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. [11]

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. [12] [13]

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. [14]

Spots, stripes

Giant Pufferfish skin pattern detail.jpg
Mbu pufferfish skin
Animal skin.jpg
Skins of a South African giraffe and Burchell's zebra

Alan Turing, [15] and later the mathematical biologist James D. Murray [16] 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. [17] These spatiotemporal patterns slowly drift, the animals' appearance changing imperceptibly as Turing predicted.

Art and architecture

Tilings

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

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. [18] 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. [19]

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. [20] 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. [21]

Patterns in Architecture: the columns of Zeus's temple in Athens Evening columns Zeus temple Athens.jpg
Patterns in Architecture: the columns of Zeus's temple in Athens

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. [22] 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. [23]

Gravity is a source of ubiquitous scientific patterns or patterns of observation. The sun rising and falling pattern each day results from the rotation of the earth while in orbit around the sun. Likewise, the moon's path through the sky is due to its orbit of the earth. These examples, while perhaps trivial, are examples of the "unreasonable effectiveness of mathematics" which obtain due to the differential equations whose application within physics function to describe the most general empirical patterns of the universe. [24]

Real patterns

Daniel Dennett's notion of real patterns, discussed in his 1991 paper of the same name, [25] provides an ontological framework aiming to discern the reality of patterns beyond mere human interpretation, by examining their predictive utility and the efficiency they provide in compressing information. For example, centre of gravity is a real pattern because it allows us to predict the movements of a bodies such as the earth around the sun, and it compresses all the information about all the particles in the sun and the earth that allows us to make those predictions.

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). [26]

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. [27]

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. [28]

A recent study from Aesthetics and Psychological Effects of Fractal Based Design [29] suggested that fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant wellbeing. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create a ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant wellbeing.

See also

Related Research Articles

<span class="mw-page-title-main">Fractal</span> Infinitely detailed mathematical structure

In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is 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, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.

<span class="mw-page-title-main">Fractal art</span> Form of algorithmic art

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

<span class="mw-page-title-main">Symmetry</span> Mathematical invariance under transformations

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

In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.

<span class="mw-page-title-main">Tessellation</span> Tiling of a plane in mathematics

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

Low-complexity art, first described by Jürgen Schmidhuber in 1997 and now established as a seminal topic within the larger field of computer science, is art that can be described by a short computer program.

<span class="mw-page-title-main">Cymatics</span> Creation of visible patterns on a vibrated plate

Cymatics is a subset of modal vibrational phenomena. The term was coined by Swiss physician Hans Jenny (1904–1972). Typically the surface of a plate, diaphragm, or membrane is vibrated, and regions of maximum and minimum displacement are made visible in a thin coating of particles, paste, or liquid. Different patterns emerge in the excitatory medium depending on the geometry of the plate and the driving frequency.

<span class="mw-page-title-main">Form constant</span> Recurringly observed geometric pattern

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

<span class="mw-page-title-main">Fractal curve</span> Mathematical curve whose shape is a fractal

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite length — and every subarc longer than a single point has infinite length.

<i>Girih</i> tiles Five tiles used in Islamic decorative art

Girihtiles 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.

<i>Girih</i> Geometric patterns in Islamic architecture

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

<span class="mw-page-title-main">Mathematical beauty</span> Aesthetic value of mathematics

Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics as beautiful or describe mathematics as an art form, or, at a minimum, as a creative activity.

<span class="mw-page-title-main">Islamic geometric patterns</span> Geometric pattern characteristic of Muslim art

Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create a representation of an important Islamic figure according to many holy scriptures.

<span class="mw-page-title-main">Mathematics and art</span> 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.

<span class="mw-page-title-main">Penrose tiling</span> 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 a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. 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 was an American professor of mathematics at the New Jersey Institute of Technology and author.

<span class="mw-page-title-main">Rep-tile</span> Shape 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.

<span class="mw-page-title-main">Patterns in nature</span> 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.

<span class="mw-page-title-main">Symmetry (geometry)</span> Geometrical property

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.

Edmund Orme Harriss is a British mathematician, writer and artist. Since 2010 he has been at the Fulbright College of Arts & Sciences at The University of Arkansas in Fayetteville, Arkansas where he is an Assistant Professor of Arts & Sciences (ARSC) and Mathematical Sciences (MASC). He does research in the Geometry of Tilings and Patterns, a branch of Convex and Discrete Geometry. He is the discoverer of the spiral that bears his name.

References

  1. ↑ Garai, Achraf (3 March 2022). "What are design patterns?". achrafgarai.com. Retrieved 1 January 2023.
  2. 1 2 3 4 5 Stewart, Ian (2001). What shape is a snowflake?. London: Weidenfeld & Nicolson. ISBN   0-297-60723-5. OCLC   50272461.
  3. ↑ Stevens, Peter. Patterns in Nature, 1974. Page 3.
  4. ↑ Kappraff, Jay (2004). "Growth in Plants: A Study in Number" (PDF). Forma. 19: 335–354. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-01-18.
  5. ↑ Crutchfield, James P; Farmer, J Doyne; Packard, Norman H; Shaw, Robert S (December 1986). "Chaos". Scientific American. 254 (12): 46–57. Bibcode:1986SciAm.255f..46C. doi:10.1038/scientificamerican1286-46.
  6. ↑ 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 . Bibcode:2016OExpr..2415478C. doi:10.1364/OE.24.015478. PMID   27410822. S2CID   34610459.
  7. ↑ von Kármán, Theodore. Aerodynamics. McGraw-Hill (1963): ISBN   978-0070676022. Dover (1994): ISBN   978-0486434858.
  8. ↑ 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.
  9. ↑ 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. Bibcode:1994CSF.....4.1323S. doi:10.1016/0960-0779(94)90084-1.
  10. ↑ French, A.P. Vibrations and Waves. Nelson Thornes, 1971.[ full citation needed ]
  11. ↑ Tolman, H.L. (2008). "Practical wind wave modeling" (PDF). In Mahmood, M.F. (ed.). CBMS Conference Proceedings on Water Waves: Theory and Experiment. Howard University, USA, 13–18 May 2008. World Scientific Publ.
  12. ↑ Ball, Philip. Shapes, 2009. pp. 68, 96-101.[ full citation needed ]
  13. ↑ 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.
  14. ↑ Stevens, Peter. 1974. Page 207.
  15. ↑ 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. S2CID   937133.
  16. ↑ Murray, James D. (9 March 2013). Mathematical Biology. Springer Science & Business Media. pp. 436–450. ISBN   978-3-662-08539-4.
  17. ↑ Ball, Philip. Shapes, 2009. pp. 159–167.[ full citation needed ]
  18. ↑ Jirousek, Charlotte (1995). "Art, Design, and Visual Thinking". Pattern. Cornell University. Retrieved 12 December 2012.
  19. ↑ Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns . New York: W. H. Freeman. ISBN   9780716711933.
  20. ↑ Adams, Laurie (2001). A History of Western Art. McGraw Hill. p. 99.
  21. ↑ Jackson, William Joseph (2004). Heaven's Fractal Net: Retrieving Lost Visions in the Humanities. Indiana University Press. p. 2.
  22. ↑ 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.
  23. ↑ Bayne, Richard E (2012). "MATH 012 Patterns in Mathematics - spring 2012". Archived from the original on 7 February 2013. Retrieved 16 January 2013.
  24. ↑ Steen, Lynn (June 1988). "The Science of Patterns". Science. 240 (4852): 611–616. doi:10.1126/science.240.4852.611 . Retrieved 2024-08-11.The author attributes Eugene Wigner for the claim for the "unreasonable effectiveness of mathematics," a partial quote which continues "[t]he miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."
  25. ↑ Dennett, D. C. (1991). Real Patterns. The Journal of Philosophy88(1), 27–51.
  26. ↑ Mandelbrot, Benoit B. (1983). The fractal geometry of nature. Macmillan. ISBN   978-0-7167-1186-5.
  27. ↑ Grenander, Ulf; Miller, Michael (2007). Pattern Theory: From Representation to Inference. Oxford University Press.
  28. ↑ "Causal Patterns in Science". Harvard Graduate School of Education. 2008. Retrieved 16 January 2013.
  29. ↑ Robles, Kelly E.; Roberts, Michelle; Viengkham, Catherine; Smith, Julian H.; Rowland, Conor; Moslehi, Saba; Stadlober, Sabrina; Lesjak, Anastasija; Lesjak, Martin; Taylor, Richard P.; Spehar, Branka; Sereno, Margaret E. (2021). "Aesthetics and Psychological Effects of Fractal Based Design". Frontiers in Psychology. 12. doi: 10.3389/fpsyg.2021.699962 . ISSN   1664-1078. PMC   8416160 . PMID   34484047.

Bibliography

In nature

In art and architecture

In science and mathematics

In computing

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