Loop
Loop is an ambiguous English-language term with multiple meanings in various fields, including mathematics, science, computing, geography, transportation, arts, and more. For specific uses, see the sections below.
Mathematics and science
Mathematics
In graph theory, a loop is defined as a degenerate edge that connects a vertex to itself, commonly referred to as a self-loop. Unlike edges between distinct vertices, a self-loop does not contribute to cycles of length greater than one in simple graphs, where loops are prohibited; however, in more general graphs allowing loops, they represent reflexive connections, such as a node influencing itself in network models. In the adjacency matrix A of an undirected graph, a self-loop at vertex i is indicated by a 1 in the diagonal entry A_{ii}, distinguishing it from simple graphs where the diagonal is zero; this placement ensures that powers of A correctly count walks, including those that revisit the vertex immediately. For degree calculations, a self-loop contributes 2 to the degree of its vertex, aligning with the undirected nature where edges are bidirectional.[1][2][3]
In algebraic topology, a loop based at a point x_0 in a topological space X is a continuous map f: [0,1] \to X such that f(0) = f(1) = x_0, representing a closed path that returns to its starting point. These loops are studied up to homotopy, where two loops are homotopic if one can be continuously deformed into the other while fixing the basepoint, forming equivalence classes that capture the space's 1-dimensional holes. The fundamental group \pi_1(X, x_0) is the group of these homotopy classes of loops, equipped with the operation of concatenation: for loops f and g, the product \cdot is the class of the loop that traverses f followed by g, adjusted for reparametrization. This structure, first systematically developed in the early 20th century, provides an algebraic invariant for distinguishing topological spaces, such as \pi_1(S^1) \cong \mathbb{Z}, where classes correspond to winding numbers around the circle.[4]
An algebraic loop is a quasigroup equipped with a two-sided identity element e, meaning for all x in the loop L, e \cdot x = x \cdot e = x, where \cdot denotes the binary operation. As a quasigroup, every pair of elements admits unique solutions to the equations a \cdot x = b (right division, x = a \backslash b) and y \cdot a = b (left division, y = b / a), ensuring cancellativity without requiring associativity. Loops generalize groups by relaxing associativity, allowing structures like Moufang loops that satisfy additional identities; for instance, the nonzero octonions form a nonassociative loop under multiplication, inheriting division from the octonion algebra's normed division property, which guarantees unique left and right inverses for nonzero elements. This framework supports applications in geometry and physics, where nonassociative multiplications model exceptional structures.[5][6][7]
The concept of algebraic loops emerged in the mid-20th century, with the term "loop" coined around the 1940s by Abraham Adrian Albert and his Chicago school colleagues, chosen to rhyme with "group" while evoking cyclic returns, building on earlier quasigroup studies from the 1920s by mathematicians like Aurel Wintner. In combinatorial designs, loops gained prominence post-1950 through connections to Latin squares and orthogonal arrays, where quasigroups model mutually orthogonal resolutions; for example, loop isotopes facilitate constructing projective planes of non-prime-power order, a key development in finite geometry during the 1960s–1970s by researchers like Dénes and Keedwell. This historical thread intertwined loops with broader 20th-century advances in abstract algebra and design theory, influencing coding and cryptography.[8][9]
Physics
In physics, closed loops play a central role in describing interactions governed by fundamental laws, particularly in electromagnetism, magnetism, wave mechanics, and control theory. These concepts emerged prominently in the early 19th century, with André-Marie Ampère's foundational work in 1820 establishing the force law between current-carrying loops following Hans Christian Ørsted's discovery of electromagnetism.[10] Ampère demonstrated that parallel currents in loops attract when in the same direction and repel when opposite, quantifying the force through experiments with spirals and magnets, which showed torque effects dependent on pole orientation.[10] His empirical formula for the force between current elements, presented on December 4, 1820, to the Académie Royale des Sciences, laid the groundwork for electrodynamics by treating magnetic phenomena as arising from electric currents rather than static poles.[10]
A key application of loops in electromagnetism is Faraday's law of induction, discovered by Michael Faraday in 1831, which states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) along the loop.[11] This law, formalized in Maxwell's equations, explains phenomena like generators and transformers, where relative motion between a conductor and magnetic field lines produces current.[11] The integral form of the law is given by
\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt},
where \Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A} is the magnetic flux through the surface S bounded by the loop C, and the negative sign reflects Lenz's law, indicating the induced current opposes the flux change.[12] In differential form, it appears as \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, linking the curl of the electric field to the time-varying magnetic field.[12] Faraday's experiments, such as rotating a copper disc near a magnet, confirmed that induction arises from flux variation, not absolute motion, building on Ampère's earlier observations.[11]
In magnetism, hysteresis loops characterize the behavior of ferromagnetic materials under cyclic magnetization, plotting magnetic flux density B against magnetic field strength H.[13] The loop forms due to irreversible domain wall motion and magnetization rotation, with key parameters including coercivity (the H field needed to demagnetize the material) and remanence (residual B after field removal).[13] The area enclosed by the B-H curve represents energy loss per cycle, dissipated as heat from irreversible processes, quantified empirically by Charles Proteus Steinmetz in 1892 as P_h = \eta B_m^{1.6} f, where P_h is hysteresis power loss, \eta is a material constant, B_m is maximum flux density, and f is frequency.[14] This Steinmetz equation enables prediction of losses in transformers and motors without exhaustive testing, emphasizing the loop's role in energy inefficiency.[14]
Standing wave loops arise in transverse waves on fixed strings, where vibrations form stationary patterns with nodes at the ends. Each loop corresponds to half a wavelength, as interference between incident and reflected waves creates antinodes in between. For a string of length L with n loops (harmonics), the wavelength satisfies \lambda = \frac{2L}{n}, where n = 1, 2, 3, \dots, yielding frequencies f_n = \frac{n v}{2L} with wave speed v = \sqrt{T/\mu} (tension T, linear density \mu). This quantization explains resonance in musical instruments like guitars, where only discrete modes sustain due to boundary conditions.
Feedback loops in physical control systems maintain stability or amplify signals through closed-path interactions, classified as negative (stabilizing) or positive (amplifying).[15] In a negative feedback loop, output is subtracted from input to reduce error, as in a thermostat regulating temperature by counteracting deviations.[15] Positive feedback adds output to input, potentially leading to instability, such as in oscillators where gain builds until saturation.[15] Stability is assessed via the Nyquist criterion, which examines the open-loop transfer function L(i\omega) in the complex plane; the closed-loop system is stable if the Nyquist plot does not encircle the point -1 + 0i, ensuring no right-half-plane poles.[16] For negative feedback, gain and phase margins (typically 30°–60° and 2–5, respectively) quantify robustness against variations.[16] These loops, often represented in block diagrams with forward path G(s) and feedback H(s), underpin applications from servomechanisms to chemical reactors.[15]
Biology
In biology, loops refer to flexible structural motifs or regulatory circuits that play crucial roles in molecular and physiological processes. Protein loops are unstructured or partially structured segments connecting secondary structural elements such as alpha helices and beta sheets, providing flexibility essential for protein function and stability.[17] These loops often mediate interactions with substrates, cofactors, or other proteins, and their conformational dynamics contribute to enzymatic catalysis.[18] For instance, in serine proteases like chymotrypsin, the catalytic loop containing the serine residue of the catalytic triad (Ser-His-Asp) positions the nucleophile for peptide bond hydrolysis, enabling efficient substrate cleavage.[19] Additionally, loops near active sites act as molecular switches, regulating access and enhancing specificity during catalysis.
DNA loops form through chromatin folding, facilitating long-range interactions that govern gene expression. In eukaryotic cells, enhancer-promoter looping brings distant regulatory elements into proximity, mediated by the insulator protein CTCF and the cohesin complex, which extrude chromatin loops to stabilize these contacts.[20] This looping model, refined through studies in the 2010s using chromosome conformation capture techniques, explains how enhancers activate specific promoters while insulating others, as seen in the beta-globin locus where CTCF sites block inappropriate interactions.[21] Chromatin loops thus provide a topological framework for transcriptional condensates, where phase-separated domains concentrate factors for efficient gene regulation.[22]
Feedback loops maintain homeostasis in physiological systems, with negative feedback exemplifying self-regulation. The hypothalamic-pituitary-adrenal (HPA) axis operates via a negative feedback loop where stress induces hypothalamic release of corticotropin-releasing hormone (CRH), stimulating pituitary adrenocorticotropic hormone (ACTH) secretion, which prompts adrenal cortisol production; elevated cortisol then inhibits CRH and ACTH to restore balance.[23] This glucocorticoid-mediated feedback prevents overactivation, ensuring adaptive stress responses without chronic elevation.[24]
Evolutionary pressures have shaped loop variations to enhance adaptive diversity, particularly in immune molecules. In antibodies, the hypervariable regions—comprising complementarity-determining region (CDR) loops—undergo somatic hypermutation and recombination, generating conformational diversity that broadens antigen recognition. For example, variations in CDR-H3 loop length and sequence in heavy chains allow antibodies to accommodate diverse epitopes, contributing to the vast repertoire of the adaptive immune system. These loop evolutions, conserved across vertebrates, underscore their role in immunological versatility.[25]
Recent advances in cryo-electron microscopy (cryo-EM) have elucidated loop conformations in viral proteins, revealing dynamic roles in infection. Between 2020 and 2025, high-resolution structures of picornavirus capsids captured intermediate states during cell entry, showing how flexible loops in VP1 and VP4 proteins rearrange to facilitate membrane penetration and genome release.[26] Similarly, cryo-EM of herpesvirus capsids highlighted loop-mediated conformational switches in portal proteins, safeguarding genome packaging and ejection.[27] These insights, up to 2025, inform antiviral strategies by targeting loop flexibility in pathogens like SARS-CoV-2 spike proteins, where loops modulate receptor binding.[28]
Computing and technology
Programming constructs
In computer programming, a loop is a control structure that repeats a block of code multiple times until a specified condition is satisfied, enabling efficient handling of repetitive tasks without duplicating instructions.[29] This construct is fundamental to algorithmic design, allowing programs to process sequences, accumulate values, or iterate over data structures. The primary types of loops include for loops, which execute a fixed number of times based on a counter or iterable; while loops, which continue as long as a condition evaluates to true; and do-while loops, which execute the body at least once before checking the condition.[30] These structures draw briefly from mathematical concepts of cycles, providing a practical foundation for iterative algorithms.
For loops are particularly suited for scenarios where the iteration count is predetermined, such as traversing an array or generating a sequence. In pseudocode, a simple for loop to compute the sum of the first n natural numbers might appear as:
total ← 0
for i ← 1 to n do
total ← total + i
end for
total ← 0
for i ← 1 to n do
total ← total + i
end for
[31] An equivalent implementation in Python uses the range function for iteration:
python
total = 0
for i in range(1, n + 1):
total += i
total = 0
for i in range(1, n + 1):
total += i
[32] In C++, a for loop follows a similar pattern with initialization, condition, and increment:
cpp
int total = 0;
for (int i = 1; i <= n; ++i) {
total += i;
}
int total = 0;
for (int i = 1; i <= n; ++i) {
total += i;
}
While loops are ideal for indefinite iterations dependent on runtime conditions, such as reading input until a sentinel value. The syntax tests the condition before each execution, ensuring the body runs only if true. A C++ example to sum positive numbers until a negative input is encountered:
cpp
int total = 0;
int num;
while (num >= 0) {
std::cin >> num;
if (num >= 0) {
total += num;
}
}
int total = 0;
int num;
while (num >= 0) {
std::cin >> num;
if (num >= 0) {
total += num;
}
}
[33] Do-while loops differ by evaluating the condition after the body, guaranteeing at least one execution—useful for menu-driven programs. In C++:
cpp
int total = 0;
int num;
do {
std::cin >> num;
if (num > 0) {
total += num;
}
} while (num > 0);
int total = 0;
int num;
do {
std::cin >> num;
if (num > 0) {
total += num;
}
} while (num > 0);
[34]
To ensure correctness, programmers rely on loop invariants—properties that remain true throughout execution—and well-defined termination conditions that guarantee finite iterations. Loop invariants, formalized by Edsger W. Dijkstra in his 1976 book A Discipline of Programming, help verify that the loop achieves its intended postcondition, such as maintaining an accumulated sum accurately after each step.[35] Termination is ensured by conditions that progressively approach falsity, like decrementing a counter toward zero. In terms of time complexity, a basic loop iterating n times performs O(n) operations, scaling linearly with input size and establishing essential context for algorithm efficiency analysis.[36]
The origins of loops trace to early electronic computers, including the ENIAC in 1945, which implemented iterative processes via plugboard wiring and switches to repeat calculations for ballistic trajectory simulations.[37] Common pitfalls include infinite loops, where the termination condition fails to evaluate to false—often due to logical errors like unupdated variables—leading to resource exhaustion and program hangs.[38] To mitigate this, developers test conditions explicitly and use debugging tools to trace iterations. Optimization techniques, such as loop unrolling, address performance overhead by replicating the loop body multiple times, reducing branch instructions and enabling better instruction scheduling, though at the cost of increased code size.[39] For instance, unrolling a loop four times minimizes control flow checks, potentially halving execution time in tight inner loops on modern processors.
Telecommunications and hardware
In telephony, the local loop refers to the physical connection between a subscriber's premises and the central office, traditionally implemented using copper twisted-pair cables that form a multi-level star network with branching at distribution points. These cables typically consist of twisted wire pairs of 0.4 mm diameter within protective sheaths, supporting voice transmission over frequencies up to 4 kHz and enabling digital services such as asymmetric digital subscriber line (ADSL) with downstream speeds up to 6.144 Mbit/s.[40] The infrastructure includes feeder cables from the central office to cross-connect points and distribution cables to subscriber points, often deployed aerially on poles or underground in ducts, and powered by -48 V DC from the central office to mitigate corrosion.[40]
The transition from analog copper loops to digital and fiber-optic alternatives has accelerated to meet broadband demands, with digital subscriber line (DSL) technologies overlaying existing twisted pairs to provide higher speeds without full replacement. Early DSL variants like ADSL, standardized by the International Telecommunication Union (ITU) in the late 1990s, achieved up to 8 Mbit/s downstream, evolving to very-high-bit-rate DSL (VDSL2) with 100 Mbit/s over shorter loops by the 2000s.[41] By 2015, ITU-T G.fast extended this to 1 Gbit/s over loops under 100 meters using advanced modulation, while hybrid approaches like fiber-to-the-x (FTTx) integrate fiber backhaul with copper drops for gigabit access up to 2025 standards. Fiber-optic deployments, such as fiber-to-the-home (FTTH), now dominate new installations for their immunity to electromagnetic interference and capacity exceeding 10 Gbit/s over 20 km without repeaters.[40]
Loopback testing in telecommunications diagnostics involves redirecting a transmitted signal back to the receiver at the test point to verify hardware integrity and link quality, commonly using loopback plugs that physically interconnect transmit and receive ports on devices like modems or network interface cards. This method isolates faults in the local device or cable without external traffic, supporting protocols such as the bit error ratio test (BERT), which generates pseudorandom bit sequences to measure error rates typically below 10^{-9} in digital loops like DSL.[42] In Ethernet-based telecom systems, loopback modes defined in IEEE 802.3 standards enable operations, administration, and maintenance (OAM) functions for remote diagnostics over twisted-pair links.[43]
Audio induction loop systems assist hearing-impaired individuals by generating low-frequency electromagnetic fields that telecoil-equipped hearing aids convert into audio signals, improving speech intelligibility in environments like theaters or classrooms. These systems, comprising loop amplifiers and wire loops installed around a space, produce magnetic fields in the 100 Hz to 5 kHz range per International Electrotechnical Commission (IEC) standards.[44] IEC 60118-4 specifies a minimum field strength of 100 mA/m (measured as magnetic field intensity) for adequate signal-to-noise ratios above 50 dB without hearing aid overload, with a frequency response ensuring intelligibility for speech.[44] The systems minimize distortion through even field distribution, as outlined in measurement methods for loop performance.[44]
Loop antennas in radio communications, particularly small loops for high-frequency (HF) reception, consist of a single-turn or multi-turn conductor forming a closed circuit, often less than 0.1 wavelength in circumference to maintain compactness. For HF bands (3-30 MHz), small loops exhibit vertically polarized radiation patterns with maximum gain in the plane of the loop and nulls perpendicular to it, providing directional sensitivity for noise reduction in receiving applications.[45] As loop size approaches 0.4λ perimeter, patterns become nearly omnidirectional with balanced vertical and horizontal polarization, and circular polarization emerges at 45° angles, enhancing reception of skywave signals affected by ionospheric Faraday rotation.[45] These antennas are favored for their low-noise figure and immunity to electric field interference when shielded or ferrite-loaded.[45]
Software and systems
The LOOP programming language is a simple register-based language designed to precisely capture the class of primitive recursive functions, serving as an educational tool for understanding computability and recursion limits in programming. Developed by Albert R. Meyer and Dennis M. Ritchie, it features bounded iteration through LOOP instructions that execute a fixed number of times based on register values, without support for unbounded loops or recursion, ensuring all computable functions terminate.[46] This structure makes LOOP ideal for teaching primitive recursion, as programs consist solely of assignments and these controlled loops, mirroring the hierarchical complexity of primitive recursive computations.[47]
Microsoft Loop is a collaborative productivity application introduced in 2021 as part of the Microsoft 365 suite, enabling real-time teamwork on notes, tasks, and wikis through flexible, shareable components that sync across apps like Teams, Outlook, and OneNote. It integrates structured content blocks—such as tables, checklists, and progress trackers—allowing users to co-create and iterate on ideas without switching tools, with changes propagating instantly to maintain alignment in distributed environments.[48] Launched at Microsoft Ignite 2021, Loop emphasizes fluid collaboration by embedding interactive elements directly into emails or chats, reducing context-switching and enhancing workflow efficiency.
In Unix-like operating systems, a loop device provides a mechanism to treat regular files as block devices, facilitating the mounting of file-based images like disk partitions or ISO files as if they were physical storage. The losetup utility manages these associations, attaching a file to a /dev/loopX device (e.g., losetup /dev/loop0 image.iso) to enable filesystem access via standard mount commands, and supports detachment with losetup -d.[49] This feature is essential for tasks like testing filesystems in virtual environments or running container images without dedicated hardware, with encryption options available via kernel modules for secure handling.
LOOPS (Lisp Object-Oriented Programming System) is an integrated object-oriented extension to the Interlisp-D environment, developed in the early 1980s at Xerox PARC to support AI applications through a unified framework combining procedural, data-oriented, object-oriented, and rule-based paradigms. It employs a prototype-based model where objects inherit from customizable prototypes, enabling dynamic method attachment and message passing for flexible knowledge representation in expert systems.[50] LOOPS influenced subsequent Lisp object systems, including CLOS, by providing built-in support for active values—objects that trigger rules on access—and modular organization via contexts, which encapsulate related objects and behaviors for scalable AI development.[51]
The continue statement in programming languages serves as a flow control mechanism within loops to prematurely end the current iteration and proceed to the next one, skipping remaining code in the loop body without exiting the loop entirely. In Java, for instance, it is used in for, while, and do-while constructs; consider a loop that processes an array but ignores even indices:
java
for (int i = 0; i < 10; i++) {
if (i % 2 == 0) {
continue; // Skips even i values
}
System.out.println(i); // Prints only odd numbers: 1, 3, 5, 7, 9
}
for (int i = 0; i < 10; i++) {
if (i % 2 == 0) {
continue; // Skips even i values
}
System.out.println(i); // Prints only odd numbers: 1, 3, 5, 7, 9
}
This enhances readability and efficiency in iterative algorithms by avoiding nested conditionals, with labeled variants allowing control over outer loops for complex nested structures.[52]
Geography
Urban districts
The Loop in Chicago serves as the city's central business district and one of its 77 officially recognized community areas, encompassing a densely developed area of approximately 1.5 square miles at the heart of downtown.[53] The core of the district is encircled by the elevated rail tracks of the Chicago 'L' system, forming a rectangular loop that was completed in 1897 to connect rapid transit lines serving the growing urban core.[53] The name "The Loop" originated from an earlier cable car line that encircled the central business area starting in 1882, predating the elevated tracks but establishing the conceptual boundary that persists today.[54] Urban development in the area accelerated in the late 19th century amid Chicago's rapid industrialization and preparations for the 1893 World's Columbian Exposition, which highlighted the city's infrastructure ambitions and spurred investments in transportation and commercial spaces.[55]
In Houston, "The Loop" commonly refers to the Inner Loop, the urban zone enclosed by Interstate 610, a 38-mile freeway that delineates the city's core neighborhoods, including downtown. Downtown Houston within this Inner Loop underwent transformative growth in the 1970s, evolving from expansive surface parking lots into a skyline of high-rises facilitated by tax increment reinvestment zones (TIRZ) established to incentivize commercial and residential development.[56] Key examples include the JPMorgan Chase Tower, completed in 1982 as the city's tallest structure at 1,002 feet, and other 1970s-era skyscrapers like Pennzoil Place, which symbolized Houston's oil-driven economic boom and urban densification efforts.[57]
Another notable example is Baltimore's Inner Loop, a proposed circumferential interstate highway system planned in the 1950s as part of the federal Interstate Highway program to encircle the city's downtown and improve access to the central business district.[58] Only a 1.4-mile segment, designated as Interstate 170 and dubbed the "Highway to Nowhere," was constructed in the 1970s through West Baltimore neighborhoods, displacing over 1,500 residents and demolishing nearly 1,000 homes and 60 businesses before opposition halted further expansion.[59] Demolition of this incomplete spur began in 2010 to reconnect divided communities, with full redevelopment efforts receiving $85.5 million in federal funding by January 2025 for green spaces, housing, and transit-oriented projects.[60][61]
Chicago's Loop holds enduring cultural significance as a nexus for finance and the performing arts, housing institutions like the Chicago Mercantile Exchange and the Theater District, where over 30 venues draw millions annually.[53] By 2025, ongoing developments, including new residential high-rises and retail openings, have reinforced its role as an economic engine, with arts and culture events generating $280 million in direct impact during the first quarter alone and boosting pedestrian traffic post-pandemic.[62][63]
Natural and geographical features
Oxbow lakes represent a classic example of U-shaped loops in natural river systems, formed through the process of meander cutoff in meandering rivers.[64] In this process, rivers develop pronounced bends due to differential erosion on the outer concave banks and sediment deposition on the inner convex banks, causing adjacent meanders to narrow over time.[64] During high-flow events such as floods, water breaches the narrow neck between two bends, creating a shortcut channel that diverts the main flow and isolates the abandoned loop.[64] The inlet to the isolated segment gradually silts up through sedimentation, transforming it into a standalone U-shaped lake disconnected from the river.[64]
A prominent illustration of this pre-cutoff stage is Horseshoe Bend along the Colorado River in Arizona, where the river forms a tight, nearly complete U-shaped meander loop approximately 1,000 feet deep.[65] This feature emerged around 5 to 6 million years ago amid the uplift of the Colorado Plateau, as the river eroded through softer Navajo sandstone layers while maintaining its meandering path against resistant rock.[65] Continued outer-bank erosion at this site exemplifies the dynamic forces that could eventually lead to an oxbow lake if the neck fully erodes.[65]
In oceanography, loop currents manifest as large-scale circulatory features, such as the Loop Current in the Gulf of Mexico, which transports warm water northward through the Yucatan Channel, loops anticyclonically, and exits via the Straits of Florida at approximately 27.6 Sverdrups.[66] This current periodically extends and becomes unstable, shedding massive anticyclonic eddies known as Loop Current Eddies (LCEs) that propagate westward, influencing regional circulation and nutrient distribution.[66] Satellite monitoring of these dynamics began in the late 1970s with early infrared sensors for sea surface temperature, evolving to continuous altimetry observations since 1992 that track eddy separation events, with notable shedings occurring seasonally in late summer and early spring.[67]
Geological loops also appear in entrenched river canyons, as seen in the Goosenecks of the San Juan River at Goosenecks State Park in Utah, where the river—a tributary of the Colorado—carves sinuous, goose-neck-like meanders over 1,000 feet deep into layered sedimentary rocks.[68] These loops formed when the river meandered lazily across a flat, low-gradient plain in an ancient depositional environment of limestones, shales, and sandstones from 300 million years ago, only to become deeply incised 15 to 20 million years ago as the Colorado Plateau uplifted, preserving the original winding pattern while the canyon downcut rapidly.[68]
Such loop formations significantly influence local hydrology and erosion dynamics. Meanders and resulting oxbows alter water flow paths, promoting floodwater storage in abandoned loops that enhances groundwater recharge and reduces downstream peak flows during high-water periods.[69] They also modulate erosion rates, with accelerated bank scouring on outer bends driving meander growth at rates tied to channel sinuosity, while cutoffs shorten the channel and redistribute energy, potentially slowing migration in reformed segments but increasing incision elsewhere.[69] These shapes can be briefly modeled using sinusoidal mathematical curves to approximate meander wavelengths and amplitudes in river planforms.[69]
Transportation
Rail and transit systems
Balloon loops are specialized track configurations in rail systems designed to reverse the direction of trains without the need for shunting operations or turntables, typically forming a circular or teardrop-shaped extension at the end of a line.[70] These loops enable efficient turnaround by allowing a train to enter, loop around, and exit in the opposite direction on the same tracks, which is particularly useful for terminal stations with limited space. Engineering specifications for balloon loops emphasize minimum curvature radii—often 500 to 1,000 feet (152 to 305 meters) depending on train length and speed—to prevent excessive wear on wheels and tracks, while grades are kept under 2% to maintain safe operations under gravity and adhesion limits.[71] In the London Underground, the Kennington Loop on the Northern line exemplifies this design; opened in 1926 as part of the City and South London Railway extension, it allows southbound trains from the Charing Cross branch to reverse direction after passengers alight at Kennington station, with the single-track loop spanning approximately 0.5 miles (0.8 km) and accommodating up to 1,000 passengers per train in under five minutes.[72]
A loop line in railway terminology refers to a secondary branch that diverges from the main line and rejoins it further along, forming a circuit that facilitates access to intermediate areas without requiring a full reversal.[73] This configuration enhances connectivity for freight or passenger services while minimizing disruptions to primary traffic flows. Passing loops, a subset often integrated into single-track sections, provide sidings where faster trains can overtake slower ones, typically 1 to 2 kilometers (0.6 to 1.2 miles) long to allow safe clearance and signaling coordination.[74] On single-track railways, these loops are essential for maintaining schedules, with signals ensuring one train waits in the loop while the other passes on the main track, as standardized in global rail engineering practices to support bidirectional operations.[75]
In the history of U.S. rail passenger service, the formation of Amtrak on May 1, 1971, marked the end of many pre-existing routes operated by private railroads, with over half of the nation's intercity trains discontinued on that date to consolidate operations under the new national corporation.[76] Modern circle routes, by contrast, thrive in integrated metro systems; the Moscow Metro's Koltsevaya (Ring) line, constructed between 1950 and 1954, forms a 12-mile (19.5 km) orbital circuit with 12 stations that interconnects all radial lines, handling over 760,000 daily passengers and enabling seamless transfers in one of the world's busiest underground networks.[77] This design prioritizes circumferential flow to alleviate congestion at central hubs, a model that has influenced global transit planning since its completion.[78]
Road and vehicular systems
Ring roads, also known as beltways or circumferential highways, encircle urban areas to facilitate traffic flow around city centers, reducing congestion on radial routes. These loop configurations allow vehicles to bypass inner-city streets, connecting suburbs and providing efficient access to surrounding regions. In the United Kingdom, the M25 motorway serves as a prominent example, forming a 117-mile orbital route around Greater London that was fully opened on October 29, 1986, by Prime Minister Margaret Thatcher.[79] The M25 handles approximately 15% of the UK's motorway traffic and incorporates smart motorway technology, first introduced in 1995 as a controlled system with variable speed limits and electronic signage to manage congestion dynamically.[80] Further enhancements, including all-lane running without permanent hard shoulders, were implemented in sections starting in 2014 to increase capacity without widening the entire roadway.[81]
In the United States, loop routes integrated into interstate systems exemplify urban planning strategies from the mid-20th century, designed to delineate and support metropolitan growth. Houston's Interstate 610, often called the 610 Loop, is a 38-mile auxiliary highway that encircles the city's inner core, with construction segments beginning in the early 1960s as part of a broader "Defense Loop" concept originally proposed in the 1930s and approved by voters in 1941 for military and civilian mobility. By the late 1960s, key portions like the West Loop were completed, enabling radial spokes such as I-10 and I-45 to intersect efficiently and fostering development of commercial and residential zones within and adjacent to the loop.[82] This design has defined Houston's spatial organization, concentrating employment centers and educational institutions inside the loop while accommodating suburban expansion outward.[83]
Spiral loops in road engineering address challenging topography by using helical or tightly curving alignments to achieve elevation gain without excessive gradients, common in mountainous regions. In Yosemite National Park, the Tioga Road (California State Route 120) exemplifies this through its series of looping switchbacks ascending from the western entrance near Crane Flat at 6,192 feet to Tioga Pass at 9,945 feet, a total gain of over 3,700 feet across approximately 46 miles.[84] These configurations, part of the road rebuilt in the 1930s and 1960s, allow vehicles to navigate the Sierra Nevada's steep terrain safely, transitioning from forested valleys to alpine meadows while minimizing environmental impact.[85] The spirals near the pass provide panoramic views of Yosemite's high country, though the road closes seasonally due to snow from late fall to spring.[86]
Innovative underground loop systems represent a contemporary evolution, utilizing tunnel networks for high-speed vehicular transport in dense urban environments. The Boring Company's Vegas Loop, developed by Elon Musk's infrastructure firm, features a series of subterranean tunnels exclusively for Tesla electric vehicles, beginning with the Las Vegas Convention Center (LVCC) Loop that opened in April 2021.[87] Spanning initially 1.7 miles with two tunnels and three stations, the system expanded to 2.1 miles and five stations by 2024, incorporating connectors to nearby resorts like Resorts World (opened 2022) and Westgate (2024).[88] In 2025, the Encore-LVCC Connector added a short 0.1-mile link, enhancing direct access between the convention center's Center Hall and the Encore resort, with the overall Vegas Loop now operational across multiple segments totaling over 10 miles as of November 2025.[89] These loops employ autonomous or semi-autonomous driving to shuttle passengers at speeds up to 40 mph, aiming to alleviate surface traffic in Las Vegas while prioritizing point-to-point travel over traditional mass transit.[90] In October 2025, the company faced a fine of nearly $500,000 for an environmental violation involving the dumping of drill fluid into the sewer system.
Amusement and recreation
Roller coasters
In roller coaster design, a vertical loop is an inversion element where the track completes a full 360-degree circle, inverting riders twice while subjecting them to significant gravitational forces. This element provides intense thrills by combining circular motion with changes in velocity and direction, requiring precise engineering to ensure rider safety and comfort. Modern vertical loops typically feature a clothoid (teardrop) shape rather than a perfect circle to manage forces more effectively.[91]
The concept of vertical loops dates back to the late 19th century, but early attempts suffered from excessive g-forces that caused discomfort and injuries, leading to their decline by the 1910s. The first modern vertical loop on a full-circuit steel roller coaster was introduced in 1976 on Revolution (originally Great American Revolution) at Six Flags Magic Mountain, marking a revival in looping designs with improved tubular steel track technology. This 70-foot-diameter loop used a clothoid profile to reduce the peak forces experienced by riders compared to earlier circular attempts.[92]
The physics of navigating a vertical loop demands a minimum entry speed to prevent riders from falling out at the top, where centripetal force must counteract gravity. For a circular loop of radius r, the minimum speed v at the bottom is given by v \geq \sqrt{gr \left(1 + \frac{2h}{r}\right)}, where g is gravitational acceleration and h is the height from the bottom to the top of the loop (typically $2r, yielding v \geq \sqrt{5gr}). This ensures the speed at the top is at least \sqrt{gr}, maintaining positive normal force on riders. Clothoid shapes, with a gradually increasing radius from bottom to top, further mitigate g-forces by allowing slower entry speeds and more consistent acceleration, peaking at around 4-5 g rather than 6 g or higher in circular designs.[93][94]
In more recent innovations up to 2025, Jurassic World VelociCoaster at Universal's Islands of Adventure (opened 2021) incorporates a non-circular dive loop, an elongated inversion that combines a half-loop ascent with a steep descent, enhancing thematic immersion while distributing forces. Examples from 2025 include the Hot Wheels Looping Coaster at Mattel Adventure Park, which features vertical loops in a family-friendly format. These designs showcase evolving loop variations beyond traditional circles, prioritizing both thrill and physiological comfort.[95][96]
Safety standards for roller coaster loops are governed by the ASTM International F24 Committee on Amusement Rides and Devices, which outlines requirements in ASTM F2291 for design, testing, operation, and maintenance to limit g-forces and ensure structural integrity. Historical accidents, such as the 1930 derailment on the Big Dipper at Krug Park in Omaha (killing four due to mechanical failure during high-speed descent), underscored the need for these standards, leading to stricter inspections and the abandonment of poorly engineered loops in the 1930s. Modern adherence to ASTM guidelines has significantly reduced such risks, with no fatal loop failures reported in the U.S. since the mid-20th century.[97][98]
Sports and games
In tennis, a loop refers to a high-arching shot or looped swing path, particularly in topspin forehands that create significant spin and dip, allowing the ball to clear the net generously before dropping sharply into the court. This technique gained prominence in the 2000s through players like Rafael Nadal, whose extreme topspin forehand—often finishing with a lasso-like loop around the head—revolutionized baseline play on clay surfaces, generating up to 3,300 revolutions per minute for enhanced control and bounce. Nadal's adoption of this looped swing, influenced by his coach Toni Nadal, emphasized wrist snap and racquet head acceleration, making it a staple for modern aggressive playstyles.[99][100]
In cross-country skiing and biathlon, loops denote circuitous trail paths that athletes repeat during races, forming the backbone of endurance events at the Olympics. The men's 12.5 km biathlon pursuit, for instance, involves multiple loops of varying lengths—typically 2.5 to 5 km each—combining freestyle skiing with shooting stages, where competitors ski the full distance while managing penalties for missed shots. These loops, designed for tactical pacing and recovery, have been a fixture since biathlon's Olympic debut in 1960, with formats testing aerobic capacity over looped terrain in events like the 2022 Beijing Games. Penalty loops, shorter 150-meter detours skied for each missed target, add strategic depth without time penalties, forcing athletes to balance speed and accuracy across the main race circuits.[101][102][103]
Basketball dribbling incorporates loops through advanced tricks like the behind-the-back loop, where the ball arcs in a curved path around the player's body to evade defenders, enhancing ball-handling fluidity in fast breaks or isolation plays. This move, part of progressive dribbling drills, involves bouncing the ball low and wide behind the hips to create a looping trajectory, often practiced in sequences with crossovers for better court vision and deception. Popularized in streetball and NBA training since the 1990s, it demands wrist control and body feints, as seen in routines by players like Jamal Crawford, who used such loops to break ankles in high-stakes games.[104][105]
The board game Loopin' Louie, released in 1992 by Milton Bradley, features looped mechanical paths central to its gameplay, where a battery-powered plane circles a central track, dipping into players' chicken coops to knock out tokens unless deflected by levers. Named Germany's Children's Game of the Year in 1994, it emphasizes quick reflexes and timing over four radial loop arms, accommodating 2-4 players aged 4 and up in short, chaotic rounds. By the 2010s, it evolved into adult variants like drinking games in Europe, but its core looped circuit design remains a hallmark of family dexterity gaming.[106][107][108]
In esports racing simulations, loop strategies involve optimizing laps on closed-circuit tracks in titles like iRacing and Gran Turismo, focusing on consistent line-taking, tire management, and pit timing to minimize lap-time variance up to 2025. In iRacing's GT Sprint series, players employ loop-aware tactics such as apex clipping on banked turns and fuel-saving in endurance loops, which have influenced professional esports leagues like the iRacing World Championship Grand Prix Series. Gran Turismo 7's online races similarly reward loop strategies through dynamic weather adaptations on looped courses, with AI-assisted setups enabling precise corner sequencing for competitive edges in 2024-2025 seasons. These approaches, blending simulation realism with real-time decision-making, underscore the genre's growth in esports, where loop efficiency can shave seconds per circuit in multi-lap events.[109][110]
Music
In music production, an audio loop refers to a repeating segment of sound, often used in electronic music to create rhythmic or melodic patterns. This technique was pioneered by composer Terry Riley in the early 1960s through his experiments with tape loops, which involved physically splicing and replaying magnetic tape to generate continuous, evolving soundscapes. Riley's innovations, such as in his 1963 piece Music for The Gift, laid foundational groundwork for minimalist and ambient genres by enabling real-time manipulation of recorded audio.[111][112]
Live looping extends this concept to performance, where musicians record and layer audio in real time using hardware devices. The Boss RC-300 Loop Station, introduced as a flagship model, features three synchronized stereo tracks, dedicated footswitches, and effects like transpose and flanger, allowing performers to build complex arrangements on stage with up to three hours of internal recording. Artists such as Ed Sheeran have popularized this method, constructing songs layer by layer—starting with a beatbox rhythm, adding guitar riffs, and overlaying vocals—to create full-band simulations during solo shows.[113][114]
Several musical groups have adopted the name Loop, most notably the British noise rock band formed in 1986 in Croydon, London, led by Robert Hampson. Active primarily from 1986 to 1991, the band blended droning space rock with repetitive, psychedelic structures, releasing influential albums like Heaven's End (1987), which peaked at number four on the UK Indie Charts and featured tracks such as "Soundhead" and "Straight to Your Heart." Their sound, characterized by heavy guitar feedback and hypnotic rhythms, influenced shoegaze and post-rock scenes before a reunion in the 2010s.[115][116]
Loop pedals and software have broader applications across genres, including folk music, where they enable solo artists to layer acoustic elements like guitar strums, harmonies, and percussion for richer arrangements. In digital audio workstations (DAWs), tools like Ableton Live's loop warping—introduced with the software's debut in 2001—allow users to stretch, compress, or synchronize audio clips to a project's tempo without pitch distortion, facilitating seamless beat-matching and creative remixing.[117][118]
By 2025, looping has evolved with AI-assisted generation integrated into DAWs, enabling automated creation of genre-specific loops and stems from text prompts or audio analysis to accelerate composition workflows. Tools in platforms like FL Studio and Ableton Live now incorporate AI for suggesting harmonic progressions and rhythmic variations, enhancing accessibility for producers while preserving artistic control.[119][120]
Film and television
In film and television, the concept of a "loop" often manifests as the time loop trope, where characters are trapped in a repeating cycle of events, typically a single day or sequence, until a personal or cosmic resolution is achieved. This narrative device gained prominence with the 1993 comedy Groundhog Day, directed by Harold Ramis, in which weatherman Phil Connors, played by Bill Murray, relives February 2 repeatedly in Punxsutawney, Pennsylvania, using the repetition to evolve from cynicism to empathy.[121] The film's success popularized the trope, influencing subsequent works by emphasizing themes of self-improvement and existential reflection within constrained temporal boundaries.[122]
Building on this foundation, action-oriented interpretations appeared in films like Edge of Tomorrow (2014), directed by Doug Liman and based on Hiroshi Sakurazaka's novel All You Need Is Kill. Here, Major William Cage, portrayed by Tom Cruise, enters a loop triggered by alien blood during an invasion, resetting each death to refine his combat skills and ultimately break the cycle. This variation shifted the trope toward high-stakes strategy and survival, diverging from Groundhog Day's introspective focus while retaining the core mechanic of iterative learning.[123]
Beyond narrative devices, "loop" refers to physical film loops in experimental cinema, particularly 16mm short films spliced into seamless cycles for continuous projection. Pioneered in the mid-20th century, these loops allowed filmmakers to explore abstraction and perception without traditional plot progression. Stan Brakhage, a key figure in American avant-garde film, employed this technique in works like Rounds (2001, though rooted in his 1960s practices), creating hand-painted 16mm loops that emphasized visual rhythm and non-linear temporality.[124] Brakhage's loops, often projected in galleries or small theaters, challenged conventional storytelling by prioritizing sensory immersion over linear time.[125]
In television, loop narratives have featured in episodic formats, adapting the trope for serialized drama. The Netflix series Russian Doll (2019), created by Natasha Lyonne, Leslye Headland, and Amy Poehler, centers on Nadia Vulvokov (Lyonne), who dies repeatedly at her 36th birthday party on March 30, 2019, resetting to the same moment in a glitch-like temporal loop.[126] The show intertwines personal trauma with metaphysical elements, using the loop to unpack themes of mortality and interconnectedness, as Nadia collaborates with another looper, Alan.[127]
Production techniques involving loops include "loop dubbing," also known as automated dialogue replacement (ADR), where actors re-record lines in a studio while viewing the footage in a continuous loop to achieve precise lip-sync and emotional matching. This post-production method, standard since the sound era, addresses on-set audio issues like noise or accents, ensuring seamless synchronization in the final edit.[128] Directors cue actors with visual "bleeps" aligned to mouth movements, repeating the loop until the performance aligns perfectly with the visuals.[129]
As of 2025, loop narratives continue to appear in streaming sci-fi anthologies, with episodes exploring temporal recursion amid broader speculative themes. For instance, the seventh season of Black Mirror on Netflix includes standalone stories delving into time loops as metaphors for digital entrapment and ethical dilemmas in AI-driven simulations.[130] These segments build on the trope's legacy, adapting it to contemporary concerns like algorithmic repetition in virtual realities.[131]
Time loops have been a recurring motif in literature, allowing authors to explore themes of regret, choice, and existential repetition. In Ken Grimwood's 1986 novel Replay, protagonist Jeff Winston dies of a heart attack at age 43 and awakens as his 18-year-old self in 1963, retaining memories of his previous life; this cycle repeats multiple times, enabling him to relive decades and experiment with alternate paths, from career ambitions to personal relationships.[132] The narrative delves into the psychological toll of endless replays, highlighting how accumulated knowledge both empowers and burdens the character across iterations.[133]
In comic books, loop concepts often intersect with multiverse structures, creating cyclical alterations to reality. DC Comics' Flashpoint (2011), written by Geoff Johns, features Barry Allen (The Flash) inadvertently reshaping the DC Universe through time manipulation, resulting in a divergent timeline that echoes multiverse loops by forcing repeated attempts to restore the original continuity.[134] This event not only reboots the DC multiverse but illustrates narrative loops where heroic interventions perpetuate alternate realities, influencing subsequent storylines like the New 52 era.[135]
Video games represent another medium where loop mechanics drive gameplay and storytelling, emphasizing repetition for progression. Nintendo's The Legend of Zelda: Majora's Mask (2000) employs a three-day time loop in the land of Termina, where player character Link must avert a cataclysmic moon crash by playing the Song of Time to reset the cycle, preserving key events through owl statues while managing escalating urgency.[136] This structure fosters strategic replay, mirroring themes of inevitability and redemption as the loop reveals deeper character backstories and side quests.[137]
Postmodern literature frequently incorporates looped narratives to challenge linear storytelling and reader perception. Mark Z. Danielewski's House of Leaves (2000) constructs a labyrinthine structure through nested manuscripts and footnotes, where the central tale of a house larger inside than out creates recursive loops in its metafictional layers, blurring boundaries between reality, fiction, and interpretation.[138] The novel's typographic experiments, such as spiraling text and mirrored passages, evoke endless cycles that trap readers in disorienting repetition, amplifying horror through formal innovation.[139]
By 2025, digital media has expanded loop narratives into interactive fiction, enabling player-driven branching cycles that evolve with choices. Games like In Stars and Time (2023), an indie RPG, feature a time loop where the protagonist relives a dungeon crawl indefinitely, using accumulated knowledge to alter outcomes and uncover emotional arcs in a pixel-art world.[140] Platforms such as itch.io host text-based interactive fiction with time loop elements, like her tears were my light (2022), where branching decisions create cyclical puzzles blending mystery and introspection.[141] These developments integrate loops with procedural generation, allowing for personalized replays that extend literary traditions into participatory experiences.
People
Call Me Loop, born Georgia Buchanan on December 23, 1990, is an English pop singer-songwriter.[142] She debuted with the single "Looking at You" in 2016, followed by independent extended plays including Call Me Loop (2017), Give 'n' Take (2018), and Drama (2019).[143] Her work has garnered support from BBC Radio 1, MTV, and publications like GQ and Official Charts.[144] In 2023, she released singles such as "Year of the Ex" before announcing in July her decision to stop releasing her own music and shift focus to songwriting for other artists, a direction she has pursued since.
Uno Loop (May 31, 1930 – September 8, 2021) was an Estonian singer, guitarist, songwriter, and music educator whose career spanned over seven decades.[145] Beginning in the early 1950s, he recorded more than 250 songs, many self-composed, and performed globally, promoting Estonian music through collaborations like the 1971 bossa nova album with Marju Kuut.[145][146] As a teacher, he influenced generations of musicians in Estonia. No major posthumous recognitions were reported between 2021 and 2025.
Victoria Loop is an English musician known for her contributions as a guitarist and brass player, particularly through live performances with the rock band Half Man Half Biscuit since the mid-2010s. She has joined the group on stage playing tenor horn, cornet, and bass guitar, earning the nickname "The 5th Biscuit" among fans.[147]
Liza Loop, active since 1975, is an American educational technology pioneer and founder of LOOP Center, Inc., which supports innovative projects in learning and digital media. While primarily recognized for introducing early computers like the Apple I into educational settings, her work encompasses broader contributions to educational media, including explorations of technology in creative and interdisciplinary contexts. As of 2025, she continues as Executive Director of LOOP Center, contributing to virtual museums on computing history.[148][149]
Other notable individuals
Steven Tyler Loop, born August 4, 2001, in Lucas, Texas, is an American football placekicker who played college football for the Arizona Wildcats from 2020 to 2024.[150] During his collegiate career, he converted 67 of 80 field goal attempts (83.8%) and demonstrated long-range accuracy with successful kicks from beyond 50 yards, including a 51-yard field goal against Oregon State in 2023.[151] Loop was selected by the Baltimore Ravens in the sixth round (186th overall) of the 2025 NFL Draft, marking him as one of two kickers chosen that year. Following the release of longtime Ravens kicker Justin Tucker in May 2025 due to an NFL investigation into allegations of sexual misconduct (resulting in a 10-game suspension), Loop became the team's primary kicker. As of November 2025, in his rookie season, he has converted 19 of 21 field goals (90.5%) through 10 games.[150][152]
In the field of theoretical physics, Carlo Rovelli (born May 3, 1956) is an Italian physicist renowned for his foundational contributions to loop quantum gravity (LQG), a non-perturbative approach to quantizing general relativity that emerged in the late 1980s.[153] Rovelli, who has held positions at institutions including the University of Pittsburgh and Aix-Marseille University, co-developed key aspects of LQG, including the use of spin networks to represent quantized spacetime geometry, as detailed in seminal works like his 1993 paper with Lee Smolin on the structure of quantum spacetime.[154] His research emphasizes background-independent formulations, resolving singularities in black holes and the Big Bang through discrete spacetime at the Planck scale.[155] Rovelli has also popularized these concepts through books such as Quantum Gravity (2004), co-authored with Ashtekar and Smolin, which outlines the mathematical framework of LQG. In 2024, he received the Lewis Thomas Prize for his writings on scientific matters.[156][157]
Abhay Ashtekar, an Indian-American physicist born July 5, 1949, is another pioneer of loop quantum gravity, having introduced the Ashtekar variables in 1986 that reformulated general relativity in terms of new canonical variables, facilitating its quantization.[158] Ashtekar's work at Pennsylvania State University has advanced loop quantum cosmology, predicting a bounce replacing the Big Bang singularity, as explored in his 2006 paper "Gravity, Geometry and the Quantum."[159] By 2025, ongoing research in LQG, including contributions from emerging scholars at conferences like Loops'24, continues to refine these models, with Ashtekar leading efforts in homogeneous-isotropic sectors of the theory.[160]
Brands and enterprises
Companies and services
Loop Mobile was a mobile network operator in India that primarily served the Maharashtra telecom circle, including Mumbai, with approximately three million subscribers at the time of its acquisition.[161] In 2014, Bharti Airtel acquired Loop Mobile's customer base and certain tower assets, integrating its operations into Airtel's network following regulatory approvals.[162]
Loop is a mobile virtual network operator (MVNO) in Bulgaria, launched in 2006 by Mobiltel (now A1 Bulgaria) as a youth-oriented brand offering affordable mobile services on the host network's infrastructure. It operates as the largest MVNO in the country, targeting younger demographics with specialized plans and promotions.[163]
Loop Internet is a fiber-optic internet service provider based in Scranton, Pennsylvania, founded in 2015 to deliver high-speed broadband in northeastern Pennsylvania.[164] The company expanded its fiber network throughout 2025, including new constructions in areas like Hazleton and Kingston, before being acquired by Greenlight Networks on October 1, 2025, to accelerate regional growth.[165][166]
Loop is a global reuse platform developed by TerraCycle, launched in 2019 to enable circular shopping through durable, reusable packaging for consumer products.[167] The service partners with major brands such as Procter & Gamble to offer refillable items like shampoos and detergents, where packaging is collected, cleaned, and reused after consumer use.[168] In 2021, Loop had secured $25 million in funding to scale its operations internationally.[168]
Loop Capital is a full-service investment bank and financial advisory firm founded in 1997 in Chicago, specializing in municipal bond underwriting and sales.[169] The firm has grown to over 260 employees across more than 20 offices, participating in over $1.6 trillion in municipal bond transactions for public sector issuers.[170] It provides brokerage, advisory, and capital-raising services to corporations, governments, and institutions.[169]
Products and initiatives
Loop reusable containers, launched by TerraCycle in 2019, represent a circular economy initiative designed to replace single-use packaging with durable, returnable alternatives for consumer goods such as laundry detergent and vitamins.[167] Customers purchase products in these containers from participating retailers, pay a refundable deposit, and return the emptied packaging via mail or drop-off points for cleaning, refilling, and reuse, thereby reducing plastic waste.[171] By 2021, the program had expanded beyond e-commerce to include in-store options at partners like Kroger and Tesco, with initial pilots in the U.S. and Europe demonstrating viability for scalable reuse systems.[172]
Microsoft Loop, introduced as a productivity app in 2021, facilitates real-time collaboration for teams through integrated workspaces that combine notes, tasks, and files across Microsoft 365 applications.[48] Recent updates have incorporated AI integrations, such as agents in Microsoft Teams for workflow automation and content generation, enabling embedding of Loop components like Pages into operations for improved efficiency.[173]
Loop earplugs, developed by the Belgian company founded in 2016, provide stylish hearing protection tailored for concerts and live events, featuring acoustic filters that reduce noise while preserving sound clarity.[174] The flagship Experience 2 model offers 17 dB noise reduction (SNR), with customizable options like the Plus variant that includes an attachable Mute accessory for additional 5 dB attenuation on demand.[175] By 2025, the product line had sold millions of units globally, emphasizing reusable, jewelry-like designs in various colors to appeal to music enthusiasts and promote ear health without muffling audio quality.[176]
Closed Loop Ventures, the venture capital arm of Closed Loop Partners, operates as an investment fund dedicated to advancing sustainability through early-stage funding in circular economy solutions, including plastics recycling, sustainable packaging, and supply chain innovations.[177] Since its inception, the fund has raised over $50 million for its second iteration in 2021, supporting portfolio companies that achieve net-positive environmental impacts while targeting market-leading returns.[178] As of 2025, its impact report highlights investments yielding significant waste diversion, such as enhanced recycling technologies, underscoring the fund's role in scaling sustainable initiatives across consumer goods sectors.[179]
Other uses
Everyday objects and mechanics
Belt loops in clothing serve as fabric reinforcements designed to secure belts around the waist, providing both functional support and structural integrity to garments like pants and trousers. These loops, typically made from the same material as the clothing or reinforced fabric, emerged as a practical feature in the early 20th century when men's pant waistlines began to drop lower, with Levi's introducing them on 501 jeans in 1922 as an alternative to traditional suspenders.[180] They became more standardized in casual and workwear by the mid-20th century, evolving from simple stitched tabs into durable, evenly spaced reinforcements that distribute belt tension evenly to prevent fabric tearing.[181]
Loop fasteners, such as the hook-and-loop system exemplified by Velcro, represent a versatile mechanical closure used in everyday items like clothing, footwear, and bags. Invented in 1941 by Swiss engineer George de Mestral, this mechanism consists of two fabric strips: one covered in tiny, rigid hooks and the other in soft, fuzzy loops, allowing the hooks to engage and disengage repeatedly with minimal force for easy fastening and release.[182] De Mestral's design, inspired by burr seeds sticking to clothing, revolutionized adjustable closures by providing a reliable alternative to buttons or zippers, with the interlocking action creating a secure hold under tension while permitting quick separation.[182]
In jewelry and crafts, wire loops form essential connections for pendants, earrings, and chain links, often created through techniques that ensure durability and permanence. Crafters typically form basic loops by bending wire around mandrels or pliers, then secure them with wrapping methods where excess wire is coiled around the base to prevent unraveling under stress.[183] For closed loops requiring enhanced strength, such as jump rings or charm attachments, soldering techniques are employed: the wire ends are butted together, fluxed, and heated with a torch to melt solder into a seamless joint, eliminating gaps and boosting tensile resistance against pulling forces.[184]
Mechanical loops in machinery, particularly timing belts in internal combustion engines, function as toothed rubber bands that loop around synchronized pulleys to maintain precise timing between the crankshaft and camshaft. This design prevents slippage by meshing the belt's teeth directly into pulley grooves, ensuring consistent power transfer and avoiding catastrophic engine misalignment that could damage valves or pistons.[185] Proper tensioning of these loops is critical, as inadequate adjustment can lead to wear, while the toothed profile inherently resists sliding under high rotational speeds.[185]
Safety applications of loops are prominent in climbing gear, where lifeline loops—such as tie-in points on harnesses—provide secure anchorages for ropes and belay systems to arrest falls. These loops, constructed from high-strength webbing, must withstand extreme forces and are regulated by UIAA 105 and EN 12277 standards, which mandate minimum static strengths of at least 15 kN for belay loops and 10 kN for leg loop attachments to ensure reliability in dynamic loads.[186] UIAA certification verifies that such loops maintain integrity without elongation or failure under repeated stress, forming the critical link in a climber's safety chain.[187] In these contexts, the physics of tension in loops dictates that even distribution of force across the material prevents localized failure, enhancing overall system stability.
Miscellaneous concepts
In knitting, the fundamental process involves creating interlocking loops of yarn to form fabric, where each stitch represents a loop pulled through a previous one using needles. This technique dates back centuries and underpins various patterns; for instance, garter stitch, one of the simplest, is produced by knitting every row, resulting in a textured fabric of horizontal ridges formed by these interlocking loops.[188][189]
The Devil's loop, also known as the Chinese rings puzzle or Baguenaudier, is a classic disentanglement toy featuring a looped handle interlocked with a series of rings threaded onto pillars, challenging users to separate them without force. Originating in ancient China over 2,000 years ago but gaining widespread popularity as a 19th-century European toy through trade routes, it exemplifies mechanical illusions where apparent linkages create puzzling constraints, often requiring up to 341 sequential moves for the standard nine-ring version.[190][191]
In electronics, loop gain refers to the amplification factor, expressed as a ratio, within a feedback circuit of an amplifier, determining stability and overall performance by quantifying how much the output signal reinforces or attenuates the input through the loop. This concept is essential for designing reliable audio and signal processing systems, where appropriate loop gain ensures controlled response without oscillation.[192]
The idiom "in the loop" denotes being kept informed or involved in ongoing communications or decisions, a phrase rooted in World War II military terminology for maintaining personnel within command-and-control feedback loops to ensure coordinated operations. It evolved from practices in aviation and radio networks, where excluding someone from the "loop" meant they were out of the information circuit, and has since entered general usage to describe inclusion in professional or social updates.[193]
As of 2025, loop economies—framed within circular economy principles—have emerged in urban planning as strategies to close resource cycles in city design, emphasizing material reuse and waste minimization beyond commercial branding initiatives. For example, initiatives in cities like Amsterdam integrate loop systems into infrastructure, such as modular building components that enable disassembly and recycling, reducing environmental impact by up to 75% in construction emissions compared to linear models.[194][195]