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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 and logical manner. There exists countless kinds of unclassified patterns, present in everyday nature, fashion, many artistic areas, as well as a connection with mathematics. A geometric pattern is a type of pattern formed of repeating geometric shapes and typically repeated like a wallpaper design.

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

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Main article: Patterns in 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

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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

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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

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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

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Dune ripple
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

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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

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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

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Mbu pufferfish skin
Skins of a South African giraffe and Burchell's zebra
Main article: Pattern formation

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

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Further information: Mathematics and art and Mathematics and architecture

Tilings

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Elaborate ceramic tiles at Topkapi Palace
Further information: Tessellation and Tile

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

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Patterns in architecture: the Virupaksha temple at Hampi has a fractal-like structure where the parts resemble the whole.
Main articles: Pattern (architecture) and Mathematics and 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.[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
See also: pattern book

Language and linguistics

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Language provides researchers in linguistics with a wealth of patterns to investigate,[22] and literary studies can investigate patterns in areas such as sound, grammar, motifs, metaphor, imagery, and narrative plot.[23]

Science and mathematics

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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.[24] 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.[25]

Gravity is a source of ubiquitous scientific patterns or patterns of observation. The rising and falling pattern of the sun 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.[26]

Real patterns

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Daniel Dennett's notion of real patterns, discussed in his 1991 paper of the same name,[27] 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 the prediction of 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 scientists to make those predictions.

Fractals

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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 coastlines and tree-shapes, which repeat their shape regardless of the magnification used by the viewer. 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).[28]

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.[29]

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.[30]

A 2021 study, "Aesthetics and Psychological Effects of Fractal Based Design",[31] 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

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References

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Bibliography

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Pattern

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from Grokipedia
A pattern is a discernible regularity, repetition, or arrangement of elements that forms a coherent , in both natural and artificial contexts. It serves as a model or template for imitation, such as in or , and encompasses artistic, musical, or literary forms that organize elements in a predictable way. In , patterns are defined as regularities or similarities that characterize sets of numbers, shapes, graphs, tables, or other objects, enabling generalization and prediction. Patterns permeate nature, manifesting as self-similar structures like fractals in coastlines, river deltas, or tree branches, and repeating forms such as spirals in galaxies, shells, or plant phyllotaxis. These natural patterns often arise from simple rules repeated iteratively, revealing underlying principles governed by physics, biology, and mathematics. In art and design, patterns function as repeating units of shape or form that organize surfaces consistently, enhancing aesthetic appeal in textiles, architecture, and visual compositions. The recognition and study of patterns form the core of mathematics, often described as the science of identifying such structures in numbers and the natural world, from ancient sequences to modern phenomena like planetary orbits. This interdisciplinary pursuit extends to science and technology, where pattern analysis drives discoveries, such as the double helix in DNA, and applications like machine learning algorithms that detect regularities in data. Understanding patterns not only predicts behaviors but also inspires innovations across fields, underscoring their role in bridging abstract theory with tangible reality.

Definitions and Concepts

Core Definition

A pattern is fundamentally a recognizable regularity or repetition in events, objects, or phenomena, characterized by a of repetition, variation, or that establishes predictability, aesthetic , or functional order across diverse domains such as , , and . This encompasses both perceptible designs, like decorative motifs, and abstract sequences, such as recurring behaviors or trends, allowing for the identification of underlying structures that facilitate understanding and prediction. The word "pattern" derives from the Middle English term patron, borrowed from Old French patron meaning "protector" or "model," which ultimately traces back to Latin patronus, denoting a patron, guardian, or exemplar to be followed. In its early usage, it referred to an original or serving as a guide; by the in English, the term had evolved to specifically indicate templates, outlines, or designs intended for replication in crafts, , or other imitative processes. Patterns can be broadly distinguished as static or dynamic: static patterns involve fixed, unchanging arrangements that provide stability and consistency, such as a woven , while dynamic patterns evolve over time or adapt to external influences, introducing variation and progression, as seen in fluctuating sequences. This highlights how patterns can either preserve order through immutability or generate through temporal change. The capacity for has been a of , conferring survival advantages by enabling early hominids to detect recurring cues in their environment, such as interpreting animal tracks to track prey or monitoring seasonal cycles to predict resource availability like migrations or growth. This innate ability, rooted in association learning, favored those individuals who could discern meaningful regularities amid noise, thereby enhancing , social cooperation, and threat avoidance in ancestral settings.

Types and Classification

Patterns can be classified based on their structural properties, particularly regarding repetition and order. Periodic patterns are those that repeat at fixed intervals, exhibiting where the entire structure can be shifted by a constant vector to overlay itself exactly. This is fundamental in and , where periodic structures form the basis for regular lattices and tessellations. Aperiodic patterns, in contrast, lack such translational repetition but maintain long-range order through quasiperiodic arrangements, meaning they are non-repeating yet exhibit structured correlations without periodic cycles. A seminal example is the , developed by in the 1970s, which uses two rhombus-shaped prototiles to cover the plane exclusively in aperiodic ways, demonstrating five-fold impossible in periodic lattices. Random patterns appear disordered at first glance but are governed by underlying statistical regularities, such as probabilistic distributions in processes, where outcomes follow predictable aggregate behaviors despite individual unpredictability. Patterns can further be distinguished by their organizational . Hierarchical patterns feature nested levels of , where subunits form larger repeating motifs, often exhibiting scale-invariance such that the pattern resembles itself at different magnifications; self-similar patterns, such as fractals, exemplify this type. Non-hierarchical patterns, conversely, lack such embedded scaling and operate on a single structural level without recursive subdivision. Additionally, patterns are categorized by origin: intentional patterns are deliberately designed by agents to achieve specific configurations, as in engineered layouts, while emergent patterns arise unplanned from interactions among components in complex systems, producing novel properties not evident in isolated elements. Recognition of patterns relies on key criteria including repetition (cyclic recurrence of elements), progression (sequential advancement like arithmetic or ), symmetry (balanced mirroring or rotational invariance), and contrast (differentiation through variation in attributes such as color or shape). These criteria enable identification across domains, from to algorithmic detection, by highlighting deviations from uniformity.

Patterns in Nature

Symmetry and Geometric Forms

serves as a fundamental geometric pattern in natural phenomena, manifesting through balanced transformations that preserve the form of organisms and structures. In , refers to arrangements where parts of an mirror or repeat in a consistent manner, promoting structural and functional . This balance arises from developmental processes governed by genetic and environmental cues, allowing organisms to adapt to their surroundings while minimizing energy expenditure in growth and maintenance. The primary types of symmetry observed in nature include , , reflectional, and . involves shifting an object along a direction without altering its appearance, as seen in repeating arrangements on a stem. occurs when an object can be rotated around a central point and appear unchanged, such as in the arms of a . Reflectional symmetry, or mirror , divides an object into identical halves across a line or plane, evident in the . combines a reflection with a translation parallel to the mirror line, appearing in certain striped patterns on animal hides. These types form the basis of geometric patterns that underpin natural designs. In nature, bilateral symmetry predominates in mobile animals, exemplified by the , where the left and right sides mirror each other across a , facilitating coordinated movement and sensory integration. Radial , by contrast, characterizes sessile or slow-moving organisms like flowers and , where body parts radiate equally from a central axis, enabling omnidirectional interaction with the environment. Flowers often exhibit radial to optimize from various angles, while use it for regeneration and defense. These forms highlight how aligns with ecological niches, from predation avoidance to resource acquisition. Mathematically, symmetry operations in nature are formalized through , where a comprises all transformations that leave an object invariant. For instance, rotational and reflectional symmetries in radial forms like correspond to , such as the dihedral group D5D_5, which includes five rotations and five reflections around a pentagonal axis. These groups provide a rigorous framework for analyzing how symmetries influence biological structures, from molecular assemblies to organismal morphology, revealing underlying regularities in evolutionary design. Symmetric patterns confer evolutionary advantages, particularly in efficiency and . Bilateral symmetry enhances forward propulsion and balance in animals, reducing drag and enabling —the concentration of sensory organs at the anterior end—for improved navigation and predation. Radial symmetry benefits sessile organisms by allowing uniform detection of threats or food from , promoting efficient resource use in stationary lifestyles. In , symmetric coloration can exploit predator limitations, as analyzing symmetrical patterns demands greater cognitive effort, thereby delaying detection and increasing survival rates. These traits underscore symmetry's role in optimizing fitness across diverse taxa.

Dynamic and Emergent Patterns

Dynamic and emergent arise from time-dependent processes and , where simple interactions among components lead to complex, evolving structures over time. These patterns often emerge without centralized control, driven by physical forces like , growth mechanisms, and environmental stresses. In natural systems, such patterns manifest in spirals, waves, turbulent flows, foams, and fractures, each governed by underlying principles that produce repeatable yet unpredictable forms. Spirals represent a prominent dynamic pattern observed across scales in , from microscopic to cosmic, frequently approximating logarithmic spirals related to the and , particularly in . In hurricanes, the swirling arms form logarithmic spiral patterns as air masses rotate inward due to the Coriolis effect and pressure gradients. Similarly, spiral galaxies exhibit arms that follow logarithmic spirals, with some approximating the (approximately 1.618). The nautilus shell exemplifies logarithmic spiral growth in biology, where the radius approximately triples with each full turn, optimizing space and . Waves and dunes emerge from erosional and depositional processes driven by wind or water, creating rhythmic, migrating patterns in arid and aquatic landscapes. Sand dunes form when transport and deposit grains, accumulating on the windward side until reaching the angle of repose (typically 30-34 degrees), at which point avalanches occur on the leeward side, propelling the dune forward at rates up to 20 meters per year in high-wind areas like the Great Sand Dunes National Park. Smaller-scale ripple patterns, often 5-20 cm in , develop through saltation—where wind lifts and bounces grains—eroding troughs and depositing crests perpendicular to the flow direction, as seen in floors. In marine environments, underwater dunes or waves similarly arise from tidal currents eroding and reshaping sediments, forming elongated ridges that migrate with seasonal water flows. Chaos theory elucidates the emergence of intricate patterns in systems highly sensitive to initial conditions, where minute variations amplify into vastly different outcomes, particularly in fluid dynamics leading to turbulent flows. This sensitivity, known as the butterfly effect, underlies the unpredictable yet patterned evolution of natural phenomena, as deterministic equations yield aperiodic behavior over time. For instance, river meanders develop through erosive feedbacks where slight bends in the channel intensify flow velocities on , causing lateral migration and cutoff events that exhibit chaotic dynamics in numerical models. Cloud formations similarly display turbulent patterns, with convective updrafts and wind shears producing billowing shapes that evolve chaotically, influenced by tiny perturbations in and . Bubbles and foams self-organize through , which minimizes the energy of liquid-air interfaces in dynamic packing arrangements. In soap films, arises from molecular attractions at the film's boundaries, pulling it into stable configurations where three films meet at 120-degree angles and four edges converge at tetrahedral (109.5-degree) junctions, as dictated by Plateau's laws. This results in polyhedral bubble clusters, such as the Weaire-Phelan structure, where disordered yet efficient packing occurs in froths like , driven by drainage and coalescence under gravity and agitation. The spherical shape of individual bubbles balances against , with radius RR related to excess pressure ΔP=4γR\Delta P = \frac{4\gamma}{R}, where γ\gamma is , enabling transient, emergent lattices in natural foams. Cracks in rocks and propagate as stress-induced patterns, emerging from mechanical instabilities under environmental loads like thermal cycling. In freeze-thaw processes, infiltrating pores or fissures expands by about 9% upon freezing, generating internal pressures that exceed rock tensile strength (typically 1-10 MPa), initiating tensile cracks perpendicular to the stress direction. These s branch and interconnect over repeated cycles, forming hierarchical networks—such as wing-like extensions from initial flaws in or —accelerating deterioration as pore volumes increase and connectivity rises. In sheets, similar cryogenic cracking occurs when seasonal thawing refreezes surface layers, producing polygonal patterns from cumulative shear and normal stresses, with crack widths growing from micrometers to centimeters over dozens of cycles.

Patterns in Art and Design

Tilings and Tessellations

Tilings and tessellations represent a fundamental application of patterns in and , involving the repetitive covering of a plane with geometric shapes that fit together without gaps or overlaps, creating visual harmony through structured repetition. These designs emphasize aesthetic unity, often drawing on mathematical precision to evoke and balance in decorative contexts. In , regular tilings are formed by a single type of repeated across the plane, with only three such configurations possible: the triangular tiling using equilateral triangles (denoted as 3^6, where six triangles meet at each vertex), the (4^4, with four squares per vertex), and the (6^3, with three hexagons per vertex). Semi-regular tilings, also known as Archimedean tilings, extend this by combining two or more types of regular polygons while maintaining identical vertex configurations throughout, resulting in eight distinct types that blend shapes like triangles, squares, and hexagons for varied yet uniform patterns. Historical examples abound in , where geometric tilings adorned mosques and palaces to symbolize infinite divine order, as seen in the 14th-century Nasrid tilework of the in , . These alicatado patterns, crafted from cut and glazed ceramic tiles, feature interlocking stars, polygons, and girih straps in vibrant colors, covering walls in the and other spaces to create mesmerizing, non-repeating expanses. Aperiodic tilings, which cover the plane without repeating periodically, introduced a revolutionary departure from traditional repetition; discovered such a set in 1974 using a thin with angles of 36° and 144° and a thick with angles of 72° and 108°, marked by matching rules to ensure non-periodic arrangements. These rhombic prototiles, often visualized in proportions, produce intricate, self-similar patterns that defy yet fill the plane completely. In modern art, elevated tilings by transforming abstract geometric forms into figurative motifs, achieving harmony through meticulous repetition and strategic use of color to delineate interlocking shapes like birds or reptiles. Inspired by Islamic designs, Escher's works, such as Reptiles (1943), employ semi-regular and hyperbolic tilings where colored regions enhance perceptual depth and unity, blending mathematical rigor with imaginative narrative. These principles underscore how tilings foster visual coherence, often leveraging underlying symmetries to guide the viewer's eye across expansive surfaces.

Architectural and Decorative Applications

In architecture, patterns have long served both aesthetic and structural purposes, with Gothic emerging in the 13th century as a defining feature of medieval European design. This intricate stonework, consisting of curvilinear and geometric divisions in windows, allowed for larger glazed areas while distributing structural loads, as seen in cathedrals like , where became a of Gothic innovation. By the late 19th century, introduced organic motifs inspired by nature, such as flowing vines, flowers, and whiplash curves, integrating them into building facades and interiors to create fluid, asymmetrical compositions that blurred the lines between structure and decoration. Architects like exemplified this in works such as the in (1893), where iron supports and panels echoed natural forms for a holistic aesthetic. Decorative arts have similarly embraced patterns for visual enhancement, with William Morris's 19th-century wallpapers revolutionizing through hand-blocked, nature-based motifs like trellises and florals, produced by his firm to promote craftsmanship amid industrialization. These designs, such as the 1862 Trellis pattern inspired by his Red House garden, emphasized repetitive yet organic elements for domestic . In contemporary applications, fractal-inspired facades draw from mathematical to create complex, irregular surfaces that enhance visual depth and environmental integration, as analyzed in modern office buildings where higher s correlate with greater aesthetic appeal. For instance, structures like the Phoenix Office in exhibit -like patterns in their neo-classical elements, achieving a of approximately 1.70 for balanced complexity. Beyond ornamentation, patterns fulfill functional roles in , such as load-bearing that combine with decorative narrative, as in the (5th century BCE), where the continuous Ionic atop the walls bore weight while depicting mythological processions. In modern contexts, perforated patterns in acoustic panels optimize sound absorption by allowing sound waves to enter and dissipate within backing materials, reducing in spaces like auditoriums; these designs often feature geometric holes arranged for both efficacy and visual interest. Such panels, as used in contemporary interiors, can achieve coefficients up to 0.85 depending on perforation density. Cultural variations highlight patterns' diverse roles in decorative applications, contrasting the bold, symbolic weaves of African kente cloth with the subtle, seasonal motifs of Japanese . Kente, originating among the of in the 17th century, uses vibrant geometric strips woven on narrow looms to encode proverbs and social values—such as the "Nkyemfre" pattern representing resilience—primarily for elite garments signifying status and heritage. In contrast, Japanese patterns, evolving from Heian-period (794–1185) robes, incorporate motifs like cranes for longevity or plum blossoms for perseverance, dyed via yuzen techniques to reflect seasonal and , with colors like denoting . These traditions underscore patterns' capacity to convey cultural narratives through decoration.

Patterns in Language and Cognition

Linguistic Structures

Linguistic structures exhibit patterns as systematic recurrences in the organization of , words, and across languages, enabling efficient communication and aesthetic expression. These patterns operate at multiple levels, from the phonetic building blocks of speech to the grammatical frameworks that govern meaning, reflecting both universal tendencies and language-specific conventions. In , patterns involve the arrangement of ; in and morphology, they dictate how elements combine to form coherent units. Such structures evolve over time, adapting to social and cognitive pressures while maintaining core regularities. Phonological patterns manifest as repetitive sound arrangements that enhance memorability and , particularly in poetic and literary forms. , the repetition of initial consonant sounds in nearby words, creates auditory cohesion, as seen in poetry where it structured verses by linking stressed s. schemes, involving matching vowel and consonant sounds at word ends, further exemplify these patterns, organizing lines into predictable auditory units. A prominent example is , a rhythmic pattern of five iambs (unstressed-stressed pairs) per line, which Shakespeare employed extensively in his sonnets and plays to mimic natural speech cadences while imposing metrical discipline. Syntactic patterns govern the linear ordering of constituents in sentences, varying systematically across languages to convey relationships like agency and action. English follows a subject-verb-object (SVO) order, where the subject precedes the verb and object, as in "The cat chased the mouse," facilitating quick identification of the actor. In contrast, Japanese adheres to a subject-object-verb (SOV) structure, placing the verb last, as in "Neko ga nezumi o otta" (cat [subject] mouse [object] chased), which allows for postpositional markers to clarify roles before the predicate resolves the clause. These word order patterns influence processing efficiency and are shaped by informational predictability, with SVO orders often emerging in languages prioritizing early disclosure of agents. Morphological patterns handle through affixation or non-concatenative processes, altering to express grammatical categories like tense or number. In such as Latin and , inflectional endings attach to stems to indicate case, number, and gender; for instance, the Latin noun "" (house) inflects as "domum" in the accusative singular via the -um ending, signaling the direct object role. like employ a root-and-pattern system, where triliteral consonantal (e.g., for writing-related concepts) interweave with vocalic or affixal patterns to derive forms; the root yields "kataba" (he wrote) in the perfective active pattern, systematically encoding verbal aspects and derivations from a shared semantic core. The evolution of linguistic patterns is evident in contact situations, where simplified pidgins develop into complex creoles with full grammars over generations. Pidgins, arising in trade or colonial contexts, initially feature minimal morphology and flexible , relying on lexical borrowing for basic communication. As native speakers acquire these as first languages, creoles emerge with expanded inflectional systems and syntactic rules; for example, Hawaiian Creole English evolved from a pidgin by incorporating tense markers like "bin" for past actions, mirroring broader processes that reconstruct innate grammatical capacities. This generational shift highlights how patterns stabilize through child , transforming rudimentary structures into robust systems.

Perceptual and Cognitive Processing

Human perception of patterns is fundamentally shaped by Gestalt principles, which describe how the brain organizes sensory input into coherent wholes. These principles include proximity, where elements close together are grouped as a unit; similarity, where elements sharing attributes like color or shape are perceived as related; and closure, where incomplete forms are mentally completed to form a whole. Originating from early 20th-century , these laws explain pattern grouping in visual scenes, enabling efficient interpretation of complex environments. A classic illustration of closure is the Kanizsa triangle illusion, where three incomplete Pac-Man-like shapes induce the perception of a bright white occluding disks, despite no explicit contours defining it. First described in , this subjective contour phenomenon demonstrates how the fills in gaps to create illusory boundaries and forms, highlighting the brain's predisposition to impose structure on ambiguous stimuli. In contrast, represents the tendency to perceive meaningful patterns in random or unrelated data, often leading to erroneous connections. Coined by Klaus Conrad in 1958 to describe a feature of , apophenia manifests in everyday cognition as , the specific illusion of seeing faces or other familiar patterns in inanimate objects, such as clouds or toast. Neuroimaging studies reveal that pareidolia activates face-selective regions like the , similar to real faces, suggesting an overactive pattern-detection mechanism evolved for social survival but prone to false positives. The neural basis of pattern recognition primarily involves the , where primary visual area V1 detects basic edges and orientations through oriented receptive fields, as established in foundational studies on cortical organization. Higher areas like V2 and V4 process repetitions and textures, integrating local features into global patterns via hierarchical feedforward computations. This ventral stream pathway supports rapid despite variations in viewpoint or lighting. Recent draws parallels with AI-inspired cognitive models, where convolutional neural networks mimic cortical layers to learn hierarchical pattern representations, informing bidirectional research on human-like in machines. Cross-cultural variations influence pattern interpretation, particularly in aesthetic contexts. Western observers often exhibit a stronger preference for symmetrical patterns in , associating them with order and harmony, as seen in classical European compositions. In contrast, Eastern traditions, such as Chinese landscape painting, favor asymmetrical balance to evoke dynamism and holistic flow, reflecting cultural emphases on over isolated forms. Empirical studies confirm these differences, with East Asian participants rating asymmetrical artworks higher in beauty due to perceptual styles attuned to relational contexts rather than bilateral .

Patterns in Science and Mathematics

Mathematical Foundations

In mathematics, patterns often emerge from structured sequences and series, which provide foundational tools for modeling repetitive or progressive structures. An is a sequence where each term after the first is obtained by adding a constant difference dd to the preceding term, given by the formula an=a+(n−1)da_n = a + (n-1)d, where aa is the first term and nn is the term number. This generates linear patterns, such as evenly spaced points on a number line, essential for understanding uniform growth or distribution in discrete systems. In contrast, a geometric progression involves multiplying each term by a constant ratio rr, yielding an=a⋅rn−1a_n = a \cdot r^{n-1}, which produces exponential patterns observed in scaling phenomena like population growth models or fractal approximations. These progressions form the basis for analyzing periodic and self-similar patterns in one-dimensional settings, with their sums—arithmetic series and geometric series—quantifying cumulative effects in pattern generation. Graph theory extends pattern analysis to networked structures, where patterns manifest as connectivity motifs in vertices and edges. Regular lattices represent highly ordered patterns, such as grid graphs where each vertex connects to a fixed number of neighbors in a Euclidean embedding, forming tessellations like square or hexagonal arrays that exhibit translational and rotational symmetries. These contrast with random graphs, exemplified by the G(n,p)G(n, p), where edges exist independently with probability pp, leading to emergent patterns like small-world properties or giant components as nn grows large. In this model, the threshold for connectivity occurs around p=ln⁡n/np = \ln n / n, highlighting phase transitions in network patterns from disconnected clusters to unified structures. Such distinctions underpin the study of structural patterns in complex systems, distinguishing predictable lattices from probabilistic random configurations. Topology provides abstract tools for preserving pattern continuity under deformations, focusing on properties invariant to stretching or without tearing. A is a continuous between topological spaces with a continuous inverse, ensuring that patterns like connectedness or holes remain intact. This preserves the continuity of patterns by maintaining open sets' images as open, allowing equivalence classes of spaces—such as a and an —to share the same topological pattern despite metric differences. In pattern description, homeomorphisms classify shapes by their intrinsic continuity, ignoring rigid transformations while capturing essential structural motifs. For two-dimensional periodic patterns, wallpaper groups classify the possible symmetry operations, comprising 17 distinct groups generated by translations, , reflections, and glide reflections. These groups, first enumerated by Evgraf Fedorov in 1891, fully describe the symmetries of repeating tilings on the plane, with notations like p1 (pure translations) to p6mm (full hexagonal symmetry). Each group dictates allowable pattern invariances, ensuring no more than these 17 exist due to crystallographic restrictions on rotation orders (2, 3, 4, or 6). This classification enables precise generation and recognition of periodic patterns in mathematical design.

Physical and Natural Sciences Applications

In the physical and natural sciences, patterns emerge from underlying mathematical principles applied to real-world phenomena, providing insights into , , and irregularity across disciplines such as physics, chemistry, and . Reaction-diffusion systems, pioneered by , exemplify how chemical reactions and can generate spatial patterns observed in natural structures like animal coats and markings. These systems are governed by coupled partial differential equations describing the concentrations of two interacting substances, typically an activator uu and an inhibitor vv: ∂u∂t=Du∇2u+f(u,v),\frac{\partial u}{\partial t} = D_u \nabla^2 u + f(u, v), ∂v∂t=Dv∇2v+g(u,v),\frac{\partial v}{\partial t} = D_v \nabla^2 v + g(u, v), where DuD_u and DvD_v are diffusion coefficients with Dv>DuD_v > D_u, and ff and gg represent reaction kinetics that promote instability leading to Turing patterns, such as spots or stripes. Turing's seminal 1952 analysis demonstrated that diffusion-driven instabilities could spontaneously form periodic patterns from homogeneous states, influencing fields from morphogenesis to material science. Experimental validations, such as those in chemical reactions like the Belousov-Zhabotinsky system, confirm these equations produce observable striped or spotted domains under controlled conditions. Fractal patterns capture the irregularities prevalent in natural landscapes and biological forms, quantifying complexity through non-integer dimensions. The , defined iteratively by the quadratic recurrence zn+1=zn2+cz_{n+1} = z_n^2 + c where zz and cc are complex numbers starting from z0=0z_0 = 0, generates intricate boundaries that model at multiple scales, as visualized in Benoit Mandelbrot's 1980 work on iterative mappings. This set's structure mirrors natural , such as the convoluted coastlines of Britain—whose length increases with measurement scale—or the branching of trees and blood vessels, where finer resolutions reveal repeated motifs. The , a measure of a set's fractal irregularity defined as the infimum of ss where the ss-dimensional is zero, quantifies this complexity; for instance, the British coastline has a of approximately 1.25, exceeding its topological dimension of 1 to reflect its roughness. Mandelbrot's framework in (1982) established as tools for analyzing such patterns, with applications in and emphasizing scale-invariance over Euclidean ideals. In , patterns of atomic arrangement in minerals and solids are described by Bravais lattices, which classify the 14 unique three-dimensional periodic arrays possible for point groups in space. Named after Auguste Bravais, these lattices—such as cubic, hexagonal, or monoclinic—form the backbone of crystal structures, where atoms occupy lattice points to produce repeating translational symmetries observed in minerals like or . Bravais's 1850 memoir systematically enumerated these lattices by considering primitive cells and centering types, linking macroscopic crystal habits to microscopic periodicity and enabling diffraction analyses that confirm structures. This classification underpins , where lattice patterns dictate properties like conductivity in semiconductors. Quantum patterns manifest in wave-like behaviors of particles, prominently through interference fringes in double-slit experiments, revealing the probabilistic nature of . In Thomas Young's demonstration with , a coherent source passing through two slits produces alternating bright and dark fringes on a screen due to constructive and destructive interference, with fringe spacing Δy=λLd\Delta y = \frac{\lambda L}{d} where λ\lambda is , LL is slit-to-screen , and dd is slit separation. Extended to quantum entities like electrons in later experiments, such as Claus Jönsson's 1961 setup, the same interference pattern emerges, confirming particles exhibit wave duality and superposition, with fringes forming even from single-particle detections over time. This pattern underscores quantum coherence, influencing technologies from to .

Patterns in Technology and Society

Computing and Algorithmic Patterns

In , patterns refer to reusable solutions to common problems in and algorithmic implementation, enabling efficient and maintainable code structures. patterns, formalized in the seminal work by , Richard Helm, Ralph Johnson, and John Vlissides—known as the ""—categorize these solutions into creational, structural, and behavioral types to promote flexibility and reusability in . For instance, the ensures that a class has only one instance and provides a global point of access to it, commonly used for managing shared resources like database connections. Similarly, the Observer pattern defines a one-to-many dependency between objects, allowing automatic updates to dependents when a subject's state changes, which is foundational for event-handling systems in graphical user interfaces. Algorithmic patterns extend this concept to generative processes, where rules iteratively produce complex structures from simple initial conditions. , introduced by biologist Aristid Lindenmayer in 1968, exemplify this by modeling developmental patterns through parallel string rewriting mechanisms. An L-system begins with an , such as a single symbol representing a stem, and applies production rules—like replacing a branch symbol with two angled branches—to simulate branching growth, yielding plant-like structures suitable for and . These systems have influenced algorithmic generation in fields like , where they efficiently create realistic foliage by iterating rules to approximate natural morphologies without exhaustive manual design. In , patterns manifest as recognition techniques that identify structures in data, particularly through neural networks trained to detect features hierarchically. Convolutional neural networks (CNNs), a key architecture for , employ convolutional layers to apply learnable filters that capture local spatial patterns, such as edges or textures in images, enabling robust detection tasks like object identification. and colleagues demonstrated this in their 1998 work on gradient-based learning for document recognition, where CNNs achieved over 99% accuracy on handwritten digit classification by stacking convolutional layers with subsampling to reduce parameters while preserving invariant features. This approach has become central to applications, from to autonomous driving, by automating the extraction of discriminative patterns from raw pixel data. For analysis, algorithmic patterns like clustering uncover hidden trends by grouping similar data points based on distance metrics. The k-means algorithm, formalized by Stuart Lloyd in 1982 as a least-squares quantization method, partitions datasets into k clusters by iteratively assigning points to the nearest and updating centroids as means, minimizing intra-cluster variance to reveal natural groupings. Widely adopted for its simplicity and scalability, k-means has been applied to vast datasets, such as customer segmentation in , where it identifies behavioral patterns from millions of records with linear relative to data size. While fractal algorithms in physical modeling draw on similar iterative principles, computing patterns emphasize practical implementation for data-driven insights.

Cultural and Social Patterns

Cultural and social patterns encompass recurring structures in human societies that shape interactions, traditions, and institutions. These patterns emerge from historical, environmental, and economic influences, often persisting across generations and adapting to contemporary changes. In indigenous cultures, systems exemplify fundamental , while cultural motifs in reveal universal narrative frameworks. Modern has amplified societal shifts like migration and , and economic cycles demonstrate predictable fluctuations in market behavior. Social patterns, particularly structures, form the backbone of many indigenous societies by defining , residence, and social roles. Matrilineal descent, where lineage and pass through the mother's line, is prevalent among groups such as the and in , empowering women with authority over decisions and resource allocation. In contrast, patrilineal descent traces affiliation through the father's line, as seen in many Arab Bedouin tribes, where it governs alliances and political within lineage groups. These systems influence dynamics and social cohesion; for instance, matrilineal societies often exhibit greater female in compared to patrilineal ones. Anthropological studies highlight how such structures evolved to adapt to ecological pressures, with matrilineal patterns more common in horticultural economies and patrilineal in pastoral ones. Cultural motifs, as recurring themes in folklore and mythology, provide insight into shared human experiences and values. The archetype, outlined by in his 1949 work The Hero with a Thousand Faces, describes a universal narrative pattern involving departure, initiation through trials, and return with newfound wisdom, appearing in myths from to Indigenous American stories. This monomyth underscores themes of transformation and resilience, influencing oral traditions in cultures like the of , where ancestral heroes embody communal ideals. Such motifs foster cultural continuity, serving as educational tools for moral and social guidance across diverse societies. Post-20th century has driven pronounced modern societal patterns, including waves of migration and accelerated . surged from 154 million people in 1990 to 304 million as of 2024, fueled by economic disparities and conflicts, creating transnational communities that blend cultural identities. trends have been equally transformative, with the global urban rising from 30% in 1950 to 57% as of 2023, particularly in the Global South where cities like and absorbed rural migrants seeking opportunities. These patterns reflect interconnected , leading to diverse urban fabrics but also challenges like informal settlements and . Economic patterns, such as boom-bust cycles, illustrate the rhythmic volatility of markets, often triggered by speculative bubbles and policy responses. The 1929 , marking the onset of the , saw the plummet 89% from its peak, resulting from overleveraged investments and banking failures that idled 25% of the U.S. workforce. Similarly, the 2008 , precipitated by the U.S. collapse, led to global GDP contraction of 0.1% and spikes exceeding 10% in affected nations, highlighting parallels in credit expansion and regulatory lapses. Historical analyses reveal these cycles recur approximately every 7-10 years, driven by endogenous factors like asset overvaluation, underscoring the need for macroeconomic safeguards.

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