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Complex

In mathematics, a complex most commonly refers to a complex number, which is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i^2 = -1.^[1] This extends the real numbers to solve equations like x^2 + 1 = 0, forming the set \mathbb{C}. Complex numbers are visualized as points in the complex plane, with the real part on the horizontal axis and imaginary part on the vertical.^[2] The term "complex" has broader applications across fields. In chemistry, it denotes coordination compounds consisting of a central metal atom bonded to ligands. In biology, protein complexes are assemblies of multiple polypeptides performing cellular functions. In psychology, a complex is a cluster of unconscious ideas and emotions influencing behavior. Other uses include geological formations, architectural structures, and concepts in arts and systems theory. See the following sections for detailed explanations.

Mathematics

Complex numbers

A complex number is an element of the number system extending the real numbers by including the imaginary unit i, defined by the property i^2 = -1. It is typically expressed in rectangular form as z = a + bi, where a and b are real numbers, with a called the real part \operatorname{Re}(z) and b the imaginary part \operatorname{Im}(z).^[3] This construction allows solutions to polynomial equations with real coefficients that have no real roots, such as x^2 + 1 = 0, whose solutions are x = \pm i.^[4] Arithmetic operations on complex numbers are defined componentwise for addition and subtraction, and extend the real number operations for multiplication and division. Specifically, for complex numbers z_1 = a + bi and z_2 = c + di:

Addition: z_1 + z_2 = (a + c) + (b + d)i
Subtraction: z_1 - z_2 = (a - c) + (b - d)i
Multiplication: z_1 z_2 = (ac - bd) + (ad + bc)i

Division requires the conjugate \overline{z_2} = c - di, yielding z_1 / z_2 = \frac{z_1 \overline{z_2}}{|z_2|^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}, where |z_2|^2 = c^2 + d^2 is the square of the modulus.^[3] These operations form a field, making the complex numbers algebraically closed under the fundamental theorem of algebra.^[4] In polar form, a complex number z = a + bi is written as z = r (\cos \theta + i \sin \theta), or equivalently z = r e^{i\theta} via Euler's formula, where the modulus r = |z| = \sqrt{a^2 + b^2} gives the distance from the origin in the complex plane, and the argument \theta = \arg(z) = \tan^{-1}(b/a) (adjusted for the correct quadrant) specifies the angle.^[3] This representation simplifies multiplication and exponentiation, as |z_1 z_2| = |z_1| |z_2| and \arg(z_1 z_2) = \arg(z_1) + \arg(z_2).^[3] The development of complex numbers traces to the 16th century with solutions to cubic equations by Cardano and Bombelli, but widespread acceptance came in the 18th and 19th centuries through Leonhard Euler, who introduced the notation i = \sqrt{-1} around 1777 and linked complex numbers to exponentials and trigonometry, and Carl Friedrich Gauss, who by 1799 provided a rigorous geometric interpretation as points in the plane and coined the term "complex numbers" in 1831.^[4] Euler's work formalized their use in solving equations like x^2 + 1 = 0, while Gauss's contributions emphasized their foundational role in algebra.^[4] Complex numbers are essential in electrical engineering for representing phasors in alternating current (AC) circuit analysis, where impedance combines resistance and reactance as a complex quantity, and in physics for modeling oscillatory systems like waves and quantum states.^[5]

Complex analysis

Complex analysis is the branch of mathematical analysis that studies functions of complex variables, focusing on their differentiability, analytic continuation, and integral properties in the complex plane. Unlike real analysis, where differentiability is a local property, complex differentiability imposes stringent conditions that lead to powerful global theorems. This field, developed in the 19th century by pioneers such as Augustin-Louis Cauchy and Bernhard Riemann, reveals deep connections between geometry, analysis, and topology, with applications in physics, engineering, and other sciences. Central to complex analysis are holomorphic functions, which are complex differentiable at every point in an open domain Ω ⊆ ℂ. A function f(z) = u(x, y) + i v(x, y), with z = x + i y, u and v real-valued, is holomorphic in Ω if the partial derivatives exist and satisfy the Cauchy-Riemann equations:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.

These equations ensure that the derivative f'(z) = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h} exists independently of the direction of approach of h in the complex plane.^[6] Holomorphic functions exhibit remarkable rigidity: if they are differentiable once, they are infinitely differentiable, and their real and imaginary parts are harmonic, satisfying Laplace's equation.^[7] A cornerstone result is Cauchy's integral theorem, which states that if f is holomorphic in a simply connected domain containing a closed contour C and its interior, then the line integral \oint_C f(z) \, dz = 0. This theorem implies path independence of integrals for holomorphic functions, contrasting sharply with real line integrals. An extension, Cauchy's integral formula, expresses f at a point inside C as f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - a} \, dz, allowing recovery of function values and derivatives from boundary data.^[8] For functions with singularities, the residue theorem provides a tool for evaluating integrals: if f is holomorphic except at isolated singularities inside a closed contour C, then \oint_C f(z) \, dz = 2\pi i \sum \operatorname{Res}(f, z_k), where the sum is over residues at those points z_k. The residue at a simple pole, for instance, is \operatorname{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z). This theorem facilitates computation of real definite integrals and sums via contour integration.^[9] Holomorphic functions admit series expansions that capture their behavior. In a disk around a point in the domain, f has a Taylor series f(z) = \sum_{n=0}^\infty a_n (z - a)^n, converging uniformly to f. Around an isolated singularity, the Laurent series f(z) = \sum_{n=-\infty}^\infty a_n (z - a)^n generalizes this, with the principal part \sum_{n<0} a_n (z - a)^n classifying the singularity (removable if all negative coefficients vanish, pole if finitely many, essential otherwise). The residue is the coefficient a_{-1}.^[10] Non-constant holomorphic functions are conformal mappings, preserving oriented angles between curves at each point where f'(z) \neq 0. This follows from the fact that the derivative f'(z) acts as a complex multiplication and scaling, which rotates and stretches without shearing. The Riemann mapping theorem asserts that any simply connected domain other than ℂ is conformally equivalent to the unit disk, underscoring the role of conformal maps in solving boundary value problems.^[6]

Chemistry

Coordination compounds

A coordination compound consists of a central metal atom or ion bonded to surrounding molecules or ions called ligands through coordinate covalent bonds, where the shared electrons are donated by the ligands. These bonds form a coordination entity, typically treated as an ion or neutral molecule with the metal at the center and ligands arranged in a specific geometry defined by the coordination number—the number of donor atoms attached to the metal.^[11] The geometry of coordination compounds depends on the coordination number and electronic factors of the metal. Common structures include octahedral for coordination number 6, as seen in hexaamminecobalt(III) ion [ \ce{Co(NH3)6^{3+}} ], where ligands occupy positions at 90° angles; tetrahedral for coordination number 4, such as in [ \ce{CoCl4^{2-}} ], with bond angles of approximately 109.5°; and square planar for coordination number 4 in d⁸ metals, exemplified by cis-[ \ce{PtCl2(NH3)2} ] (cisplatin). These geometries influence the physical and chemical properties of the complexes.^[12] Bonding in coordination compounds is described by theories such as valence bond theory and crystal field theory. Valence bond theory posits that the metal's atomic orbitals hybridize with ligand orbitals to form sigma bonds; for inner orbital octahedral complexes like [ \ce{Co(NH3)6^{3+}} ], d²sp³ hybridization occurs using two 3d orbitals, the 4s orbital, and three 4p orbitals, resulting in low-spin configurations. Crystal field theory treats ligands as point charges that split the five degenerate d orbitals of the metal into a lower-energy t₂g set (d_{xy}, d_{xz}, d_{yz}) and a higher-energy e_g set (d_{x²-y²}, d_{z²}) in octahedral fields, with the splitting energy denoted as Δ_o; this splitting determines magnetic properties and color based on whether Δ_o exceeds the electron pairing energy.^[13]^[14] Nomenclature follows IUPAC recommendations, where ligands are named alphabetically before the central metal, with anionic ligands ending in "-ido" (e.g., chlorido) and neutral ligands retaining modified names (e.g., ammine for NH₃); the metal's oxidation state is indicated in Roman numerals. For instance, [ \ce{Cu(NH3)4^{2+}} ] \ce{SO4^{2-}} is named tetraamminecopper(II) sulfate, with "tetraammine" denoting four NH₃ ligands and the counterion sulfate following the coordination entity. This systematic naming ensures clarity in describing composition and charge.^[11] The foundational understanding of coordination compounds emerged from Alfred Werner's work, for which he received the Nobel Prize in Chemistry in 1913; Werner proposed that atoms exhibit a specific coordination number for bonding, elucidating the structures of inorganic complexes and resolving discrepancies in isomerism.^[15] Stability of coordination compounds is assessed through formation constants, where the overall constant β_n is the product of stepwise constants K_i for successive ligand additions (e.g., β_4 = K_1 × K_2 × K_3 × K_4 for [ \ce{Cu(NH3)4^{2+}} ], indicating higher stability with larger β values). The chelate effect further enhances stability in complexes with multidentate ligands, such as ethylenediamine forming five-membered rings, due to a favorable entropy increase from fewer free ligand molecules in solution compared to equivalent monodentate ligands like ammonia.^[16]^[17]

Molecular complexes

Molecular complexes, a cornerstone of supramolecular chemistry, are defined as assemblies of discrete molecular entities held together primarily by non-covalent interactions, including hydrogen bonding, π-π stacking, hydrophobic effects, and van der Waals forces. These interactions enable the formation of dynamic, reversible structures that mimic biological systems and allow for self-assembly into higher-order architectures. Unlike coordination compounds, which feature strong dative bonds between metals and ligands, molecular complexes emphasize weaker, non-covalent associations that confer adaptability and responsiveness to external stimuli.^[18]^[19] Key types of molecular complexes include inclusion complexes and charge-transfer complexes. Inclusion complexes involve a host molecule enclosing a guest within its cavity through non-covalent encapsulation, as exemplified by cyclodextrins—cyclic oligosaccharides that form stable host-guest pairs with hydrophobic drugs or molecules, enhancing solubility and stability via van der Waals and hydrophobic interactions. Charge-transfer complexes arise from electron donor-acceptor interactions, where partial electron transfer between components produces intense colors observable in the visible spectrum; a classic example is the complex between picrate ions (from picric acid) and electron-rich donors like aromatic amines, where the vivid coloration stems from the excitation of electrons from the donor's highest occupied molecular orbital to the acceptor's lowest unoccupied molecular orbital. Similarly, tetracyanoethylene (TCNE) forms such complexes with aromatic donors, yielding deeply colored solids due to this charge-transfer mechanism.^[20]^[21]^[22] Prominent among molecular complexes are crown ethers and cryptands, which exemplify selective ion binding through preorganized cavities lined with donor atoms. Crown ethers, such as 18-crown-6, feature a cyclic polyether structure with six oxygen atoms that coordinate potassium ions (K⁺) in a near-perfect size match, forming a stable pseudocrown complex stabilized by electrostatic and ion-dipole interactions; this selectivity arises from the cavity diameter aligning optimally with the ion's ionic radius, enabling applications in ion transport and sensing. Cryptands, three-dimensional analogs like [2.2.2]-cryptand, extend this concept by encaging cations in a spherical cage, providing even higher binding affinities due to the "cryptate effect," which enhances kinetic and thermodynamic stability through multiple chelate interactions.^[23]^[24] Supramolecular polymers represent another vital class, consisting of monomeric units linked into linear or networked assemblies via directional non-covalent bonds, such as multiple hydrogen bonds or metal-ligand coordinations, yielding materials with polymer-like mechanical properties but reversible connectivity. These structures often propagate unidirectionally due to the geometry of the interactions, allowing for stimuli-responsive disassembly. In applications, molecular complexes facilitate drug delivery systems, where cyclodextrin inclusion complexes enable controlled release and improved bioavailability of poorly soluble therapeutics, and catalysis, where host-guest assemblies mimic enzyme pockets to accelerate reactions with high specificity. These advancements trace back to foundational work in supramolecular chemistry, for which Jean-Marie Lehn shared the 1987 Nobel Prize in Chemistry with Donald J. Cram and Charles J. Pedersen.^[25]^[26]^[24]

Biology

Protein complexes

Protein complexes are stable assemblies of multiple polypeptide chains, or subunits, that interact non-covalently or covalently to form functional units capable of executing coordinated biological roles within the cell. These multi-subunit structures, often referred to as having quaternary structure, enable emergent properties not achievable by individual proteins, such as enhanced stability, regulation, or catalytic efficiency. A classic example is hemoglobin, a tetrameric protein complex composed of two alpha and two beta subunits that collectively transport oxygen in vertebrates.^[27] The assembly of protein complexes is primarily driven by non-covalent interactions, including hydrophobic forces, hydrogen bonds, and electrostatic interactions, which stabilize the quaternary arrangement, while covalent disulfide bonds contribute in certain cases, such as in extracellular complexes. Hydrophobicity plays a central role at subunit interfaces, promoting burial of non-polar residues to minimize solvent exposure and enhance stability. For instance, the mitochondrial ATP synthase, or F₀F₁ complex, exemplifies large-scale assembly with eight distinct subunits in prokaryotes such as E. coli and up to 18 distinct subunits comprising 29 polypeptide chains in humans, organized into a rotary motor that couples proton translocation to ATP synthesis through these interactions.^[28]^[29] Prominent examples of protein complexes include the ribosome, a ribonucleoprotein assembly of ribosomal RNA and approximately 80 proteins that catalyzes protein translation, and the proteasome, a cylindrical complex with a 20S core particle containing 28 subunits responsible for ubiquitin-dependent protein degradation to maintain cellular homeostasis. These structures highlight the diversity of complex sizes and functions, from macromolecular machines to regulatory hubs.^[30]^[31] Dynamics within protein complexes often involve allosteric regulation, where binding of a ligand at one site induces conformational changes that affect activity at distant sites, optimizing function. In hemoglobin, oxygen binding triggers a transition from the low-affinity tense (T) state to the high-affinity relaxed (R) state, facilitating cooperative oxygen uptake and release through subunit rearrangements. Such mechanisms underscore the role of complexes in responsive biological processes.^[32] Structural determination of protein complexes has advanced through techniques like X-ray crystallography, which provides atomic-resolution insights into ordered assemblies, and cryo-electron microscopy (cryo-EM), which excels at visualizing large, dynamic complexes in near-native states. The 2017 Nobel Prize in Chemistry recognized cryo-EM's development by Jacques Dubochet, Joachim Frank, and Richard Henderson, enabling breakthroughs in resolving structures like the ribosome and ATP synthase at resolutions below 3 Å. These methods have revolutionized understanding of complex architecture and function.^[33]^[34]

Cellular complexes

Cellular complexes represent integrated assemblies of proteins, lipids, and other molecules that function as coordinated machines within cellular compartments, such as plasma membranes, organelle membranes, or the cytoplasm, enabling essential processes like energy transduction and signal propagation. These structures often involve dozens to hundreds of subunits, forming dynamic, multi-component units that integrate inputs from multiple pathways to produce specific cellular outputs. Unlike isolated protein complexes, cellular complexes emphasize their embedded roles in broader cellular architecture, where they interact with membranes or cytoskeletal elements to maintain functionality.^[35] Prominent examples include the respiratory complexes of the electron transport chain in mitochondria and photosystems in chloroplasts. The mitochondrial electron transport chain comprises four main respiratory complexes (I–IV) embedded in the inner mitochondrial membrane: Complex I (NADH dehydrogenase) oxidizes NADH and pumps protons; Complex II (succinate dehydrogenase) feeds electrons from the Krebs cycle; Complex III (cytochrome bc1) conducts the Q cycle for electron transfer to cytochrome c; and Complex IV (cytochrome c oxidase) reduces oxygen to water while pumping additional protons, collectively generating a proton gradient for ATP synthesis.^[36] In photosynthetic eukaryotes, photosystems I and II form analogous light-harvesting and electron-transfer machines in the thylakoid membranes of chloroplasts; photosystem II splits water to release oxygen and electrons, while photosystem I uses light energy to reduce NADP+, driving carbon fixation. These complexes exemplify how cellular assemblies couple energy capture to biochemical gradients across diverse organelles.^[37]^[38] In signaling contexts, cellular complexes facilitate rapid information relay through receptor-ligand interactions and downstream cascades. G-protein-coupled receptors (GPCRs), the largest family of membrane receptors, form transient complexes with extracellular ligands such as hormones or neurotransmitters; upon binding, they undergo conformational changes that activate heterotrimeric G-proteins, catalyzing GTP exchange and dissociating Gα and Gβγ subunits to modulate effectors like adenylyl cyclase, thereby amplifying signals via second messengers such as cAMP. Membrane rafts further enhance compartmentalization by serving as cholesterol- and sphingolipid-enriched microdomains (10–200 nm) that cluster receptors (e.g., TCR or BCR in immune cells) with kinases and adaptors, isolating signaling events to prevent cross-talk and ensure spatiotemporal precision in responses like cytokine production or degranulation.^[39]^[40] These cellular complexes exhibit remarkable evolutionary conservation across eukaryotes, reflecting their ancient origins in bacterial ancestors via endosymbiosis, with core subunits of respiratory complexes and photosystems preserved from prokaryotes to mammals and plants. Disruptions in their assembly or function, such as mutations altering mitochondrial complex I activity or aberrant GPCR signaling, are implicated in diseases including cancer, where they promote metabolic reprogramming, uncontrolled proliferation, and resistance to therapy through elevated reactive oxygen species or dysregulated growth cascades.^[41]^[38]^[42]^[43]

Psychology

Psychological complexes

In psychology, a complex refers to a cluster of unconscious ideas, emotions, memories, and perceptions that form an autonomous pattern within the psyche, often operating independently of the conscious ego and influencing behavior in subtle or disruptive ways. This concept emphasizes emotionally charged constellations that can distort perception and decision-making, rooted in personal experiences or archetypal influences.^[44]^[45] The notion of psychological complexes was developed by Carl Gustav Jung in the early 20th century, particularly through his word association experiments conducted around 1907, which revealed how certain stimuli triggered emotional disturbances indicative of underlying unconscious structures. Unlike Sigmund Freud's emphasis on repression as the primary mechanism for unconscious content, Jung viewed complexes as natural, affect-toned formations that could possess a degree of autonomy, contrasting with the ego's control and potentially leading to neurotic symptoms if unintegrated. A seminal example from Jung's framework is the mother complex, which encompasses unconscious attitudes toward the mother figure that shape relational patterns, such as dependency or rebellion in adult life, often amplifying archetypal maternal influences.^[46]^[44] Other notable examples include Alfred Adler's inferiority complex, introduced in 1907, which describes a pervasive feeling of inadequacy stemming from perceived physical or social deficits, driving compensatory behaviors like overachievement or withdrawal. Freud's Oedipus complex, though predating Jung, relates as a familial dynamic involving unconscious desires toward parents, briefly illustrating early psychoanalytic parallels to complex theory. These examples highlight how complexes cluster around core themes, varying in intensity and impact across individuals. Complexes manifest through indirect channels such as dreams, where they appear as symbolic motifs reflecting unresolved tensions; Freudian slips or errors in speech and action that betray unconscious influences; and projections, wherein internal complex-driven emotions are attributed to external people or situations. Activation often occurs when contemporary triggers resonate with the complex's archetypal core, leading to heightened emotional responses or behavioral patterns that disrupt conscious intentions.^[44]^[47] In modern analytical psychology, complexes are approached through therapeutic integration, where the goal is to bring unconscious material into awareness via techniques like active imagination or dream analysis, fostering assimilation to enhance psychological wholeness and reduce autonomous interference. This process, central to Jungian therapy, views complexes not merely as pathologies but as opportunities for personal growth, with contemporary practitioners emphasizing their role in individuation.^[48]^[49] Modern psychological concepts related to or analogous to Jungian complexes include those in trauma and personality disorders. In the context of trauma, complex post-traumatic stress disorder (CPTSD) manifests as persistent emotional disturbances triggered by adverse events, arising from prolonged or repeated trauma such as chronic abuse or violence, leading to symptoms including emotional dysregulation, negative self-concept, and interpersonal difficulties.^[50] Unlike standard PTSD from single incidents, CPTSD involves a broader syndrome of disturbances in self-organization, often requiring integrated therapeutic approaches to address the entrenched emotional patterns.^[51] Patterns analogous to psychological complexes appear in personality disorders like borderline personality disorder (BPD), characterized by unstable relational patterns, intense fear of abandonment, and alternating idealization-devaluation in relationships, stemming from early invalidating environments.^[52] These patterns form a complex interplay of affective instability and identity diffusion, contributing to chronic relational turmoil and impulsivity.^[53] Cognitively, schema therapy identifies maladaptive schemas as enduring patterns resembling psychological complexes, formed in childhood through unmet emotional needs and activated by situational triggers to perpetuate dysfunctional thoughts and behaviors.^[54] These schemas, such as abandonment or defectiveness, function like emotional clusters that bias perception and coping, with therapy aiming to reframe them for healthier adaptation.^[55] Neuroscientific research on trauma-related responses in PTSD links them to heightened amygdala activation, as evidenced by fMRI studies in PTSD patients showing hyperresponsivity to trauma cues, indicating impaired fear regulation and emotional processing.^[56] This activation correlates with symptom severity, where the amygdala's overengagement disrupts prefrontal control, perpetuating the cycle of distress in trauma-related conditions.^[57] Cultural variations influence the expression of psychological complexes, with collectivist societies often centering on family-oriented dynamics that emphasize group harmony and interdependence, potentially fostering enmeshed relational patterns, while individualist societies highlight personal inferiority feelings tied to achievement and autonomy.^[58] In collectivist contexts, such family complexes may prioritize collective well-being over individual expression, contrasting with the self-focused inferiority motifs prevalent in individualist cultures.^[59]

Geology

Geological formations

In geology, a complex refers to a heterogeneous assemblage of rocks derived from multiple genetic origins, including igneous, sedimentary, and metamorphic types, typically lacking distinct layering and exhibiting intricate structural relationships. These units often represent polyphase histories of deformation, metamorphism, and intrusion, forming coherent but compositionally varied bodies within the crust.^[60] Common types include gneiss complexes, which consist of banded, high-grade metamorphic rocks dominated by alternating layers of quartzofeldspathic and mafic minerals, resulting from intense regional metamorphism. For instance, the Ancient Gneiss Complex in Eswatini comprises Archean gray gneisses southeast of the Barberton greenstone belt, showcasing early crustal evolution through tonalitic magmatism and subsequent deformation. Another key type is ophiolite complexes, which are uplifted sections of ancient oceanic crust and upper mantle, featuring a characteristic sequence from peridotites and gabbros to sheeted dikes and pillow basalts, often obducted onto continental margins during tectonic convergence.^[61]^[62] Geological complexes primarily form through orogenic processes, where continental collisions and subduction drive the mixing, folding, and metamorphism of diverse rock suites over millions of years. In the Alpine-Himalayan orogenic belt, such complexes emerge from the ongoing convergence of the African, Arabian, and Eurasian plates, producing thrust sheets and metamorphic piles that record Cenozoic to Mesozoic tectonic events.^[63] The investigation of these complexes relies on field mapping to identify lithological boundaries and structural fabrics, complemented by geochronology to establish timelines of igneous, metamorphic, and deformational episodes. Techniques like U-Pb dating of accessory minerals, such as zircon in gneisses or calcite in veins, provide precise ages for events within orogenic settings, revealing sequences of crustal growth and recycling. A notable example is the Lewisian Complex in northwestern Scotland, an Archean-Proterozoic cratonic fragment of gneissic and schistose rocks up to 3 billion years old, serving as the basement to younger sedimentary covers and illustrating multi-phase Laxfordian deformation.^[64]^[65]

Tectonic complexes

A tectonic complex refers to a large-scale assemblage of disparate terranes—distinct crustal fragments with unique geologic histories—joined together by major fault systems, often in the context of plate boundary interactions. These structures, such as accretionary complexes, form at convergent margins where oceanic sediments, volcanic rocks, and crustal slices are scraped off subducting plates and welded onto the overriding plate through tectonic suturing. This process creates heterogeneous rock packages bounded by thrust faults and shear zones, reflecting prolonged episodes of deformation over millions of years.^[66]^[67] The formation of tectonic complexes is driven by key tectonic processes, including subduction and continental collision. During subduction, the downgoing oceanic plate experiences off-scraping of its sedimentary cover and upper crustal layers, which are accreted as imbricated thrust sheets, building wedge-shaped prisms. In collision settings, convergence between continental blocks compresses and folds these accreted materials into fold-thrust belts, where shortening is accommodated by low-angle thrusts and ductile shearing, often under high-pressure conditions that produce metamorphic assemblages like blueschists. These belts represent the peripheral architecture of orogenic systems, with deformation propagating outward from the collision zone.^[68]^[69] A prominent example is the Franciscan Complex of coastal California, a Mesozoic accretionary complex formed along the ancient subduction zone beneath the North American plate from the Late Jurassic to the Eocene. It comprises a mélange of metamorphosed oceanic basalts, radiolarian cherts, and graywackes, with blueschist-facies rocks indicating subduction depths of 20–40 km before exhumation. The complex's terranes were sutured by faults during ongoing convergence, preserving evidence of episodic accretion over 150 million years.^[70]^[71] Tectonic complexes play a central role in the Wilson cycle, the recurring process of supercontinent assembly and disassembly driven by plate tectonics. During the closing phase of an ocean basin, subduction and collision incorporate terranes into continental margins, forming complexes that contribute to supercontinent coalescence, as seen in the assembly of Pangaea. Rifting phases then fragment these landmasses, initiating new subduction zones and perpetuating the cycle over 400–500 million years.^[72]^[73] Active tectonic complexes pose significant seismic hazards due to ongoing fault activity at plate boundaries. For instance, the San Andreas fault system in California, part of a transform boundary offsetting subduction-related complexes like the Franciscan, accommodates 35–40 mm/year of lateral slip between the Pacific and North American plates, generating frequent earthquakes, including the 1906 magnitude 7.9 event that caused widespread destruction. Such zones remain prone to large ruptures, with potential magnitudes up to 8.0, underscoring the need for hazard mitigation in populated areas.^[74]^[75]

Arts, entertainment, and media

Fictional works

In science fiction literature, the term "complex" frequently denotes secretive, labyrinthine facilities that isolate characters and amplify psychological tension. A seminal example is Michael Crichton's The Andromeda Strain (1969), where the Wildfire complex—a highly secure, underground laboratory in Nevada—serves as the primary setting for a team of scientists racing to contain an extraterrestrial microbe. The novel explores themes of psychological isolation as the researchers, cut off from the outside world under decontamination protocols, grapple with the strain's deadly effects and the moral complexities of their containment decisions.^[76] In film, such complexes often represent corporate or governmental hubs of villainy in dystopian narratives. The 2016 film Resident Evil: The Final Chapter, directed by Paul W. S. Anderson, centers on the Hive, the sprawling underground headquarters of the Umbrella Corporation beneath Raccoon City. This fortified complex houses bioweapon experiments that unleash zombies and mutants, positioning it as the antagonist's base where protagonist Alice infiltrates to deploy an antivirus antidote amid betrayals and high-stakes combat. The Hive's multi-level design, filled with traps and undead hordes, underscores the isolation of survivors in a post-apocalyptic world.^[77] Video games similarly employ complexes as expansive, explorable environments rife with conspiracy. In the Deus Ex series, beginning with the 2000 original developed by Ion Storm, Area 51 emerges as a pivotal secret military installation in Nevada, commandeered by the shadowy Majestic 12 organization for augmentations and global control schemes. Players navigate its bunkers, laboratories, and ventilation systems, uncovering plots involving nanotechnology and political intrigue, which heighten the sense of personal and societal isolation in a cyberpunk future.^[78] Across these works, "complex" symbolizes dystopian labyrinths concealing illicit experiments, evoking hidden societal depths and conspiratorial threats in science fiction. This motif has culturally impacted the genre by reinforcing narratives of institutional secrecy and human vulnerability, influencing subsequent stories in literature, film, and gaming that probe the perils of technological overreach.^[79]

Music and publications

Complex magazine, a prominent publication in youth culture, was founded in 2002 by fashion designer Marc Ecko alongside creative director Hugh McFall, initially as a bimonthly print magazine targeting young men with content on hip-hop, streetwear, sneakers, music, and pop culture trends.^[80] The magazine quickly established itself as a key voice in urban lifestyle, blending interviews with artists and influencers, style features, and cultural commentary to capture the intersection of hip-hop and emerging fashion scenes.^[81] In the 2010s, Complex underwent a significant evolution toward digital media, ceasing its print edition after the December 2016/January 2017 issue to prioritize video content, online articles, and social platforms amid shifting consumer habits.^[81] This transition allowed the brand to expand its reach. BuzzFeed acquired Complex Networks in 2021 for approximately $300 million, integrating it into a broader digital entertainment portfolio before selling it to the live-shopping platform NTWRK in 2024 for $108.6 million, with investment from Universal Music Group.^[82]^[83] In late 2024, Complex relaunched its quarterly print magazine, marking a return to physical editions while maintaining its digital dominance; as of November 2025, this has continued with Issue No. 2 (Spring 2025, featuring Travis Scott) and Issue No. 3 (Fall 2025, with covers including Clipse and Central Cee).^[84]^[85]^[86] The publication has played a pivotal role in influencing urban fashion and music trends, particularly within hip-hop, by spotlighting streetwear brands, artist collaborations, and cultural shifts that popularized items like oversized tees, throwback jerseys, and luxury sneakers in the 2000s and beyond.^[87] Its coverage helped elevate hip-hop's symbiotic relationship with high fashion, bridging underground scenes with mainstream designers and fostering trends adopted by artists such as Kanye West and Pharrell Williams.^[88] Complex's editorial focus on authenticity and innovation has made it a tastemaker, with features often dictating what resonates in sneaker culture and rap aesthetics.^[89] Complementing its print and digital offerings, Complex Networks operates a podcast network featuring in-depth interviews with musicians, actors, and cultural figures, including series like 360 With Speedy, which explores personal stories from artists such as rappers and producers, and That's Deep, a conversational show hosted by comedian Zack Fox delving into existential topics with stars.^[90]^[91] Other programs, such as Idea Generation and Complex Subject, provide long-form discussions on creativity and artist legacies, like the Spotify-exclusive series on the late Pop Smoke, enhancing Complex's role in music discourse.^[92]^[93]

Other uses

Architectural complexes

An architectural complex is defined as a unified ensemble of interconnected buildings, facilities, and infrastructure designed and constructed to fulfill a cohesive purpose, such as residential living, industrial operations, or administrative functions, typically integrated on a shared site to enhance efficiency and functionality.^[94]^[95] Common types include apartment complexes, which consist of multiple residential buildings or units clustered together under unified management to house numerous households in urban or suburban settings.^[96] Military complexes represent another category, exemplified by the Pentagon in Arlington, Virginia, completed in 1943 as the headquarters of the U.S. Department of Defense, featuring a vast, pentagonal structure that centralizes command, offices, and support facilities for national security operations.^[97]^[98] Design principles for architectural complexes emphasize functional zoning to allocate spaces by purpose—separating residential, recreational, and service areas for optimal use—alongside efficient circulation systems that facilitate seamless pedestrian and vehicular movement within the site.^[99] Sustainability has become integral since the early 2000s, with green complexes incorporating energy-efficient materials, renewable energy sources, and biophilic elements like vertical greenery to reduce environmental impact, as demonstrated by Milan's Bosco Verticale towers (completed 2014), which integrate over 900 trees and 20,000 plants across two residential high-rises to improve air quality and urban biodiversity.^[100]^[101]^[102] Historically, the Palace of Versailles in France stands as a seminal example from the 17th century, where King Louis XIV expanded a royal hunting lodge into an expansive complex of palace buildings, formal gardens, and outlying structures between 1661 and 1710, embodying absolute monarchy through symmetrical planning and integrated spatial hierarchy.^[103]^[104] In modern contexts, architectural complexes, particularly housing developments, have raised concerns over urban sprawl, as low-density expansions on city outskirts consume land and increase infrastructure demands, while also fueling gentrification in revitalized areas, where influxes of higher-income residents drive up rents and displace longstanding communities.^[105]^[106]^[107]

Systems theory

In systems theory, a complex system is defined as a collection of many interacting components or agents whose collective behavior gives rise to emergent properties that cannot be simply deduced from the individual parts. These systems are characterized by their adaptive nature, where interactions lead to patterns and outcomes that evolve over time, such as in ant colonies where simple pheromone-based rules enable efficient foraging and nest-building without centralized direction, or in economies where agent decisions result in market fluctuations and growth cycles.^[108]^[109] Central to complex systems are key properties including nonlinearity, where minor perturbations can trigger outsized responses, such as the butterfly effect in weather patterns; self-organization, allowing spontaneous order to emerge from local interactions without external imposition; and feedback loops, which can either reinforce changes (positive feedback) or maintain stability (negative feedback). For instance, Earth's climate system exemplifies these traits through chaotic attractors—geometric structures in phase space that bound unpredictable yet constrained trajectories, complicating long-term forecasting despite deterministic underlying equations. Similarly, neural networks in the brain form complex systems where billions of interconnected neurons generate emergent cognitive functions like learning and decision-making via dynamic, self-organizing patterns.^[108] The foundational theories of complexity science were significantly advanced by the Santa Fe Institute, established in 1984 to foster interdisciplinary exploration of emergent phenomena across disciplines like physics, biology, and social sciences. Practical applications of these concepts include modeling pandemics as complex adaptive systems, as seen in analyses of COVID-19 transmission, where nonlinear spread dynamics, adaptive behavioral responses, and feedback from interventions like lockdowns illustrate the challenges of predicting and controlling such systems.^[108]^[110]

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