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Cycle

A cycle is a recurring sequence of events, states, or phenomena that returns to its initial condition after completion of the full round, often exhibiting periodicity or repetition in natural, social, or abstract systems.^[1]^[2] Originating etymologically from the Greek kyklos meaning "circle" or "wheel," the concept underpins diverse domains including thermodynamics, where it describes closed processes like the Carnot cycle enabling heat-to-work conversion; biology, as in the Calvin cycle of photosynthesis fixing carbon dioxide into organic compounds; and economics, via business cycles of expansion and contraction driven by investment fluctuations and demand shifts.^[3] Notable characteristics include inherent stability in balanced systems yet vulnerability to perturbations, as empirical data from long-term observations reveal cycles can amplify through feedback loops or dampen via regulatory mechanisms, with real-world examples like solar cycles influencing Earth's climate via sunspot activity over approximately 11-year intervals. Controversies arise in interpretive applications, such as debates over the predictability of economic cycles, where causal analyses emphasize supply-side factors over Keynesian demand stimuli, challenging institutionalized models prone to overreliance on aggregated data masking micro-level incentives.

Etymology and Conceptual Foundations

Etymology

The English word cycle derives from Late Latin cyclus, which in turn comes from Ancient Greek κύκλος (kúklos), signifying "circle" or "wheel".^[3]^[1] This Proto-Indo-European root, related to terms for turning or wheeling, underscores the concept of circular motion or repetition inherent in the term.^[3] It entered Middle English around the late 14th century as cicle, initially denoting a fixed interval of years, such as in astronomical or calendrical contexts like the "cycle of the sun" or lunar periods.^[3] By the 17th century, the sense expanded to a recurring series of events or operations, reflecting the metaphorical extension from physical circularity to temporal or procedural repetition.^[3] In technical applications, cycle appeared in mechanics by 1841 to describe a complete series of operations in an engine, and by the late 19th century, it abbreviated references to bicycle (from 1881) and similar wheeled vehicles, aligning with its vehicular connotation from the Greek root.^[3] This evolution highlights a consistent thread from geometric and astronomical origins to broader periodic phenomena, without conflation with unrelated Latin-derived terms like circus (ring).^[3]

Philosophical and Logical Definitions

In philosophy, a cycle is conceptualized as a recurrent pattern or sequence of events that returns to an originating state, often observed in natural phenomena and extended to metaphysical or historical interpretations. Ancient philosophical traditions, including Indo-Hellenic and Chinese thought, articulated cyclical views of time and cosmic processes, wherein qualitative changes occur periodically but the underlying structure repeats, as evidenced in Pythagorean notions of metempsychosis and Stoic ekpyrosis, a periodic conflagration and renewal of the cosmos.^[4] This contrasts with linear conceptions of time prevalent in Judeo-Christian frameworks, where cycles emphasize repetition influenced by observable rhythms like seasonal changes, rather than irreversible progression.^[5] A prominent modern philosophical articulation of cyclicality is Friedrich Nietzsche's doctrine of eternal recurrence, introduced in works like The Gay Science (1882) and Thus Spoke Zarathustra (1883–1885), which posits that the universe and all events within it recur eternally and identically due to finite matter and infinite time, serving as a thought experiment to affirm life amid nihilism.^[6] Nietzsche derived this from physical principles of energy conservation and recurrence, though he framed it ethically rather than empirically verifiable, challenging individuals to live as if every action repeats infinitely.^[7] In logic, a cycle refers to circular reasoning (circulus in probando), a fallacy where the premises of an argument presuppose the truth of the conclusion, creating a self-reinforcing loop devoid of independent evidence.^[8] This defect arises because the argument's components are propositionally equivalent or mutually dependent, failing to provide justification beyond tautology, as analyzed in formal treatments where equivalence between premises and conclusion renders the inference vacuously true but uninformative.^[9] Logicians distinguish this from benign circularity in axiomatic systems, emphasizing that pragmatic cycles undermine justificatory aims by begging the question.^[10]

Natural Sciences

Biological Cycles

Biological cycles refer to recurring, self-regulating processes in living organisms and ecosystems that maintain homeostasis, enable reproduction, and facilitate energy and nutrient transfer. These cycles operate at multiple scales, from intracellular metabolic pathways to organismal rhythms and global biogeochemical loops involving biotic components. Empirical observations, such as phase durations measured in controlled experiments, demonstrate their precision and necessity for survival; disruptions, like mutations in cell cycle regulators, lead to uncontrolled proliferation as seen in cancers.^[11]^[12] At the cellular level, the cell cycle governs growth and division in eukaryotic cells, consisting of interphase (G1, S, and G2 phases) followed by mitosis and cytokinesis. During G1, the cell grows and synthesizes proteins; S phase replicates DNA, doubling the genome from 2C to 4C content; G2 prepares for division with further growth and checkpoint verification; and the M phase separates chromosomes into two daughter cells. This cycle typically lasts 24 hours in mammalian cells under optimal conditions, regulated by cyclins and cyclin-dependent kinases (CDKs) that ensure fidelity, with checkpoints halting progression if DNA damage is detected.^[13]^[14]^[11] Metabolic cycles, such as the Krebs cycle (also known as the tricarboxylic acid or TCA cycle), form the core of aerobic respiration in mitochondria, oxidizing acetyl-CoA derived from carbohydrates, fats, and proteins to produce ATP, NADH, and FADH2. The cycle comprises eight enzymatic steps, starting with citrate formation and regenerating oxaloacetate, yielding two CO2 molecules per turn and driving the electron transport chain for up to 30-32 ATP per glucose molecule. Discovered by Hans Krebs in 1937 through pigeon muscle extracts, it integrates with glycolysis and oxidative phosphorylation, with deficiencies causing metabolic disorders like those in congenital lactic acidosis.^[15]^[16] Organismal rhythms include the circadian cycle, an endogenous ~24-hour oscillation synchronized by light-dark cues via the suprachiasmatic nucleus in mammals, influencing sleep-wake patterns, hormone release (e.g., melatonin peaking at night), and gene expression in ~43% of the mammalian genome. Core clock genes like PER and CLOCK form feedback loops repressing their own transcription, with desynchronization from shift work or jet lag linked to impaired cognition and increased disease risk, as evidenced by longitudinal studies showing 20-50% higher metabolic disorder incidence.^[17]^[18]^[19] Reproductive cycles in mammals vary by species: most non-primate females exhibit estrous cycles (e.g., ~21 days in cattle, with proestrus follicle development, estrus ovulation, and diestrus luteal phase), while humans and some primates have menstrual cycles (~28 days), featuring endometrial shedding if unfertilized. These are driven by hypothalamic-pituitary-gonadal axis hormones—FSH/LH surges inducing ovulation—and ensure seasonal or continuous breeding aligned with environmental cues, with failure rates in synchronization contributing to infertility in 10-15% of couples per WHO data.^[20]^[21] Biogeochemical cycles sustain biotic communities through nutrient recycling; the carbon cycle shuttles ~7.8 × 10^17 grams annually via photosynthesis (fixing 120 Gt C/year by plants) and respiration/decomposition, with oceanic uptake buffering atmospheric CO2 at ~50 Gt/year. The nitrogen cycle converts N2 (78% atmosphere) to bioavailable forms via bacterial fixation (e.g., Rhizobium in legumes, 100-200 kg N/ha/year), nitrification, and denitrification, preventing deficiencies that limit primary productivity, as quantified in ecosystem models showing 90% of fixed N recycled internally. These biotic-mediated loops, measured through isotope tracing, underpin food webs but are altered by anthropogenic inputs like fertilizers, elevating N2O emissions by 20-30% since pre-industrial times.^[22]^[23]^[21]

Physical and Environmental Cycles

Physical and environmental cycles refer to recurring processes that drive the transfer of energy and matter within Earth's geosphere, hydrosphere, atmosphere, and biosphere, maintaining system equilibrium through physical mechanisms like solar radiation, gravity, and tectonic activity.^[24] These cycles operate on timescales from daily to millennial, influencing weather patterns, nutrient distribution, and long-term climate stability without reliance on biological intermediaries in their core physical phases.^[25] The hydrological cycle circulates water globally via evaporation from surface waters—primarily oceans, which hold 96.5% of Earth's water—transpiration, condensation, precipitation, infiltration, and runoff, powered by solar energy that evaporates approximately 505,000 km³ annually.^[26] This process redistributes heat and moisture, with atmospheric water vapor comprising less than 1% of total water but enabling cloud formation and precipitation that returns 90% of evaporated ocean water as rain over oceans.^[27] Disruptions, such as altered evaporation rates from temperature changes, can amplify regional droughts or floods through feedback loops in atmospheric circulation.^[28] Biogeochemical cycles integrate physical transport with chemical transformations, exemplified by the carbon cycle, where carbon moves between reservoirs via weathering, volcanic outgassing, ocean uptake, and sedimentation, with fast exchanges through atmospheric CO₂ diffusion and slow geological burial over millions of years.^[29] Key physical processes include ocean-atmosphere exchange, where surface waters absorb CO₂ via solubility pumps, and rock weathering that sequesters carbon as bicarbonate ions.^[30] The nitrogen cycle physically transports N₂ (78% of atmosphere) through lightning fixation and runoff, while denitrification releases it back to the air; phosphorus cycles via physical erosion of apatite-rich rocks into soils and sediments, limiting primary productivity in marine environments due to low solubility.^[31] The rock cycle physically recycles crustal materials among igneous, sedimentary, and metamorphic forms through melting under mantle heat, cooling and crystallization, weathering and erosion by wind and water, deposition, compaction, and metamorphism from pressure and temperature in subduction zones.^[32] These transformations, spanning millions of years, are propelled by plate tectonics, which exposes rocks to surface erosion and recycles them via subduction, preventing indefinite accumulation of any single rock type.^[33] Astronomical physical cycles arise from Earth's orbital dynamics: daily rotation produces diurnal temperature and light variations, while 23.5° axial tilt generates seasonal insolation contrasts, with higher latitudes experiencing extremes.^[34] Milankovitch cycles further modulate climate through eccentricity (orbit shape, ~100,000-year period varying solar input by up to 25%), obliquity (tilt fluctuations, 41,000 years), and precession (wobble, 26,000 years), correlating with glacial cycles via altered hemispheric summer radiation.^[35]

Formal Sciences

Mathematical Cycles

In graph theory, a cycle is defined as a nontrivial closed path in which no vertex is repeated except for the initial and terminal vertices, which coincide.^[36] This structure forms the basis for distinguishing acyclic graphs, such as trees, from cyclic ones, with applications in network analysis and combinatorial optimization./03:_Graph_Theory/12:_Moving_Through_Graphs/12.03:_Paths_and_Cycles) In abstract algebra, a cyclic group is a group generated by a single element, meaning every member can be expressed as a power of that generator./15:_Group_Theory_and_Applications/15.01:_Cyclic_Groups) Finite cyclic groups, such as the integers modulo n under addition, exhibit periodic order equal to the generator's smallest positive power yielding the identity.^[37] These groups serve as fundamental building blocks in the classification of abelian groups via the fundamental theorem. Permutation cycles arise in group theory as a decomposition of elements in the symmetric group S_n, where a k-cycle swaps k elements in a circular manner, and disjoint cycles commute./05:_Permutation_Groups/5.01:_Definitions_and_Notation) The cycle type, determined by the lengths of these disjoint cycles, uniquely identifies conjugacy classes within S_n, influencing properties like the number of fixed points and parity. In dynamical systems, a limit cycle is an isolated closed trajectory in phase space toward which nearby trajectories converge as time progresses, characteristic of nonlinear oscillations./03:_Nonlinear_Systems/3.08:_Limit_Cycles) Stable limit cycles underpin phenomena like predator-prey models, while their existence can be analyzed via criteria such as Poincaré-Bendixson theorem for planar systems, excluding continua of periodic orbits./03:_Nonlinear_Systems/3.08:_Limit_Cycles)

Computational Cycles

In computer architecture, computational cycles primarily manifest as clock cycles, which serve as the fundamental timing mechanism synchronizing operations within a central processing unit (CPU). A clock cycle represents the interval between consecutive pulses generated by the CPU's oscillator, enabling coordinated execution of instructions across hardware components.^[38] Clock speeds, measured in hertz (Hz) or multiples like gigahertz (GHz), quantify the number of such cycles per second; for instance, a 3 GHz processor completes 3 billion cycles each second, though effective performance also hinges on instructions processed per cycle (IPC) rather than raw frequency alone.^[39] The instruction cycle, often termed the fetch-decode-execute cycle, structures how CPUs process machine instructions over multiple clock cycles. This cycle begins with fetching the instruction from memory using the program counter, followed by decoding to interpret the opcode and operands, and execution to perform the operation, such as arithmetic or data movement. Additional phases like memory access or write-back may extend the cycle in complex architectures, with pipelining allowing overlap of stages across instructions to enhance throughput.^[40] Modern processors, such as those employing superscalar designs, execute multiple instructions per cycle by parallelizing these stages, mitigating limitations of sequential cycling.^[41] In algorithmic contexts, computational cycles refer to repetitive structures like loops in data structures or graphs, necessitating detection to prevent infinite computation or errors. Floyd's cycle-finding algorithm, utilizing two pointers advancing at differing speeds (tortoise and hare), identifies cycles in linked lists or functional graphs in linear time and constant space, originating from analyses of iterative processes in the 1960s.^[42] Such cycles arise in applications like garbage collection or blockchain validation, where confirming acyclicity ensures termination. Within the theory of computation, cycles appear in models like finite automata, where state transition loops encode periodic acceptance behaviors. Deterministic finite automata exhibit cycles that can be enumerated using algebraic methods over finite fields, aiding analysis of language recognition and synchronization.^[43] In Turing machines, non-halting computations often stem from cyclic state repetitions, underscoring undecidability results like the halting problem, though cycle detection remains feasible in restricted models.^[44]

Anthropological and Historical Cycles

Anthropologists and historians have proposed various theories positing recurrent patterns in human societies, often analogizing civilizations to biological organisms undergoing phases of growth, maturity, and decay driven by internal dynamics such as social cohesion, elite overreach, or generational shifts. These frameworks contrast with linear progress narratives by emphasizing empirical observations of repeated rises and falls across disparate cultures, attributing causality to factors like diminishing group solidarity or failure to adapt to challenges rather than inevitable advancement. While not universally predictive, such theories draw on historical records spanning millennia, including the collapses of Bronze Age societies around 1200 BCE and the fall of the Western Roman Empire in 476 CE, to argue for structural regularities in societal trajectories.^[45]^[46] One foundational model originates with the 14th-century Arab scholar Ibn Khaldun in his Muqaddimah (1377), which describes dynastic cycles lasting approximately three to four generations (about 120 years) fueled by asabiyyah, or tribal solidarity. Nomadic groups with strong asabiyyah conquer sedentary urban civilizations weakened by luxury and internal division, establishing new rulers who initially maintain austerity and cohesion; however, succeeding generations urbanize, accumulate wealth, and erode solidarity through corruption and taxation, leading to vulnerability against fresh nomadic challengers. This pattern, observed in North African and Middle Eastern history from the Berber dynasties to the Abbasid Caliphate's decline by the 9th century, underscores causal realism in how entropy in social bonds precipitates collapse absent renewal mechanisms.^[47]^[48] In the 20th century, Oswald Spengler extended organic analogies in The Decline of the West (1918–1922), viewing distinct cultures—such as Classical (circa 1100 BCE–100 BCE), Magian (circa 0–1100 CE), and Faustian (Western, from circa 1000 CE)—as entities with fixed lifespans of roughly 1,000 years, progressing through spring-like creative infancy, summer vitality, autumn intellectualism, and winter-phase civilization marked by materialism, imperialism, and sterility. Spengler cited morphological parallels, like the shift from Gothic architecture's upward aspiration to modern megacities' soulless expansion, as evidence of predetermined morphological decline rather than contingent events, predicting Western "Caesarism" by 2000 CE followed by 200 years of authoritarian consolidation before exhaustion.^[45]^[49] Arnold Toynbee's A Study of History (1934–1961) analyzed 21 civilizations, proposing a challenge-response mechanism where growth arises from creative minorities addressing environmental or social stresses—evident in the Roman response to Punic Wars (264–146 BCE)—but breakdown occurs when elites fail to inspire, leading to internal proletarian alienation and external invasions, as in the Byzantine Empire's fall to Ottomans in 1453 CE. Unlike rigid biologism, Toynbee allowed for potential "break-out" via spiritual renewal, though he documented empirical disintegration in most cases, such as the Mayan collapse around 900 CE amid resource depletion and schism.^[46]^[50] More granular cycles appear in William Strauss and Neil Howe's generational theory (Generations, 1991; The Fourth Turning, 1997), which identifies saecula of 80–100 years in Anglo-American history, comprising four "turnings": High (post-crisis institution-building, e.g., 1946–1964), Awakening (cultural upheaval, e.g., 1964–1984), Unraveling (institutional decay, e.g., 1984–2008), and Crisis (societal regeneration via ordeal, e.g., 2008–2030?). Drawing on archetypes like Prophet, Nomad, Hero, and Artist generations recurring every saeculum—supported by alignments with events from the American Revolution (1773–1794) to the Great Depression/WWII (1929–1946)—the model posits behavioral feedbacks where youthful idealism yields to midlife pragmatism, fostering periodic resets to avert stagnation, though critics note its post-hoc fitting to U.S.-centric data.^[51]^[52]

Economic Cycles

Economic cycles, also known as business cycles, refer to recurring fluctuations in aggregate economic activity, typically measured by indicators such as real gross domestic product (GDP), employment, industrial production, and sales.^[53] These cycles consist of four phases: expansion, characterized by rising output, employment, and income; peak, the highest point of activity; contraction or recession, marked by declining activity; and trough, the lowest point before recovery.^[54] The National Bureau of Economic Research (NBER) in the United States defines a recession as a significant decline in economic activity that is widespread across the economy and persists for more than a few months, emphasizing depth, diffusion, and duration over a simple rule like two consecutive quarters of negative GDP growth.^[53] In the U.S., business cycles have been documented since 1854, with 34 cycles identified through 2019, including expansions averaging about 65 months and contractions averaging 11 months in the post-World War II era up to 2019.^[55] Historical examples include the Great Depression (August 1929 peak to March 1933 trough, with GDP falling 26.7%) and the brief 2020 recession (February 2020 peak to April 2020 trough, driven by pandemic lockdowns, with GDP contracting 19.2% annualized in Q2).^[53] Post-1945 cycles show increasing stability, with recessions becoming shorter and less severe on average, though outliers like the 2007-2009 Great Recession (December 2007 peak to June 2009 trough, involving a 4.3% GDP drop and housing/financial collapse) highlight vulnerabilities.^[53] Empirical studies attribute part of this moderation to improved monetary policy and structural shifts, such as smaller inventory swings and better supply chain management, though variance in output growth has declined since the 1980s.^[56] Theories explaining economic cycles diverge on whether they stem primarily from exogenous shocks or endogenous market processes amplified by policy. Monetary theories, including the Austrian business cycle theory (ABCT), posit that central bank-induced credit expansions distort interest rates below natural levels, fueling unsustainable investments (malinvestments) in longer-term projects, leading to booms followed by inevitable busts as resources reallocate.^[57] ABCT, developed by economists like Ludwig von Mises and Friedrich Hayek, views cycles as non-random artifacts of fractional-reserve banking and fiat money systems, with historical evidence from credit-fueled bubbles like the 1920s U.S. expansion preceding the Depression.^[57] In contrast, Keynesian approaches emphasize demand deficiencies, where animal spirits or investment volatility cause underutilization of capacity, advocating fiscal and monetary stimulus to restore equilibrium, as seen in policy responses to the 2008 crisis.^[58] Empirical evidence on causes mixes exogenous shocks—such as financial disruptions (e.g., 2008 banking failures), technology shifts, or international events (e.g., oil shocks in 1973-1975)—with endogenous factors like credit cycles and inventory adjustments.^[59] Vector autoregression models show demand shocks explaining much of output variance in recessions, while supply shocks dominate expansions, but monetary policy's role in amplifying credit booms remains debated, with studies finding central bank actions correlate with cycle peaks.^[60] Longer-term cycles, like purported Kondratiev waves of 40-60 years tied to technological innovations, lack robust statistical validation and predictive power, often fitting data post-hoc without causal mechanisms beyond wars or gold standard shifts.^[61] Overall, cycles reflect inherent capitalist dynamics of innovation, error correction, and resource reallocation, with interventions potentially prolonging distortions rather than eliminating fluctuations.^[62]

Engineering and Technology

Thermodynamic and Mechanical Cycles

A thermodynamic cycle consists of a series of processes through which a thermodynamic system passes, returning it to its initial state while achieving net heat transfer or work output.^[63] These cycles underpin the efficiency of devices converting thermal energy into mechanical work, such as engines and turbines, governed by the first and second laws of thermodynamics that limit maximum conversion based on temperature differences.^[64] In engineering, they are analyzed under ideal assumptions like reversibility and perfect gases, though real implementations involve irreversibilities reducing efficiency.^[65] The Carnot cycle serves as the theoretical benchmark for heat engine efficiency, comprising two isothermal and two adiabatic processes between hot reservoir temperature T_h and cold reservoir temperature T_c (in Kelvin).^[66] Proposed by Sadi Carnot in his 1824 work Reflections on the Motive Power of Fire, it yields maximum thermal efficiency \eta = 1 - \frac{T_c}{T_h}, independent of working fluid but requiring reversibility unattainable in practice due to friction and heat losses.^[67] For example, with T_h = 1000 K and T_c = 300 K, \eta = 0.7 or 70%, though real engines achieve 30-50% due to non-ideal processes.^[68] Practical thermodynamic cycles approximate Carnot performance while accommodating real fluids and machinery. The Otto cycle models spark-ignition internal combustion engines, featuring isentropic compression, constant-volume heat addition via combustion, isentropic expansion, and constant-volume heat rejection; its efficiency \eta = 1 - \frac{1}{r^{\gamma-1}} (where r is compression ratio, \gamma \approx 1.4 for air) increases with r but is limited by knock to r \approx 10, yielding \eta \approx 0.56.^[69] The Diesel cycle, for compression-ignition engines, replaces constant-volume heat addition with constant-pressure, allowing higher r > 15 and \eta > 0.4, though slower combustion reduces peak power compared to Otto.^[70] The Rankine cycle drives steam power plants, involving boiler heat addition (evaporation), turbine expansion, condenser rejection, and pump compression; it handles phase changes in water, with efficiencies of 30-40% in supercritical plants exceeding 600°C.^[71] The Brayton cycle powers gas turbines and jet engines via isentropic compression, constant-pressure combustion, isentropic expansion, and constant-pressure cooling, with \eta rising with pressure ratio but constrained by turbine inlet temperatures around 1700 K.^[72] Mechanical cycles in engineering refer to the kinematic sequences in reciprocating or rotary machines that implement thermodynamic processes, often as open cycles where working fluid is renewed per cycle rather than recycled.^[73] In four-stroke internal combustion engines, the mechanical cycle aligns with Otto or Diesel thermodynamics across intake (fluid entry), compression, power (expansion), and exhaust (rejection) strokes, driven by crankshaft rotation completing one cycle every two revolutions.^[74] Unlike closed thermodynamic models assuming fixed fluid, these open mechanical cycles expel combustion products, necessitating approximations like air-standard analysis for efficiency prediction, where real fuel-air mixtures and variable composition reduce modeled performance by 10-20%.^[69] Two-stroke engines condense this into one revolution per cycle, trading efficiency for power density but increasing emissions due to incomplete scavenging.^[73] Cycle analysis integrates mechanical losses like friction (5-10% of work) and pumping work, with overall engine efficiency reflecting both thermodynamic limits and mechanical design.^[74]

Transportation Vehicles

Cycles, particularly bicycles and tricycles, serve as efficient human-powered transportation vehicles, offering low-cost mobility without reliance on fuel or complex infrastructure. The bicycle's design enables speeds of up to 20-25 km/h for average riders on flat terrain, making it suitable for short- to medium-distance urban commuting.^[75] Tricycles extend this utility by providing enhanced stability and cargo capacity, often employed in delivery and passenger services in densely populated areas.^[76] The precursor to the modern bicycle emerged in 1817 when German inventor Baron Karl von Drais introduced the draisine, a wooden two-wheeled device propelled by the rider's feet while seated, intended to aid travel amid horse shortages following the 1816 "Year Without a Summer."^[77] This evolved through various designs, culminating in the safety bicycle patented by John Kemp Starley in 1885, featuring equal-sized wheels, a diamond-shaped frame, and chain-driven rear wheel for improved balance and efficiency over predecessors like the high-wheeled penny-farthing.^[78] The addition of pneumatic tires in 1888 by John Boyd Dunlop further enhanced ride comfort and speed, spurring widespread adoption.^[79] By the late 19th century, bicycles revolutionized personal transport, with U.S. production rising from approximately 30,000 units in 1890 to 100,000 by 1894, reflecting demand for affordable alternatives to horse-drawn carriages.^[80] Globally, over 100 million bicycles are manufactured annually as of recent estimates, supporting their role in reducing traffic congestion and emissions in cities.^[81] In contemporary usage, bicycles account for significant transport shares in regions like the Netherlands, where cycling comprises a substantial portion of daily trips, and in developing countries for affordable commuting.^[75] With over 1 billion bicycles in circulation worldwide, they facilitate cargo transport via adapted models and promote health benefits through physical activity, though infrastructure limitations hinder broader uptake in car-dependent areas.^[75] Tricycles, including pedicabs and freight variants, dominate informal economies in Asia and Africa, carrying loads up to 200-300 kg for vending and logistics.^[76] Despite competition from motorized vehicles, cycles remain vital for sustainable, equitable mobility, with global market value exceeding USD 100 billion in 2024.^[82]

Arts, Entertainment, and Media

Literature and Mythology

In ancient mythologies, time was frequently conceptualized as cyclical rather than linear, reflecting patterns of renewal, decay, and recurrence observed in natural phenomena such as seasons and celestial movements. Egyptian cosmology distinguished neheh, the cyclical time tied to the sun's daily passage through the sky and underworld, from djet, a more linear eternal dimension, with the Nile's annual flooding symbolizing rebirth and continuity.^[83] Similarly, Indo-Hellenic traditions portrayed history as repeating in eras of prosperity and decline, akin to Chinese cyclical models where dynasties rose and fell in predictable patterns, as analyzed in comparative studies of ancient temporal frameworks.^[4] This motif appears in broader ancient narratives, where civilizations anticipated periodic cataclysms—floods or cosmic upheavals—followed by restarts, underscoring a worldview where human progress looped rather than advanced irreversibly.^[84] Mythological cycles often embodied these temporal loops through stories of creation, destruction, and regeneration. In Hindu cosmology, time unfolds in vast kalpas—each lasting 4.32 billion years—comprising four yugas of declining virtue, ending in dissolution (pralaya) before a new cycle begins, a structure preserved in texts like the Puranas dating to around 300–1500 CE. Norse mythology depicts Ragnarök as an apocalyptic battle culminating in the world's fiery submersion, yet survivors repopulate a renewed earth, illustrating recurrent cosmic renewal in Eddic poems compiled in the 13th century from older oral traditions. The Ouroboros, an ancient serpent devouring its tail symbolizing eternity and self-perpetuation, spanned Egyptian, Greek, and Norse myths from at least the 14th century BCE, embodying inexhaustible cyclicality.^[85] In literature, "cycles" also denote interconnected narrative series centered on heroic or mythical figures, expanding single epics into comprehensive sagas. The Greek Epic Cycle, a collection of dactylic hexameter poems from the 8th to 6th centuries BCE, encompassed the Trojan War's full arc—from divine origins in the Cypria to Odysseus's homecoming in the Odyssey and aftermath in the Telegony—with surviving fragments totaling about 60 lines amid lost works by authors like Lesches and Arctinus.^[86] ^[87] Irish mythology organizes tales into four cycles: the Mythological Cycle, detailing five invasions culminating in the Tuatha Dé Danann's arrival and battles like Moytura around the 1st millennium BCE; the Ulster Cycle, focusing on hero Cú Chulainn's exploits in pseudo-historical Ulster circa 1st century CE; the Fenian Cycle, chronicling Fionn mac Cumhaill's warrior band in 3rd-century settings; and the Kings Cycle, blending legend with early Irish history.^[88] ^[89] These medieval compilations, such as the Lebor Gabála Érenn (11th century), frame Ireland's origins as successive waves of settlement, mirroring broader Indo-European patterns of migratory myth-making.^[90] Such cycles influenced later literary traditions, where recurring motifs of fate, heroism, and downfall reinforced causal patterns of human endeavor. In medieval European literature, the Matter of Britain formed a cycle around King Arthur, weaving chronicles like Geoffrey of Monmouth's Historia Regum Britanniae (1136 CE) with romances such as Chrétien de Troyes' works (late 12th century), portraying chivalric rise and tragic fragmentation. Norse literary cycles, including the Völsunga Saga (13th century), trace the Nibelung hoard and Sigurd's lineage across generations, echoing mythic inevitability in ring-based curses and generational vendettas. These structures highlight literature's use of cycles not merely for continuity but to explore deterministic recurrences, grounded in oral precedents rather than invented linearity.

Music

In music, a cycle most commonly refers to a song cycle, a cohesive set of individually complete songs intended for sequential performance, unified by a shared poetic theme, narrative progression, or recurring musical motifs such as leitmotifs.^[91]^[92] These works emerged prominently in the early 19th-century German Lied tradition, with Franz Schubert's Die schöne Müllerin (1823), setting Wilhelm Müller's poems to depict a young miller's unrequited love and descent into despair, serving as a foundational example comprising 20 songs.^[93] Robert Schumann's Dichterliebe (1840), based on Heinrich Heine's texts, further exemplifies the form through 16 songs exploring love's emotional arc, often with piano accompaniment mirroring vocal lines for textual emphasis.^[93] Cyclic form extends this concept to instrumental compositions, particularly in multi-movement works like symphonies and sonatas from the late Classical period onward, where a principal theme, motif, or harmonic element recurs across movements to create structural unity and thematic transformation.^[94]^[95] Ludwig van Beethoven employed cyclic techniques extensively, as in his Symphony No. 5 (1808), where the iconic "fate" motif from the first movement reappears in varied guises in the finale, linking disparate sections through rhythmic and intervallic recall.^[94] César Franck's Symphony in D minor (1886–1888) exemplifies 19th-century French cyclicism, with a motto theme introduced in the opening movement that undergoes development and returns in the finale, enhancing the work's organic cohesion.^[95] Large-scale cycles include operatic tetralogies, such as Richard Wagner's Der Ring des Nibelungen (composed 1848–1874, premiered in full 1876), a four-opera narrative unified by leitmotifs representing characters, objects, and ideas, which recur and evolve across the cycle to drive dramatic continuity.^[96] In broader music theory, cycles denote repeating patterns organizing temporal perception, as in ostinato-based forms or modal cycles, but these structural devices underpin rather than define the primary cyclic genres.^[96] Song cycles and cyclic forms persist in 20th- and 21st-century compositions, adapting to modern idioms while retaining their emphasis on thematic interconnection over isolated movements.^[93]

Film and Visual Media

Films depicting cycles often explore themes of repetition, inevitability, and transformation through narrative structures like time loops or recurring motifs, reflecting philosophical concepts such as eternal recurrence or karmic progression. A prominent example is Groundhog Day (1993), directed by Harold Ramis, in which the protagonist Phil Connors, played by Bill Murray, relives February 2 indefinitely, using the loop to evolve from selfishness to empathy, grossing over $105 million worldwide and influencing subsequent genre works. Similarly, Edge of Tomorrow (2014), directed by Doug Liman and based on Hiroshi Sakurazaka's novel, features Major Bill Cage (Tom Cruise) repeating a battle against aliens, with each iteration granting tactical knowledge, earning $370 million globally and praised for its action-oriented cycle mechanics. Other films employ cyclical narratives to examine causality and choice, such as Run Lola Run (1998), directed by Tom Tykwer, where Lola (Franka Potente) navigates three rapid iterations of 20 minutes to save her boyfriend, emphasizing how minor variations alter outcomes, which premiered at the Sundance Film Festival and received critical acclaim for its kinetic style. In Source Code (2011), directed by Duncan Jones, Colter Stevens (Jake Gyllenhaal) relives eight minutes on a train to avert a bombing, blending sci-fi with thriller elements and grossing $147 million. Documentaries have visualized economic cycles to explain macroeconomic patterns. Ray Dalio's How the Economic Machine Works (2013), a 30-minute animated explainer produced by Bridgewater Associates, delineates productivity, credit expansion, and deleveraging phases driving boom-bust sequences, viewed millions of times and cited in policy discussions for its data-driven model based on historical U.S. data from 1913 onward. Canadian filmmaker Alanis Obomsawin's Riding the Tornado (1984), produced by the National Film Board of Canada, chronicles Alberta's oil boom-and-bust from the 1970s to early 1980s, interviewing residents on resource dependency's cyclical volatility, highlighting empirical boom peaks in 1979 followed by 1980s crashes due to oil price drops.^[97] These works prioritize verifiable data over narrative speculation, contrasting fictional loops' existential focus.

References

[1]
https://www.merriam-webster.com/dictionary/cycle
[2]
CYCLE Definition & Meaning - Dictionary.com
any complete round or series of occurrences that repeats or is repeated. a round of years or a recurring period of time, especially one in which certain events ...Motorcycle · Bicycle · TricycleMissing: encyclopedia | Show results with:encyclopedia
[3]
Cycle - Etymology, Origin & Meaning
Cycle originates from late 14th-century Old French and Late Latin, from Greek kyklos meaning "circle, wheel." It means a recurring series of events or to ...
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[PDF] Two Modes of Cyclicality in the Ancient World - BYU ScholarsArchive
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The Cyclical Time of the Body and its Relation to Linear Time
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Nietzsche's Idea of Eternal Recurrence - ThoughtCo
Apr 30, 2025 · Put simply, it's the theory that existence recurs in an infinite cycle as energy and matter transform over time.
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Eternal Recurrence: What Did Nietzsche Really Mean?
not this instant or that, but the overall package.
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Circular Reasoning Fallacy | Definition & Examples - Scribbr
May 1, 2023 · The circular reasoning fallacy is an argument that assumes the very thing it is trying to prove is true. It simply repeats the conclusion.
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[PDF] Circular reasoning
Moreover, circular arguments ought to include those in which a premise (or conjunction of premises) is propositionally equivalent to the conclusion, not just.
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Circular reasoning - ScienceDirect.com
Circularity is a defect in reasoning because it undermines correct attempts to justify a claim or an action.
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An Overview of the Cell Cycle - Molecular Biology of the Cell - NCBI
The phases of the cell cycle. The cell grows continuously in interphase, which consists of three phases: DNA replication is confined to S phase; G1 is the gap ...
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6.2: The Cell Cycle - Biology LibreTexts
Dec 22, 2021 · The cell cycle is an ordered series of events involving cell growth and cell division that produces two new daughter cells.
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Phases of the cell cycle (article) - Khan Academy
The cell cycle is composed of interphase (G₁, S, and G₂ phases), followed by the mitotic phase (mitosis and cytokinesis), and G₀ phase.
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The Cell Cycle - Phases - Regulation - TeachMePhysiology
Jun 8, 2025 · The cell cycle is a 4-stage process consisting of Gap 1 (G1), synthesis (S), Gap 2 (G2), and mitosis (M), which a cell undergoes as it grows and divides.
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Physiology, Krebs Cycle - StatPearls - NCBI Bookshelf
Nov 23, 2022 · The tricarboxylic acid (TCA) cycle, also known as the Krebs or citric acid cycle, is an important cell's metabolic hub.
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The citric acid cycle | Cellular respiration (article) - Khan Academy
The citric acid cycle is a closed loop; the last part of the pathway reforms the molecule used in the first step. The cycle includes eight major steps.
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Circadian Rhythms | National Institute of General Medical Sciences
May 20, 2025 · Circadian rhythms include some of the physical, mental, and behavioral changes an organism experiences over a 24-hour cycle.
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Physiology, Circadian Rhythm - StatPearls - NCBI Bookshelf
Circadian rhythm is the 24-hour internal clock in our brain that regulates cycles of alertness and sleepiness by responding to light changes in our environment.
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Molecular mechanisms and physiological importance of circadian ...
Nov 25, 2019 · Circadian physiology is generally the result of cyclic expression of clock-controlled genes downstream of the core molecular oscillator.<|separator|>
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Mammal - Reproduction, Gestation, Lactation | Britannica
Oct 2, 2025 · Reproductive patterns in placental mammals are diverse, but in all cases a secretory phase is present in the uterine cycle, and the endometrium ...
[21]
The nitrogen cycle (article) | Ecology | Khan Academy
Bacteria play a key role in the nitrogen cycle. Nitrogen enters the living world by way of bacteria and other single-celled prokaryotes, which convert ...The Nitrogen Cycle · Nitrogen Cycling In Marine... · Attribution
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Intro to biogeochemical cycles (article) - Khan Academy
Biogeochemical cycles important to living organisms include the water, carbon, nitrogen, phosphorus, and sulfur cycles.
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4.3.3: Nutrient Cycles - Biology LibreTexts
Jul 28, 2025 · The first is the carbon cycle. Plants acquire carbon through photosynthesis. The nitrogen and phosphorus cycles are also discussed, and plant ...The Carbon Cycle · Carbon Cycles Quickly... · Nitrogen Cycle · Nitrogen Fixation
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Energy and Matter Cycles | MyNASAData
Apr 3, 2019 · There are many ways in which the energy, water, and biogeochemical cycles (cycles of the elements that involve life, chemicals, and the solid ...
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[PDF] TEACHER BACKGROUND: BIOGEOCHEMICAL CYCLES
Biogeochemical cycles are intricate processes that transfer, change and store chemicals in the geosphere, atmosphere, hydrosphere, and biosphere.
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Water cycle | U.S. Geological Survey - USGS.gov
The water cycle describes where water is on Earth and how it moves. Human water use, land use, and climate change all impact the water cycle.Oceans and Seas and the... · Water Cycle Diagrams · Evaporation and the Water...
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The Hydrologic Cycle - NOAA
Mar 24, 2023 · The Hydrologic Cycle · Evaporation · Transpiration · Condensation · Precipitation · Runoff.
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The water cycle | National Oceanic and Atmospheric Administration
Sep 26, 2025 · The water cycle is often taught as a simple, circular cycle of evaporation, condensation, and precipitation. Although this can be a useful model ...<|control11|><|separator|>
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DOE Explains...the Carbon Cycle - Department of Energy
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The Carbon Cycle - NASA Earth Observatory
Jun 16, 2011 · Carbon flows between the atmosphere, land, and ocean in a cycle that encompasses nearly all life and sets the thermostat for Earth's climate ...Changes in the Carbon Cycle · The Fast Carbon Cycle · Studying the Carbon Cycle
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Biogeochemical cycles - The Physical Environment
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The Rock Cycle - National Geographic Education
Jun 4, 2025 · Each of these rocks are formed by physical changes—such as melting, cooling, eroding, compacting, or deforming—that are part of the rock cycle.
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Rock cycle - Understanding Global Change
The rock cycle describes the processes through which the three main rock types (igneous, metamorphic, and sedimentary) transform from one type into another.
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Milankovitch (Orbital) Cycles and Their Role in Earth's Climate
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[PDF] graph theory: basic definitions and theorems
Definition 9. A cycle is a nontrivial circuit in which the only repeated vertex is the first/last one. Date: December 3, ...
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[PDF] Math 403 Chapter 4: Cyclic Groups 1. Introduction
Definition: A group G is cyclic if there is some g ∈ G with G = <g>. Here g is a generator of the group G. Recall that <g> means all “powers” of g which can ...<|separator|>
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Clock Cycle - an overview | ScienceDirect Topics
The clock cycle is the fundamental unit of time in computer systems, representing the interval during which a single tick of the system clock occurs. All ...Introduction to Clock Cycle in... · Clock Cycle in Instruction... · Clock Cycle in...
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4.1: Instruction Cycles - Engineering LibreTexts
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The fetch-decode-execute cycle - Ada Computer Science
The fetch-decode-execute cycle is the basic operation of modern computer systems, repeated for every instruction, involving fetching, decoding, and executing.
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[PDF] Algorithmic Problems
Floyd's cycle detection algorithm uses just two variables, so O(1) memory usage! Finding n can be done in less than s+L steps. Finding the start of the cycle ...
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CYCLE ANALYSIS FOR DETERMINISTIC FINITE STATE AUTOMATA
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[PDF] Introduction to the Theory of Computation Some Notes for CIS511
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Oswald Spengler: Why Civilizations Die Like Organisms
Jun 11, 2025 · He didn't see history as a straight line of progress, but as a series of self-contained cycles, each civilization following a predetermined ...
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The Cycle Of Civilizations - NOEMA
Jan 27, 2023 · Writing his “Study of History” in the mid-20th century, he counted some 22 such civilizations that had arisen over the last 6,000 years, from ...
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The Rise And Fall Of Empires: Ibn Khaldun's Theory Of Social ...
May 8, 2015 · Empires are like organisms, and their life trajectories can be plotted like points on a bell curve from their beginnings to their deaths. He ...
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Ibn Khaldun's Four Stages of Empire: Where Is America Now?
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Culture and Civilization — Oswald Spengler's Approach to History
He distinguished between two main stages: culture, which is the first stage, and civilization, which is the second. Every culture has its own civilization. At ...
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[PDF] ARNOLD J. TOYNBEE THE CYCLE OF CIVILIZATIONS Page 1
Whether the world pays attention or not, Toynbee's theory of the. Cycle of Civilizations remains the best single historical theory for explaining. 26 Toynbee ...
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Strauss-Howe generational theory: Is revolution coming to America?
Nov 16, 2023 · Generations argues the development of human civilization is heavily affected by and even mirrors the transition between different generations of human beings.
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The Cyclical Nature of History: Understanding Generational Turning ...
Jun 15, 2025 · The Strauss-Howe theory suggests history has four turnings: High, Awakening, Unraveling, and Crisis, occurring roughly every 80-100 years.
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Business Cycle Dating | NBER
The NBER's definition emphasizes that a recession involves a significant decline in economic activity that is spread across the economy and lasts more than a ...Sions and ContractionsFrequently Asked QuestionsAnnouncements
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Business Cycle Dating Committee Announcement June 8, 2020
Jun 8, 2020 · A recession begins when the economy reaches a peak of economic activity and ends when the economy reaches its trough. Between trough and peak, ...
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Introduction to U.S. Economy: The Business Cycle and Growth
Oct 3, 2024 · Between 1945 and 2019, the end of the most recent business cycle, the average expansion has lasted about 65 months, and the average recession ...
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[PDF] Has the Business Cycle Changed? Evidence and Explanations
The empirical results in this paper suggest that improved monetary policy accounted for only a small fraction of the reduction in the vari- ance of output ...
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[PDF] The Austrian Theory of Business Cycles: Old Lessons for Modern ...
This paper reviews the "Austrian" theory of the business cycle first proposed by Friedrich. Hayek in the 1920s. His theory claimed that credit creation by ...
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What Is Keynesian Economics? - Back to Basics
Keynesian economists justify government intervention through public policies that aim to achieve full employment and price stability.<|separator|>
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All About the Business Cycle: Where Do Recessions Come From?
Mar 1, 2023 · The NBER also focuses on monthly assessments of the business cycle, while real GDP is reported quarterly.
[60]
What Drives Business Cycles? | Richmond Fed
The main business cycle shock in the artificial data replicates the empirical finding of a relatively small effect on inflation in actual data.
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[PDF] The Nature and Causes of Business Cycles
whereby they are generated. This paper presents in nontechnical language the main re- sults of modern research on the nature and causes of business.
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7.9: Thermodynamic Cycles - Engineering LibreTexts
Aug 5, 2022 · A thermodynamic cycle is a closed system that executes a series of processes that periodically return the system to its initial state.Key Features and Examples · Classifying Cycles · Quantifying Cycle Performance
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[PDF] 1 Thermodynamic Cycles - Wiley-VCH
The concept of a thermodynamic cycle is based on depicting thermodynamic processes which involve the transfer heat and work. This is achieved by altering.
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8 Analysis of Thermodynamic Cycles – Thermo 101
A thermodynamic cycle is a series of processes that pass through equilibrium thermodynamic states, and return to the initial state.
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4.6: The Carnot Cycle - Physics LibreTexts
Mar 2, 2025 · The total work done by the gas in the Carnot cycle is given by W = W 1 + W 2 − W 3 − W 4 . The first part of the figure shows a graph ...
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Carnot Cycle - Chemistry LibreTexts
Feb 16, 2025 · The efficiency of the carnot engine is defined as the ratio of the energy output to the energy input.The Carnot Cycle · P-V Diagram · T-S Diagram · Efficiency
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The Carnot Efficiency | EGEE 102 - Dutton Institute
From the Carnot Efficiency formula, it can be inferred that a maximum of 64% of the fuel energy can go to generation. To make the Carnot efficiency as high ...
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This article explores various thermodynamic cycles, including basic, Carnot, Rankine, Otto, Diesel, Brayton, refrigeration, and vapor-compression cycles.
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[PDF] Otto and Diesel Cycles
Diesel Cycle. • air standard model for a CI engine. SI engines. • Air-fuel mixture compressed to below fuel auto-ignition temperature.
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In this section, we present a summary of the main power cycles, namely Rankine, Brayton, Kalina, Stirling, Diesel, Otto and Ericsson cycles. 4.2.1 Rankine cycle.
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Jan 12, 2023 · Otto and Diesel Cycles are the thermodynamic cycles used in combustion vehicles' engines. Theoretical engines are more efficient at higher ...
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Engine Cycle - an overview | ScienceDirect Topics
The engine cycle refers to the series of events that occur during the operation of internal combustion (IC) engines, which can be classified into two-stroke and ...
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[PDF] Chapter23 - Rose-Hulman
Review the operation of reciprocating engines. • Solve problems based on the Otto and Diesel cycles. • Solve problems based on the Brayton cycle and the Brayton.<|control11|><|separator|>
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Cycling Statistics - RunRepeat
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Get to Know the Seven Types of Tricycles - Motrike
Mar 27, 2023 · Rickshaw trikes are common bicycles with three wheels used for local public transport. These vehicles, also known as trishaws, pedicabs, or ...
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Director's Corner: The Bicycle: Invented 200 Years Ago & Still ...
Sep 5, 2017 · Two hundred years ago, on June 12, 1817, in Mannheim, Germany, Baron Karl von Drais introduced the world's first bicycle. It was made of wood ...
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Happy Birthday, JK Starley: Inventor of the Modern Bicycle
Dec 14, 2012 · In 1885, John Kemp Starley invented the “Rover Safety Bicycle,” debuting a revolutionary ride that had two similar-sized wheels.
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The Bicycle's Bumpy History
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The global bicycle market size was valued at USD 108.74 billion in 2024 and is projected to grow from USD 116.56 billion in 2025 to USD 245.33 billion by 2032.
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Thinking Critically about Time: A Cyclical View of Knowledge and ...
They think it is cyclical: human beings progress, they are wiped out by worldwide disasters such as floods or meteor strikes, and then the survivors begin again ...
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The ancient symbol that spanned millennia - BBC
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EPIC CYCLE FRAGMENTS - Theoi Classical Texts Library
THE EPIC CYCLE was a series of old epic style poems composed between the C8th and 6th B.C. Only fragments of the ten poems survive, one of which describes the ...
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Epic Cycle - Livius.org
Apr 30, 2020 · The epic cycle tells the story of the world from the Creation to the return of the heroes of the Trojan War.
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Mythological Cycle Stories
The Mythological Cycle includes five invasions, the Tuatha De Danann, the Sons of Mil, the battles of Moytura and Tailtiu, and exile stories like Bran and Mael ...
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Four Cycles Of Irish Mythology - Myths & Folklore From Ireland
Aug 20, 2024 · The four cycles of Irish mythology are: The Mythological Cycle, The Finn Cycle, The Ulster Cycle, and The Historical cycle.<|separator|>
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What is a song cycle? - Edinburgh International Festival
Sep 28, 2023 · A song cycle is a collection of songs linked by a central theme, such as birth, love, or death, or by repeated musical motifs.
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Song cycles | Yale University Library
A song cycle is a group of complete songs, for solo or ensemble voices, that may relate events, impressions, or be unified by mood.
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[PDF] song cycles: a brief history with examples
A song cycle is a series of songs relating to the same poetic subject, forming one musical composition, based on a pre-existing text.
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The idea of cyclic form (Chapter 1) - Mendelssohn, Time and Memory
The term 'cyclic', as applied to music since c. 1750, can be used to describe: a work where part of one movement is recalled in another (examples include Haydn ...
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5.3: Cyclic forms - Humanities LibreTexts
Jul 15, 2023 · When the section that repeats is longer than an ostinato then the form is considered to be cyclic. Like ostinatos the cycles that are repeated can be melodic, ...<|control11|><|separator|>
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[PDF] Cycles in Music - MusMat Research Group
In music, a cycle is a property where a modeling reiterates itself by transformation, organizing our sense of time, and is related to bodily experience.
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Riding the Tornado - NFB
This documentary focuses on boom-and-bust economic cycles, most notably that of Alberta oil during the '70s and early '80s.<|control11|><|separator|>