Fact-checked by Grok 2 weeks ago

Transformation

A transformation is a process or function that changes the form, structure, position, or state of an object, system, or concept, often while preserving certain properties. The term is used across various disciplines, including mathematics, natural sciences, arts, business, and society.^[1] In mathematics, a transformation is a function that maps elements from one geometric space or set to another, often preserving specific properties such as distances, angles, or orientation.^[2] In the context of geometry, transformations alter the position, size, or orientation of figures on a plane or in space without necessarily changing their intrinsic shape, encompassing operations like translations, rotations, reflections, and dilations.^[3]^[4] These mappings are typically one-to-one correspondences between sets of points, ensuring each point in the original figure (preimage) corresponds uniquely to a point in the transformed figure (image).^[4]^[2] Geometric transformations are classified into rigid (isometric) types, which preserve both lengths and angles—such as translations (slides), rotations (turns around a point), and reflections (flips over a line)—and non-rigid types like dilations, which scale figures by a factor relative to a center point.^[5]^[3] In linear algebra, transformations extend to vector spaces, where a linear transformation T: \mathbb{R}^n \to \mathbb{R}^m satisfies T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) and T(c\mathbf{u}) = cT(\mathbf{u}) for vectors \mathbf{u}, \mathbf{v} and scalar c, enabling representations via matrices for computations in higher dimensions.^[6] These concepts underpin applications in computer graphics, physics simulations, and engineering, where transformations model movements, symmetries, and projections.^[7] In natural sciences, transformations take on specialized meanings; for example, in biology, bacterial transformation involves the uptake of exogenous DNA by cells, altering their genetic traits.^[8] In chemistry, chemical transformations refer to reactions that change the composition or structure of molecules.^[9] In physics, Lorentz transformations describe how space and time coordinates change between inertial frames in special relativity, preserving the invariance of physical laws.^[10] Overall, the study of transformations provides frameworks for understanding change, symmetry, and processes across fields, with rigorous definitions often ensuring bijectivity (one-to-one and onto) in mathematical contexts to preserve structural integrity.^[2]^[5]

Mathematics

Geometric transformations

A geometric transformation is a bijective mapping that assigns to every point in a Euclidean space a new position, thereby changing the position, orientation, or size of geometric figures while potentially preserving specific properties such as distances or angles.^[11] These transformations are fundamental in Euclidean geometry, where they enable the study of congruence and similarity between shapes.^[12] The primary types of geometric transformations include translations, rotations, reflections, and dilations. A translation shifts every point of a figure by the same fixed distance and direction, preserving distances, angles, and orientation; for example, moving a triangle 5 units to the right maintains its exact shape and size.^[13] Rotations turn a figure around a fixed point by a specified angle, also preserving distances and angles; rotating an equilateral triangle 120 degrees around its centroid maps it onto itself.^[14] Reflections flip a figure over a line, acting as a mirror image while preserving distances and angles, such as reflecting a square across its diagonal to produce a congruent copy.^[11] Dilations, or scalings, enlarge or reduce a figure from a fixed center point by a scale factor k, preserving angles but scaling distances by |k|; for instance, a dilation with k=2 doubles the side lengths of a rectangle while keeping its angles at 90 degrees.^[15] Transformations are classified as isometries if they preserve distances between points, ensuring that congruent figures remain congruent after mapping.^[14] Translations, rotations, and reflections are isometries, as they maintain not only distances but also areas and angles, forming the rigid motions of the plane.^[12] In contrast, dilations are non-isometric similarity transformations, which preserve angles and the ratios of distances but alter absolute sizes, useful for studying proportional figures.^[16] Historically, geometric transformations trace their roots to Euclidean geometry as outlined in Euclid's Elements around 300 BCE, where basic congruences were established through implicit mappings without explicit transformation theory.^[17] The modern framework emerged in the 19th century, particularly with Felix Klein's 1872 Erlangen Program, which unified geometries by classifying them according to their underlying groups of transformations that leave certain properties invariant.^[18] This approach elevated transformations to the core of geometry, viewing symmetry groups—collections of transformations that map a figure to itself—as central to understanding spatial invariances, such as the dihedral group for regular polygons.^[19] In applications, geometric transformations are essential in computer graphics for manipulating 2D and 3D models during rendering, such as applying rotations to orient objects in a virtual scene or dilations to adjust scales in animations.^[20] These operations enable efficient placement and viewing of complex graphics by composing multiple transformations into a single mapping.^[21]

Algebraic transformations

Algebraic transformations encompass a broad class of mappings in mathematics that operate on algebraic structures such as vector spaces, functions, and groups, often preserving key properties like linearity or symmetry. These transformations are fundamental in abstract algebra and analysis, enabling the reformulation of problems into more tractable forms without altering their essential characteristics. Unlike geometric transformations that focus on spatial changes, algebraic ones emphasize functional and structural mappings, typically represented through equations or matrices./03%3A_Linear_Transformations/3.01%3A_Definition_of_a_Linear_Transformation) Linear transformations, also known as linear maps, are functions T: V \to W between vector spaces V and W over a field (such as the real or complex numbers) that satisfy T(\alpha \mathbf{u} + \beta \mathbf{v}) = \alpha T(\mathbf{u}) + \beta T(\mathbf{v}) for all scalars \alpha, \beta and vectors \mathbf{u}, \mathbf{v} \in V. This linearity ensures that the transformation preserves vector addition and scalar multiplication, making it a cornerstone of linear algebra. The kernel of T, denoted \ker(T), is the set of vectors in V mapped to the zero vector in W, i.e., \ker(T) = \{ \mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0} \}, which forms a subspace of V. Similarly, the image of T, or range, denoted \operatorname{im}(T), is the subspace of W consisting of all vectors T(\mathbf{v}) for \mathbf{v} \in V. The rank-nullity theorem states that for finite-dimensional spaces, \dim(V) = \dim(\ker(T)) + \dim(\operatorname{im}(T)), relating the dimensions of these subspaces to the domain's dimension./16%3A_Kernel_Range_Nullity_Rank)^[22]^[23] In finite-dimensional spaces with chosen bases, linear transformations admit matrix representations. Specifically, for bases \{\mathbf{e}_i\} of V and \{\mathbf{f}_j\} of W, the transformation T is represented by a matrix A such that T(\mathbf{v}) = A \mathbf{v}, where \mathbf{v} is the coordinate vector of the input. The columns of A are the coordinates of T(\mathbf{e}_i) in the codomain basis, and operations like composition of transformations correspond to matrix multiplication. This matrix form facilitates computations, such as finding eigenvalues that reveal scaling behaviors under the transformation. Geometric interpretations, such as rotations represented by orthogonal matrices, illustrate these algebraic structures in spatial contexts.^[24]^[25] Integral transforms extend algebraic transformations to functions, converting problems in one domain (e.g., time) to another (e.g., frequency) via integration. The Fourier transform decomposes a function f(t) into its frequency components, defined as

\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt,

which is particularly useful for signal analysis by representing signals as sums of sinusoids. This transform is invertible, allowing reconstruction of the original function, and its properties, like convolution becoming multiplication in the frequency domain, simplify computations in analysis. The Laplace transform, F(s) = \int_{0}^{\infty} f(t) e^{-s t} \, dt for s in the complex plane, transforms differential equations into algebraic ones, aiding solutions of initial value problems by converting derivatives to multiplications by s. Both transforms are linear and operate on function spaces, preserving algebraic structure while enabling analytical insights./09%3A_Transform_Techniques_in_Physics/9.05%3A_Properties_of_the_Fourier_Transform)^[26]^[27] In group theory, algebraic transformations form groups under composition, known as transformation groups or symmetry groups, which capture invariances under repeated applications. A transformation group G acting on a set X consists of bijections g: X \to X such that composition and the identity preserve the structure, often modeling symmetries like those in geometric figures or algebraic equations. For instance, the general linear group GL(n, \mathbb{R}) comprises invertible linear transformations on \mathbb{R}^n, highlighting how group actions classify symmetries in vector spaces. These structures underpin advanced topics like representation theory, where transformations are realized as matrices acting on representations.^[28]^[29] Affine transformations, a generalization of linear ones incorporating translations, find applications in cryptography as substitution ciphers. An affine cipher encrypts plaintext letters via E(x) = (a x + b) \mod m, where a is coprime to modulus m (e.g., 26 for English alphabet), ensuring invertibility for decryption. This linear-algebraic mapping scrambles messages while remaining computationally simple, though vulnerable to frequency analysis; modern variants enhance security through composition with other operations. In optimization, linear transformations standardize problems in linear programming, such as converting inequalities to equalities via slack variables or negating objectives to switch maximization to minimization, facilitating solution by methods like the simplex algorithm. These reformulations preserve optimality without altering the feasible region's geometry.^[30]^[31]^[32]

Natural sciences

Biology and medicine

In biology and medicine, transformation refers to processes involving significant cellular or genetic alterations, particularly in microorganisms, cancer development, and therapeutic interventions. Genetic transformation is a fundamental process in bacteria where cells take up exogenous DNA from the environment, leading to heritable changes in their genotype and phenotype. This phenomenon was first demonstrated in 1928 by British bacteriologist Frederick Griffith through experiments with Streptococcus pneumoniae, where heat-killed virulent bacteria transformed live non-virulent strains into lethal ones when co-injected into mice, suggesting the transfer of a "transforming principle."^[33] In 1944, Oswald Avery, Colin MacLeod, and Maclyn McCarty confirmed that this transforming principle was DNA, not protein, by purifying DNA from virulent pneumococci and showing it induced stable transformation in non-virulent strains, laying the groundwork for understanding DNA as the genetic material. This discovery enabled subsequent research into horizontal gene transfer mechanisms, which contribute to bacterial evolution, antibiotic resistance, and genetic diversity. The process of bacterial genetic transformation occurs in distinct steps, primarily in naturally competent species like Streptococcus and Bacillus. First, competence development is induced by environmental signals such as nutrient limitation or quorum sensing, triggering the expression of genes for DNA uptake machinery.^[34] Next, free DNA binds to specific receptors on the cell surface, often protected by competence pseudopili. DNA uptake follows, where single-stranded DNA is translocated across the membrane via ATP-driven transporters, while the complementary strand is degraded. Finally, the internalized DNA integrates into the host genome through homologous recombination, mediated by proteins like RecA, resulting in stable genetic modification.^[34] These steps are tightly regulated and occur in about 1-10% of cells under optimal conditions, highlighting transformation's role as an adaptive strategy. In oncology, malignant transformation describes the conversion of normal cells into cancerous ones, driven by accumulated genetic mutations that disrupt cellular homeostasis. This process involves activation of oncogenes—such as RAS or MYC, which promote uncontrolled proliferation—and inactivation of tumor suppressor genes, notably TP53, which encodes the p53 protein that halts the cell cycle in response to DNA damage.^[35] Mutations in p53 occur in over 50% of human cancers and contribute to genomic instability during multistep carcinogenesis, allowing cells to evade apoptosis and acquire invasive properties.^[36] For instance, in colorectal cancer, sequential mutations in APC (tumor suppressor), KRAS (oncogene), and TP53 lead to adenoma-to-carcinoma progression, as outlined in the Vogelstein model. Therapeutic transformation leverages genetic modification for medical benefit, particularly in gene therapy, where viral vectors deliver corrective DNA to target cells. Adeno-associated virus (AAV) and lentiviral vectors are commonly used to transduce non-dividing cells, such as airway epithelia in cystic fibrosis (CF), caused by CFTR gene mutations leading to defective chloride transport.^[37] As of 2018, clinical trials showed AAV vectors expressing functional CFTR could restore partial protein function in nasal epithelia, improving mucociliary clearance without severe adverse effects, though challenges like immune responses and transduction efficiency persist.^[38] More recent advancements as of 2025 include ongoing clinical trials of novel approaches such as prime editing for mutations like G542X, CRISPR-based in vivo editing, and inhaled mRNA therapies, with the FDA clearing the next phase of the RCT2100 trial in November 2025.^[39]^[40] In immunology, transformation also occurs during B-cell activation, where naive B cells respond to antigens by undergoing blast transformation—rapid proliferation and differentiation into antibody-secreting plasma cells or memory cells. Antigen binding to the B-cell receptor, often with T-cell help via CD40L and cytokines, induces metabolic reprogramming and clonal expansion, amplifying humoral immunity against pathogens.^[41] This process is exemplified in germinal centers, where somatic hypermutation refines antibody affinity, essential for effective vaccine responses and long-term immunity.

Physics

In physics, transformations describe how physical quantities and laws change under shifts between reference frames or alterations in system symmetries, ensuring the invariance of fundamental principles. In classical mechanics, the Galilean transformation governs the relation between coordinates in inertial frames moving at constant relative velocity v, given by x' = x - v t and t' = t, preserving absolute time and enabling Newton's laws to hold equivalently in all such frames. This formulation, implicit in Newton's framework of relative motion, underpins the principle of relativity for low-speed phenomena where velocities are negligible compared to the speed of light c.^[42] Special relativity extends this through Lorentz transformations, which account for the constancy of c and the relativity of simultaneity. For motion along the x-axis, the coordinates transform as

\begin{align} x' &= \gamma (x - v t), \\ t' &= \gamma \left(t - \frac{v x}{c^2}\right), \end{align}

where \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} is the Lorentz factor. These equations, derived from the postulates of relativity and light speed invariance, mix space and time, leading to effects like time dilation and length contraction, and ensure Maxwell's equations remain form-invariant across frames.^[43] In the limit v \ll c, \gamma \approx 1, recovering the Galilean case.^[44] Symmetry transformations further reveal deep connections in physical laws via Noether's theorem, which associates continuous symmetries of the action with conserved quantities. For instance, spatial translation invariance yields momentum conservation, while time translation invariance implies energy conservation; rotational symmetry conserves angular momentum. Formulated in the context of variational principles, the theorem states that if the Lagrangian is invariant under an infinitesimal transformation \delta q = \epsilon K(q), a conserved current arises, with the Noether charge Q = \int j^0 d^3x generating the symmetry. These include time translations (t \to t + \epsilon), spatial translations (x \to x + \epsilon), and rotations, forming the Poincaré group in relativity, which unifies conservation laws across mechanics and field theories. In quantum mechanics, transformations act via unitary operators on the Hilbert space of states, preserving probabilities and inner products. The time evolution operator U(t) = e^{-i H t / \hbar}, where H is the Hamiltonian, is unitary (U^\dagger U = I), evolving states as |\psi(t)\rangle = U(t) |\psi(0)\rangle while maintaining \langle \psi | \psi \rangle = 1. Symmetries correspond to unitary representations of groups, such as rotation operators U(R) = e^{-i \mathbf{J} \cdot \mathbf{n} \theta / \hbar} for angular momentum \mathbf{J}, ensuring observables transform covariantly. This framework, central to Dirac's formulation, links classical symmetries to quantum conservation via Noether-like identities. Phase transitions in thermodynamics represent transformations between macroscopic states of matter, driven by changes in temperature, pressure, or fields, without altering chemical composition. First-order transitions, like solid-to-liquid melting, involve latent heat and discontinuous entropy or volume, as classified by the order of thermodynamic derivatives that exhibit discontinuities. Second-order transitions, such as the ferromagnetic-paramagnetic shift at the Curie point, feature continuous first derivatives but divergent second derivatives like specific heat. Ehrenfest's scheme, based on singularities in the Gibbs free energy, distinguishes these by the highest derivative order showing non-analyticity, providing a foundational thermodynamic perspective distinct from microscopic statistical mechanics.^[45]

Chemistry

In chemistry, chemical transformations refer to processes in which atoms within molecules are rearranged, leading to the formation of new substances with altered composition, constitution, or configuration. These transformations occur when molecules collide with sufficient energy, often induced by heat, light, or other forms of energy input, resulting in the breaking and forming of chemical bonds. Unlike physical changes, which preserve molecular identity, chemical transformations fundamentally alter the identity of the substances involved.^[46]^[47] Chemical transformations are classified into several major types based on the nature of the reactants and products. Synthesis (or combination) reactions involve two or more substances combining to form a single product, such as the formation of water from hydrogen and oxygen. Decomposition reactions break down a compound into simpler substances, often requiring energy input like heat or electricity. Single displacement reactions occur when one element replaces another in a compound, typically involving metals and acids, while double displacement reactions exchange ions between two compounds, commonly seen in precipitation or neutralization processes. Redox (oxidation-reduction) reactions feature the transfer of electrons between species, underlying many transformations in both inorganic and organic chemistry. These classifications aid in predicting reaction outcomes and designing synthetic pathways.^[48]^[49] The underlying mechanisms of chemical transformations consist of a series of elementary steps, each representing a single molecular event, such as bond breaking or forming. These steps proceed through transition states, fleeting high-energy configurations where bonds are partially formed and broken, representing the highest energy point on the reaction pathway. The rate of these steps depends on the activation energy (E_a), the energy barrier that must be overcome for the reaction to occur. This relationship is described by the Arrhenius equation:

k = A e^{-E_a / RT}

where k is the rate constant, A is the pre-exponential factor reflecting collision frequency and orientation, R is the universal gas constant, and T is the absolute temperature. Derived from empirical observations and theoretical insights into molecular collisions, this equation, first formulated by Svante Arrhenius in 1889, quantifies how increasing temperature exponentially accelerates reaction rates by providing more molecules with sufficient energy to reach the transition state.^[50]^[51] Beyond reactive changes, chemical transformations encompass phase transformations, particularly in solid-state chemistry, where the arrangement of atoms or molecules shifts without altering the chemical composition. A prominent example is polymorphic transitions, in which a substance adopts different crystal structures, such as the α-to-β transition in quartz or various forms of carbon like graphite and diamond. These transitions are driven by thermodynamic factors like temperature or pressure and are critical in materials science for tailoring properties such as mechanical strength, electrical conductivity, or bioavailability in pharmaceuticals. For instance, polymorphic forms of active pharmaceutical ingredients can influence dissolution rates and therapeutic efficacy, necessitating careful control during synthesis and storage.^[52] Many chemical transformations are accelerated by catalysts, substances that lower the activation energy by providing an alternative reaction pathway, thereby increasing reaction rates without being consumed. In industrial applications, heterogeneous catalysts—such as metal surfaces or zeolites—facilitate large-scale processes; a seminal example is the Haber-Bosch process, developed in the early 20th century, which employs iron-based catalysts promoted by potassium and aluminum oxides to synthesize ammonia (NH₃) from nitrogen (N₂) and hydrogen (H₂) gases at 400–500°C and 150–300 atm. This transformation has revolutionized fertilizer production, enabling global food security but requiring significant energy input. In more specialized contexts, enzymes serve as biological catalysts with exquisite specificity, though synthetic mimics in organocatalysis are increasingly used to replicate such efficiency in non-biological transformations.^[53]^[54] To ensure chemical transformations are sustainable, green chemistry principles emphasize designing processes that minimize environmental harm while maximizing efficiency. Developed by Paul Anastas and John Warner in 1998, these principles include preventing waste at the source, maximizing atom economy (incorporating as many reactant atoms as possible into the product), and selecting safer chemicals and renewable feedstocks. For sustainable transformations, this approach reduces hazardous byproducts, as seen in solvent-free reactions or biocatalytic methods that lower energy demands compared to traditional routes. By integrating these guidelines, chemical manufacturing—responsible for a significant portion of industrial emissions—can transition toward circular economies, with metrics like E-factor (waste per unit of product) serving as key indicators of progress.^[55]

Earth sciences

In Earth sciences, transformations encompass the profound changes in Earth's materials and structures driven by geological, climatic, and human-induced processes. These alterations shape the planet's crust, redistribute sediments, and influence global biogeochemical cycles, often operating over millions of years but accelerating under contemporary influences. Metamorphic transformations involve the alteration of pre-existing rocks—igneous, sedimentary, or earlier metamorphic—through intense heat, pressure, and reactive fluids, without reaching melting point. This metamorphism induces mineral recrystallization, where atoms rearrange into more stable forms, and often produces foliation, a layered texture from aligned platy minerals like mica under directed stress. For instance, sedimentary shale, rich in clay, undergoes low-grade metamorphism to form slate, a hard, fine-grained rock with pronounced foliation that enhances its durability. These changes typically occur deep in the crust or at tectonic boundaries, increasing rock density and altering physical properties.^[56] Tectonic transformations arise from the rigid movement of lithospheric plates, powering the reconfiguration of continents and ocean basins via the Wilson cycle. This cycle begins with continental rifting, where tensional forces split supercontinents, creating rift valleys and new oceanic crust through seafloor spreading at mid-ocean ridges, as seen in the East African Rift. As plates converge, oceanic lithosphere subducts into the mantle at trenches, recycling crust and forming volcanic arcs, eventually closing basins through compression. The final stage involves continental collision, uplifting mountain ranges like the Himalayas through crustal shortening and thickening, a process that has repeated over 500-million-year intervals.^[57] Surface transformations via weathering and erosion dismantle bedrock into transportable sediments, preparing materials for sedimentary rock formation. Physical weathering mechanically fragments rocks without chemical change; frost wedging exemplifies this, as water seeps into cracks, freezes and expands by 9%, prying apart formations in cold climates. Chemical weathering decomposes minerals through reactions with water and atmospheric gases; hydrolysis, for example, breaks down feldspar in granite into clay minerals by ion exchange with H⁺ and OH⁻ ions. Erosion then mobilizes these particles—via rivers, wind, or gravity—depositing them in basins and sculpting landscapes.^[58] Climate-driven transformations in the carbon cycle highlight oceanic CO₂ sequestration as a key regulator of atmospheric composition. Oceans absorb roughly 25% of anthropogenic CO₂ emissions through air-sea diffusion, where dissolved CO₂ forms carbonic acid (H₂CO₃), dissociating into bicarbonate (HCO₃⁻) and hydrogen ions (H⁺), which has acidified surface waters by 0.1 pH units since pre-industrial times. The biological pump enhances this by phytoplankton fixing CO₂ into organic matter, which sinks as particulate export to the deep ocean, isolating carbon for centuries. Cumulatively, oceans have sequestered 105 ± 20 PgC from 1960 to 2019, though warming may reduce future uptake efficiency.^[59] Anthropogenic activities profoundly transform soils, altering their structure, chemistry, and ecosystem services. Agricultural intensification, through tillage, irrigation, and fertilizer use, compacts soil pores, accelerates erosion rates by up to 100 times natural levels, and disrupts nutrient cycling, leading to degradation and loss of organic matter. Urbanization exacerbates this via soil sealing with concrete and asphalt, which impervious surfaces prevent infiltration and promote runoff, while introducing contaminants like heavy metals that elevate toxicity. These human-induced changes outpace natural pedogenesis, reducing soil fertility and carbon storage capacity in affected regions.^[60]^[61]^[62]

Arts and entertainment

Music

In music, transformation refers to techniques that alter musical elements such as harmony, melody, timbre, or structure to create variation, development, or new compositional material. These methods span historical practices and modern technologies, enabling composers to manipulate sounds in ways that evoke emotional or narrative shifts while maintaining coherence. Harmonic, melodic, and timbral transformations, for instance, draw on theoretical frameworks to redefine relationships between notes or sounds, often informed briefly by mathematical concepts like group theory in analyzing harmonic progressions.^[63] Harmonic transformations, particularly in Neo-Riemannian theory, focus on smooth modulations between triads by reinterpreting chord relations without traditional tonal hierarchies. This approach, pioneered by David Lewin, uses operations such as parallel (P), which shifts a major triad to its parallel minor or vice versa by altering the third; relative (R), which exchanges the third and fifth to move between major and minor triads sharing the same root; and leading-tone (L), which replaces the root with the leading tone to effect a dominant-to-tonic-like shift. These transformations model voice-leading efficiency in late-Romantic and post-tonal music, emphasizing contextual relations over absolute pitches.^[63] Melodic transformations play a central role in serialism, where composers like Arnold Schoenberg employed inversion, retrograde, and transposition to derive all material from a single twelve-tone row, ensuring equality among pitches. In Schoenberg's twelve-tone technique, inversion mirrors the row's intervals around a central axis, retrograde reverses the sequence, and transposition shifts the entire row by a fixed interval, preventing tonal dominance and promoting structural unity. This method, detailed in Schoenberg's writings, revolutionized atonal composition by treating the row as a transformative matrix for thematic development.^[64] Timbral transformations in electronic music production involve altering a sound's quality or texture through effects like distortion, which introduces harmonic overtones by clipping waveforms, or reverb, which simulates spatial acoustics to add depth and decay. These techniques, explored in compositional studies, allow producers to morph acoustic sources into synthetic ones, creating evolving sonic landscapes in genres like ambient and techno. For example, distortion can thicken a guitar tone for aggressive timbres, while reverb expands a dry vocal into an ethereal wash, fundamentally reshaping perceptual qualities without changing pitch or rhythm.^[65] Historically, Richard Wagner exemplified thematic transformations through leitmotifs in his operas, particularly Der Ring des Nibelungen, where recurring motifs evolve to reflect dramatic changes. A leitmotif like the "Rhinegold" theme undergoes melodic, harmonic, and rhythmic alterations—such as inversion or fragmentation—to signify narrative progression, from purity to corruption. This organic development integrates music with drama, influencing later composers in how transformations convey psychological depth.^[66] In contemporary practice, software tools facilitate MIDI transformations for algorithmic composition, enabling automated manipulations of note data like pitch, duration, and velocity. Platforms such as Ableton Live's MIDI Tools allow users to apply operations like arpeggiation, randomization, or scale remapping to generate variations from seed patterns, while environments like Opusmodus support rule-based systems for complex, generative structures. These digital methods democratize advanced transformations, blending human intuition with computational precision in real-time production.^[67]^[68]

Literature and film

In literature, character arcs often depict the protagonist's evolution from personal flaws to resolution, symbolizing internal transformation through narrative progression. A seminal example is Ebenezer Scrooge in Charles Dickens's A Christmas Carol (1843), where the miserly, isolated figure undergoes a profound moral shift after supernatural visitations, emerging generous and empathetic by the story's end. This arc illustrates how external catalysts can prompt self-reflection and redemption, a trope that underscores themes of personal growth in Victorian literature.^[69] Plot transformations in literature frequently involve sudden shifts or genre blends that disrupt conventional narratives, heightening tension and exploring existential themes. Franz Kafka's The Metamorphosis (1915) exemplifies this through Gregor Samsa's inexplicable transformation into a giant insect, blending elements of absurdism and early surrealism to critique alienation in modern society; the plot pivots from mundane routine to familial rejection, forcing a reevaluation of identity and utility. This narrative device, often likened to precursors of magical realism, emphasizes irreversible change as a metaphor for societal pressures on the individual.^[70] Thematically, transformations in literature symbolize personal growth or broader societal shifts, with Ovid's Metamorphoses (8 CE) serving as a foundational text that profoundly influenced Western literary traditions. The epic weaves over 250 myths of physical and emotional change—such as Daphne's metamorphosis into a laurel tree to evade pursuit—driven by divine whims, highlighting the tension between mutable forms and enduring essences. This work inspired later authors like Shakespeare, who drew on its motifs in plays such as A Midsummer Night's Dream and Titus Andronicus, embedding transformation as a lens for examining love, power, and fate across centuries.^[71] In film, visual transformations rely on innovative techniques to convey metamorphosis, blending practical effects with emerging CGI for visceral impact. David Cronenberg's The Fly (1986) pioneered practical makeup and puppetry for its protagonist's grotesque evolution, using progressive latex suits on Jeff Goldblum—starting with subtle skin discoloration and escalating to full-body prosthetics with mechanical aids for insect-like movements—earning an Academy Award for Best Makeup. These effects emphasized gradual decay, contrasting later CGI-driven shape-shifting in superhero films like Marvel's Avengers: Infinity War (2018), where digital morphing simulates fluid body alterations, enhancing spectacle while exploring heroic reinvention.^[72]^[73] Modern literature and film increasingly feature gender transformations to interrogate identity politics, reflecting evolving discourses on fluidity and self-determination. Virginia Woolf's Orlando (1928), though early modernist, laid groundwork for contemporary works by depicting a character's centuries-spanning shift from male to female, challenging binary norms and influencing later explorations of queer identity. In film, Sebastián Lelio's A Fantastic Woman (2017) portrays a trans woman's journey through grief and discrimination following her partner's death, using subtle cinematography to highlight emotional and social transitions amid societal constraints. Recent media representations, such as headlines on Elliot Page's coming out in 2020, further amplify these themes, though often marred by misgendering that underscores ongoing struggles for authentic depiction in public narratives.^[74]^[75]^[76]

Visual arts

In the visual arts, transformation manifests through techniques and concepts that alter form, perception, and reality, often drawing on mathematical principles to evoke depth or metamorphosis. During the Renaissance, artists pioneered perspective transformations to create illusions of three-dimensional space on flat surfaces. Linear perspective, formalized by Filippo Brunelleschi and elaborated by Leon Battista Alberti in his 1435 treatise Della Pittura, uses converging lines to simulate depth, as seen in Leonardo da Vinci's The Last Supper (1495–1498), where architectural elements recede toward a vanishing point, transforming a two-dimensional mural into a lifelike scene. This technique revolutionized painting by enabling viewers to experience spatial transformation as if stepping into the artwork. Surrealism extended transformation into realms of the subconscious, where forms dissolve and recombine in defiance of natural laws. Salvador Dalí's The Persistence of Memory (1931) features melting clocks draped over landscapes, symbolizing the fluidity of time and psychological metamorphosis, influenced by Sigmund Freud's theories on dreams. Similarly, M.C. Escher's lithographs, such as Bond of Union (1956), employ impossible geometries—tessellations that morph seamlessly from one impossible shape to another—challenging Euclidean space and inspiring later mathematical art. These works transform viewer perception, turning static images into dynamic explorations of reality's instability. In contemporary digital art, transformation leverages computational tools to manipulate and generate images, blurring lines between human creation and machine intervention. Adobe Photoshop, introduced in 1990, enables pixel-level alterations, as in artist Refik Anadol's data-driven installations that morph architectural visuals into abstract flows using algorithmic processing. AI-generated art further amplifies this, with tools like DALL-E (2021) transforming textual prompts into novel visuals through generative adversarial networks, as demonstrated in works by artists like Mario Klingemann, who creates evolving portraits that shift identities in real-time. These digital metamorphoses democratize transformation, allowing instantaneous reconfiguration of forms previously bound by physical media. Sculpture and installation art incorporate transformation through material and environmental interactions, often kinetic or perceptual. Anish Kapoor's reflective sculptures, such as Cloud Gate (2006) in Chicago's Millennium Park, use polished stainless steel to distort and multiply surrounding views, transforming public space into a mirrored, ever-shifting infinity. Kinetic installations, like Jean Tinguely's self-destructing machines from the 1960s, physically alter through motion and decay, embodying entropy as artistic process. Conceptual art pushes transformation toward the body and performance, where physical or symbolic changes critique identity and society. Marina Abramović's Rhythm 0 (1974) invited viewers to alter her body using provided objects, transforming passive sculpture into interactive vulnerability and exploring power dynamics. Body artists like Orlan use surgical interventions as art, such as her 1990s "The Reincarnation of Saint ORLAN" series, where cosmetic procedures morph her appearance to challenge beauty norms, documented in medical and artistic records. These performances redefine transformation as corporeal and ephemeral, prioritizing process over permanent form.

Business and technology

Business transformation

Business transformation refers to the fundamental and holistic reconfiguration of an organization's processes, culture, structures, and business models to enhance performance, adapt to market shifts, and achieve sustainable growth. This often involves aligning multiple internal elements—such as strategy, structure, systems, shared values, skills, style, and staff—through frameworks like the McKinsey 7-S model, which emphasizes interconnected changes to drive organizational effectiveness.^[77]^[78] Unlike incremental improvements, business transformation seeks to boost revenue, reduce costs, and improve customer satisfaction by addressing systemic inefficiencies and external pressures.^[79] A key aspect of business transformation is digital transformation, which integrates advanced technologies like cloud computing and artificial intelligence to redefine operations and customer engagement. For instance, Netflix pioneered this shift in 2007 by launching its streaming service alongside its DVD rental model, gradually pivoting to a digital-first platform that now accounts for the majority of its revenue and global subscriber base of 301.6 million as of Q3 2025.^[80]^[81] This move not only disrupted traditional media but also exemplified how digital integration can create scalable, data-driven business models. Organizational transformation complements this by focusing on structural changes, such as mergers, acquisitions, and the adoption of agile methodologies, which gained momentum post-2020 due to the COVID-19 pandemic's disruptions. The pandemic accelerated agile practices, enabling faster decision-making and adaptability in volatile environments, with companies restructuring teams into cross-functional units to enhance responsiveness.^[82]^[83] Despite these opportunities, business transformation faces significant challenges, including employee resistance to change, inadequate change management, and difficulties in measuring return on investment (ROI). Research indicates that up to 70% of transformations fail due to cultural barriers and lack of sustained leadership support, underscoring the need for robust communication and training programs.^[84] Metrics like ROI often prove elusive, as short-term costs can overshadow long-term gains, with leaders struggling to link initiatives to quantifiable outcomes such as cost savings or revenue growth.^[85] Notable case studies illustrate successful implementations. In the 1990s, IBM underwent a profound turnaround under CEO Lou Gerstner, shifting from hardware manufacturing to a services-oriented model, which rescued the company from near-collapse and generated billions in new revenue streams through consulting and IT solutions.^[86] More recently, amid 2025 ESG regulations like the EU's Corporate Sustainability Reporting Directive, companies such as global fashion brands have transformed supply chains for sustainability, integrating ESG metrics into operations to comply with disclosure requirements and reduce environmental impact while enhancing brand value.^[87] These examples highlight how targeted transformations can align business strategy with regulatory and societal demands.

Computing and information technology

In computing and information technology, transformation refers to the systematic alteration of data, models, or systems to meet specific functional, performance, or compatibility requirements. This encompasses processes that convert raw inputs into usable outputs, often leveraging standardized tools and algorithms to ensure efficiency and reliability. Data transformation is a core component of data processing pipelines, particularly in extract, transform, load (ETL) workflows used for integrating disparate data sources into databases or data warehouses. In ETL, data is first extracted from sources such as relational databases or APIs, then transformed through operations like cleaning, aggregation, filtering, and format conversion to align with target schemas, before being loaded into storage systems. Apache Spark, an open-source distributed processing engine, facilitates large-scale ETL by enabling in-memory computations and fault-tolerant transformations on massive datasets, supporting languages like Scala, Java, Python, and R. For instance, Spark's DataFrame API allows declarative transformations such as joining datasets or applying user-defined functions, which scale horizontally across clusters to handle petabyte-scale data. This approach contrasts with traditional batch processing by incorporating stream processing for real-time transformations via Spark Streaming. Model transformations in software engineering involve converting abstract representations into executable code, a key practice in model-driven architecture (MDA). MDA, developed by the Object Management Group (OMG), separates platform-independent models (PIMs)—often created using the Unified Modeling Language (UML)—from platform-specific models (PSMs) to generate code for diverse environments. In this process, UML diagrams, such as class or sequence diagrams, are transformed into source code for languages like Java or C# using automated tools that apply mapping rules defined in standards like the OMG's Meta-Object Facility (MOF). Seminal work in MDA emphasizes forward engineering, where PIMs are refined into PSMs and then code-generated, promoting reusability and reducing manual coding errors. Tools like Eclipse Modeling Framework implement these transformations by parsing UML models and producing boilerplate code, enabling rapid prototyping in domains like enterprise applications. In artificial intelligence and machine learning, feature transformations normalize input data to improve model training and performance, addressing issues like varying scales across variables. A prominent example is z-score normalization (standardization), which rescales features to have zero mean and unit variance, mitigating dominance by high-magnitude features in algorithms like gradient descent-based optimization. The transformation is defined as z = \frac{x - \mu}{\sigma} where x is the original feature value, \mu is the mean, and \sigma is the standard deviation of the feature. This technique, rooted in statistical standardization, is widely applied in preprocessing for neural networks and support vector machines, as it preserves the data's distributional shape while enhancing convergence speed. Libraries like scikit-learn integrate this via the StandardScaler class, which computes \mu and \sigma from training data and applies the transformation to both training and test sets to avoid data leakage. Cryptographic transformations ensure data security and integrity through irreversible mappings, with hash functions serving as a fundamental mechanism. The Secure Hash Algorithm 2 (SHA-256), standardized by the National Institute of Standards and Technology (NIST), processes arbitrary input data to produce a 256-bit fixed-length digest, enabling verification of unaltered transmission or storage. SHA-256's design provides high resistance to collision attacks—estimated at 128 bits—and preimage resistance, making it suitable for applications like digital signatures, message authentication codes (HMAC), and blockchain integrity checks. For data integrity, a recipient recomputes the hash on received data and compares it to the sender's provided digest; any discrepancy indicates tampering. NIST recommends SHA-256 as a minimum for interoperability in protocols requiring hash functions, due to its balance of security and computational efficiency on modern hardware. Emerging trends in 2025 highlight quantum transformations in algorithms, where quantum circuits manipulate qubits to perform computations infeasible on classical systems, such as optimizing complex optimizations or simulating molecular interactions. Quantum algorithms like the Uhlmann transformation enable efficient state fidelity estimation by converting density matrices, with query complexities scaling logarithmically in dimension, as demonstrated in recent circuit implementations. These transformations leverage superposition and entanglement for exponential speedups in tasks like machine learning feature mapping. Concurrently, edge computing data flows are evolving with real-time transformations at the network periphery, processing IoT-generated data locally to reduce latency and bandwidth demands. In 2025, edge platforms process an estimated 75% of enterprise-generated data from approximately 21.1 billion connected IoT devices, contributing to the global data volume of 181 zettabytes through decentralized analytics, incorporating AI-driven transformations for anomaly detection in streams.^[88]^[89]^[90] This shift supports hybrid cloud-edge architectures, enhancing scalability in applications like autonomous vehicles and smart manufacturing.

Society and other fields

Linguistics

In linguistics, transformations refer to systematic rule-based operations that alter linguistic structures to generate varied forms while preserving underlying meaning. Transformational grammar, introduced by Noam Chomsky in his 1957 work Syntactic Structures, posits that sentences derive from an abstract deep structure representing semantic relations, which undergoes transformations to yield the observable surface structure.^[91] These transformations include operations such as phrase movement, deletion, and insertion, enabling the generation of diverse syntactic forms from a finite set of rules.^[91] This framework revolutionized syntactic theory by emphasizing generative rules over taxonomic description, influencing subsequent developments in universal grammar. A classic example of a syntactic transformation is the passive construction, which rearranges elements from an active deep structure to a passive surface structure. For instance, the active sentence "The cat chased the mouse" transforms via passive rules—moving the object to subject position, inserting the auxiliary "was," and postposing the original subject with "by"—to produce "The mouse was chased by the cat."^[91] Such transformations account for syntactic ambiguities and relations across sentence types, as seen in relative clause formation where a noun phrase is extracted and subordinated.^[92] Morphological transformations involve rule-governed alterations at the word level, primarily through affixation and inflection, to create new forms or modify grammatical properties. Affixation adds prefixes (e.g., "un-" in "unhappy") or suffixes (e.g., "-ness" in "happiness") to roots or stems, enabling derivation of novel lexical items.^[93] Inflection, by contrast, applies affixes to indicate categories like tense (e.g., "walk" to "walked"), number (e.g., "cat" to "cats"), or case, without changing word class, thus transforming base forms into contextually appropriate variants.^[93] These processes are constrained by language-specific rules, ensuring productivity in word formation.^[93] Historical linguistics examines transformations as sound changes that systematically evolve languages over time, particularly evident in the Indo-European family. Grimm's law, a set of consonant shifts occurring around the 1st millennium BCE, transformed Proto-Indo-European stops into fricatives or aspirates in Proto-Germanic—for example, PIE *p in "pater" became Germanic *f in "father."^[94] Other shifts, such as Verner's law, further refined these changes by voicing fricatives under specific stress conditions, illustrating regular, exceptionless patterns that underpin comparative reconstruction.^[94] Vowel shifts, like the Great Vowel Shift in English (ca. 1400–1700 CE), raised and diphthongized Middle English long vowels, transforming "bite" from /bi:tə/ to modern /baɪt/.^[94] In computational linguistics, transformations facilitate parsing algorithms in natural language processing models, where syntactic rules convert input strings into hierarchical structures for analysis.^[95] These techniques, rooted in generative grammar, enable efficient dependency and constituency parsing in applications like machine translation.^[95]

Psychology and personal development

In psychology, transformation refers to profound changes in an individual's cognitive, emotional, and behavioral patterns, often driven by developmental processes, therapeutic interventions, or responses to adversity. These transformations enable personal growth, adaptation, and resilience, shifting from maladaptive habits to more integrated self-concepts. Key frameworks emphasize staged progression and motivational shifts, supported by empirical research in developmental and clinical psychology.^[96] Cognitive transformation is exemplified by Jean Piaget's theory of cognitive development, which outlines four stages through which children progress from sensorimotor exploration (birth to 2 years) to preoperational thinking (2 to 7 years), marked by egocentrism where the child struggles to perceive perspectives other than their own, concrete operational reasoning (7 to 11 years) involving logical thought about tangible objects, and finally formal operational abilities (11 years and beyond) enabling abstract and hypothetical reasoning.^[96] This progression represents a fundamental transformation from intuitive, self-centered cognition to advanced problem-solving, influencing lifelong learning and adaptability.^[97] Therapeutic transformations, particularly through cognitive behavioral therapy (CBT), involve reframing negative thoughts to alter emotional responses and behaviors, as pioneered by Aaron Beck in the 1960s and 1970s.^[98] Techniques like exposure therapy, where individuals confront feared stimuli gradually to reduce avoidance, facilitate habituation and cognitive restructuring, leading to decreased anxiety and improved functioning in conditions such as phobias and PTSD.^[99] Personal growth models, such as Abraham Maslow's hierarchy of needs proposed in 1943, describe transformation as a motivational ascent from fulfilling basic physiological and safety needs to achieving esteem and ultimately self-actualization, where individuals realize their potential through creativity and autonomy.^[100] Trauma recovery highlights post-traumatic growth (PTG), a concept developed by Richard Tedeschi and Lawrence Calhoun in 1996, wherein individuals experience positive psychological changes—such as enhanced relationships, new life appreciations, and personal strength—following severe adversity like bereavement or illness. Recent research underscores how such growth correlates with brain adaptations, including changes in areas like the prefrontal cortex that support emotional regulation and resilience, as evidenced in studies up to 2024 examining electrophysiological markers post-trauma.^[101] These transformations often emerge through deliberate reflection and support, contrasting with mere symptom relief. Addiction transformation frequently involves 12-step programs, originating from Alcoholics Anonymous in 1935, which guide participants through structured steps emphasizing surrender, inventory, amends, and ongoing spiritual maintenance to achieve sobriety and holistic recovery. Complementing this, motivational interviewing (MI), developed by William Miller and Stephen Rollnick in the 1980s, enhances engagement by resolving ambivalence through empathetic dialogue and evoking intrinsic motivation for change, proven effective in increasing treatment adherence and abstinence rates in substance use disorders.^[102] Together, these approaches foster enduring behavioral transformations, reducing relapse by addressing both cognitive distortions and social supports.^[103]

Philosophy and religion

In philosophy, transformation often manifests as a dialectical process of historical and conceptual development, most notably in the work of Georg Wilhelm Friedrich Hegel. Hegel's dialectic, outlined in his Phenomenology of Spirit (1807), posits that ideas and historical progress unfold through a dynamic interplay of thesis (an initial proposition), antithesis (its negation or opposition), and synthesis (a higher reconciliation that preserves and transcends the prior elements). This method illustrates transformation not as mere change but as the realization of Geist (spirit or mind) toward absolute knowledge, driving societal and intellectual evolution from alienation to unity.^[104]^[105] Existential philosophy further explores personal transformation through the rejection of inauthenticity and embrace of freedom, as articulated by Jean-Paul Sartre in Being and Nothingness (1943). Sartre describes "bad faith" (mauvaise foi) as a form of self-deception where individuals deny their radical freedom by conforming to external roles or expectations, such as a waiter reducing himself to his profession. Authentic transformation occurs when one confronts this freedom through conscious choice, transcending objectification to create an authentic self amid the absurdity of existence. This shift from bad faith to authenticity represents an ethical imperative for existential becoming.^[106]^[107] In religious contexts, transformation frequently involves sudden or profound spiritual conversions that realign one's life with divine purpose, exemplified in Christianity by the Apostle Paul's experience on the road to Damascus. As recounted in Acts 9:1-19 of the New Testament, Saul (later Paul), a persecutor of early Christians, encountered a blinding light and the voice of the resurrected Jesus, leading to his immediate repentance, baptism, and commission as an apostle to the Gentiles. This metamorphosis from adversary to proponent symbolizes divine grace overriding human opposition, marking a pivotal model of Christian conversion as total renewal. Scholarly analyses emphasize its role in Paul's theology of justification by faith, transforming personal enmity into universal mission.^[108]^[109] Eastern religious traditions, particularly Buddhism, conceptualize transformation as enlightenment that liberates one from the cycle of suffering (dukkha). In Zen Buddhism, a branch of Mahayana, satori refers to a sudden, intuitive awakening to one's true nature, dissolving the illusion of separate self and revealing interconnected reality. This experiential insight builds on the foundational Noble Eightfold Path, detailed in the Dhammacakkappavattana Sutta (the Buddha's first sermon), which encompasses right view, intention, speech, action, livelihood, effort, mindfulness, and concentration. By cultivating these interdependent practices, practitioners uproot craving and ignorance—the roots of suffering—achieving nirvana as the ultimate transcendence of samsara (rebirth).^[110]^[111] Contemporary interpretations extend these metaphysical transformations into modern paradigms like transhumanism, which envisions technology as a means to surpass biological limitations and achieve a form of transcendence. By 2025, transhumanist discourse, influenced by thinkers like Nick Bostrom, emphasizes enhancements such as neural implants and genetic editing to overcome aging, mortality, and cognitive bounds, echoing religious quests for immortality but grounded in scientific progress. This movement posits human evolution as an ongoing project, where technological integration fosters a post-human state of enhanced agency and wisdom, though it raises ethical questions about equity and the essence of humanity.^[112]^[113]

References

[1]
Geometric Transformations
An Euclidean transformation is either a translation, a rotation, or a reflection. We shall discuss translations and rotations only.Missing: definition | Show results with:definition
[2]
[PDF] Notes on transformational geometry - Jeremy Martin
Mar 25, 2013 · Let's formally define what a transformation is: Definition 1. A transformation of a space S is a map φ from S to itself which is 1-1 and onto. ( ...
[3]
[PDF] Translations, Rotations, Reflections, and Dilations
Page 2. In geometry, a transformation is a way to change the position of a figure. Page 3. In some transformations, the figure retains its size and only its ...
[4]
[PDF] Chapter 6 Transformation and the Coordinate Plane
A transformation is a one-to-one correspondence between two sets of points, S and S , such that every point in set S corresponds to one and only one point in ...
[5]
[PDF] Section 5.2. Geometric Transformations
Jul 3, 2020 · Definition. A transformation that preserves lengths and angles is an isometric transformation. A transformation that preserves angles is an ...
[6]
Linear Transformations
A linear transformation is a transformation T : R n → R m satisfying T ( u + v )= T ( u )+ T ( v ) T ( cu )= cT ( u ) for all vectors u ,
[7]
CoordinateTransformations - Intelligent Motion Lab
Transformations are functions that map n-D vectors to other n-D vectors: T:Rn→Rn. They can represent geometric operations, which are caused by movement or ...Vectors and coordinates · Transformations · Scaling · Working with coordinate...
[8]
[PDF] isometries of the plane - UChicago Math
Jul 21, 2009 · What is an isometry? An isometry is a distance-preserving transformation. In this paper, we consider isometries of the plane C. Definition ...
[9]
[PDF] GEOMETRIC TRANSFORMATIONS IN OLYMPIADS 1. Some types ...
Some types of geometric transformations isometries: are transformations that preserve all distances. Isometries pre- serve lengths, areas, and angles.Missing: definition | Show results with:definition
[10]
[PDF] Isometries of the plane and linear algebra - Keith Conrad
It is clear that the three kinds of isometries pictured above (translations, rotations, reflections) are each invertible (translate by the negative vector, ...
[11]
[PDF] Classifying Isometries - MIT Mathematics
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, ...
[12]
Dilations - Ximera - The Ohio State University
Aug 30, 2025 · Let's add dilations to our list of transformations. The first thing to know about dilations is that they still preserve angles. If I start ...
[13]
[PDF] Transformational Geometry - UCSB Math
Aug 12, 2011 · (Also called rigid transformations.) The following types of translations are isometries: Translation. Reflection. Rotation. A dilation is not an ...
[14]
[PDF] 11.03.04: The Parameters of Transformational Geometry
Isometries. Isometries are transformations, or movements, in the Euclidean plane that preserve the distance between points. In my textbook, they are referred ...
[15]
[PDF] The Evolution of Group Theory: A Brief Survey - Israel Kleiner
Mar 14, 2004 · The aim of this so-called. Erlangen Program was the classification of geometry as the study of invariants under various groups of ...
[16]
Geometric Transformations - Brown Math
When you discuss symmetry and groups, it is important to remember that the elements of those groups are often geometric transformations. A geometric ...
[17]
Introduction to Computer Graphics, Section 2.3 -- Transforms
... geometric transformations can be used to place graphics objects into a coordinate system. ... geometric transforms are a central concept in computer graphics.Viewing and Modeling · Translation · Combining Transformations · Scaling
[18]
[PDF] Coordinate Spaces & Transformations - Computer Graphics
Transformations In Computer Graphics. • Common uses of linear ... • Discrete conformal geometry. • Hyperbolic geometry. • Clipping. • Directional lights. • …
[19]
LTR-0050: Image and Kernel of a Linear Transformation - Ximera
We define the image and kernel of a linear transformation and prove the Rank-Nullity Theorem for linear transformations.
[20]
[PDF] Linear Transformations and the Rank-Nullity Theorem
We will start with the initial definition. Definition 1 A function T : V → W from a vector space V to a vector space W is called a linear transformation if.<|control11|><|separator|>
[21]
[PDF] Math 221: LINEAR ALGEBRA - Emory University
Oct 26, 2020 · Let V and W be vector spaces, and T : V → W a linear transformation. 1. The kernel of T (sometimes called the null space of T) is defined to be.
[22]
[PDF] Affine Transformations - UT Computer Science
Geometric transformations will map points in one space to points in another: (x',y',z') = f(x,y,z). These transformations can be very simple, such as.
[23]
[PDF] EE 261 - The Fourier Transform and its Applications
In the modern formulation of partial differential equations, the Fourier transform ... definition, it's reasonable to say that one of the patterns is “low ...Missing: authoritative | Show results with:authoritative
[24]
Differential Equations - Laplace Transforms - Pauls Online Math Notes
Apr 5, 2019 · The Definition – In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the ...
[25]
[PDF] Transformation Groups for Beginners S. V. Duzhin B. D. ...
Sep 1, 2002 · The notion of a symmetry group is a source of numerous interesting examples of transformation groups. Let us consider some of them. Problem ...
[26]
[PDF] 7 Symmetry and Group Theory - Penn Math
These symmetries can be pictured by how they transform the colored triangle in Figure 42. Symmetries of the square. A square is in some sense “more symmetric” ...
[27]
[PDF] Affine Ciphers
An affine cipher, (like a shift cipher), is an example of a substitution cipher: In encryption using a substitution cipher, each time a given letter occurs ...<|control11|><|separator|>
[28]
Affine transformation in cryptography | Request PDF - ResearchGate
Aug 10, 2025 · The affine transformation is the generalized shift cipher. The shift cipher is one of the important techniques in cryptography.
[29]
[PDF] Linear Programming Notes V Problem Transformations
Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material constraints.
[30]
The Significance of Pneumococcal Types | Epidemiology & Infection
The Significance of Pneumococcal Types. Published online by Cambridge University Press: 15 May 2009. Fred Griffith. Show author details ...
[31]
Bacterial transformation: distribution, shared mechanisms and ...
Feb 10, 2014 · In this Review, we highlight divergent and common principles that govern the transformation process in different bacteria. We discuss how this ...
[32]
Oncogenes, Tumor Suppressor Genes, and DNA Repair Genes
Aug 31, 2022 · For example, TP53 is an important tumor suppressor gene. It codes for the p53 protein, which helps keep cell division under control. Inherited ...
[33]
Mutations in the p53 Tumor Suppressor Gene - PubMed Central - NIH
Mutations in the p53 gene were shown to occur at different phases of the multistep process of malignant transformation, thus contributing differentially to ...
[34]
Gene Therapy for Cystic Fibrosis
The study indicated that the CF gene therapy was safe and resulted in a small improvement in lung function. Multiple clinical trials in the U.S. are currently ...
[35]
AAV gene therapy for cystic fibrosis: current barriers and recent ...
One of the most promising and enduring vectors is the adeno-associated virus (AAV), which has been shown to be safe in several clinical trials. A number of ...
[36]
Defining antigen-specific plasmablast and memory B cell subsets in ...
Feb 15, 2017 · Overall, our data supports a model where antigenic stimulation drives naïve or resting memory B cells to undergo a rapid clonal expansion.
[37]
[PDF] Newton's Principia : the mathematical principles of natural philosophy
TRANSLATED INTO ENGLISH BY ANDREW MOTTE. TO WHICH IS ADDKTV. NEWTON S SYSTEM OF THE WORLD ;. With a Portrait taken from the Bust ...
[38]
[PDF] ON THE ELECTRODYNAMICS OF MOVING BODIES - Fourmilab
This edition of Einstein's On the Electrodynamics of Moving Bodies is based on the English translation of his original 1905 German-language paper. (published as ...
[39]
[PDF] Albert Einstein - Relativity: The Special and General Theory - Ibiblio
This system of equations is often termed the. “Galilei transformation.” The Galilei transforma- tion can be obtained from the Lorentz trans- formation by ...
[40]
A Look Back at the Ehrenfest Classification. Translation and ... - ar5iv
A translation of Paul Ehrenfest's 1933 paper, entitled “Phase transitions in the usual and generalized sense, classified according to the singularities of the ...
[41]
Chemical and Physical Transformations of Matter - NCBI - NIH
A chemical transformation can occur when molecules collide with sufficient energy. Typically, reaction happens when the molecules are heated or irradiated to ...
[42]
Chemical Reactivity - MSU chemistry
A transformation resulting in a change of composition, constitution and/or configuration of a compound ( referred to as the reactant or substrate ).
[43]
CH104: Chapter 5 - Chemical Reactions - Chemistry
Chemical reactions can be classified by type. Combination reactions (also called synthesis reactions) build or make a substance from other substances.
[44]
[PDF] 9. Chemical Reactions
Chemical reactions are classified by simplicity/complexity of reactants/products, and by types of reactants/products, such as combination, decomposition, and ...
[45]
chapter iv chemical kinetics
The Arrhenius equation (4.15) is the classic model. (4.15). where ko is the called the frequency factor, or the pre-exponential factor, Ea is the activation ...
[46]
Chapter 1. Introduction | NC State Extension Publications
Apr 17, 2025 · The equation, proposed by Svante Arrhenius (see Figure 1-2) in 1889, was based on the work of Dutch chemist Jacobus H. van't Hoff. Arrhenius ...
[47]
Polymorphic Phase Transformations in Crystalline Solid Dispersions
Apr 7, 2022 · This study extends the fundamental understanding of the CPP's effect on PPTs through temperature–pressure-simulated extrusion (TPS-E) employing the model ...
[48]
Reaction Mechanisms, Kinetics, and Improved Catalysts for ...
Apr 6, 2022 · The Haber–Bosch (HB) process is the primary chemical synthesis technique for industrial production of ammonia (NH3) for manufacturing ...
[49]
Theoretical Approach toward a Mild Condition Haber–Bosch ...
Nov 7, 2023 · In summary, we illustrate the high feasibility of achieving a mild condition Haber–Bosch process on the ferrierite zeolite catalyst with ...
[50]
Basics of Green Chemistry | US EPA
Feb 4, 2025 · The 12 Principles of Green Chemistry · 1. Prevent waste · 2. Maximize atom economy · 3. Design less hazardous chemical syntheses · 4. Design safer ...Missing: transformations | Show results with:transformations
[51]
What are metamorphic rocks? | U.S. Geological Survey - USGS.gov
Metamorphic rocks form when rocks are subjected to high heat, high pressure, hot mineral-rich fluids or, more commonly, some combination of these factors.Missing: recrystallization | Show results with:recrystallization
[52]
2 Plate Tectonics – An Introduction to Geology - OpenGeology
Describe the Wilson Cycle, beginning with continental rifting, ocean basin creation, plate subduction, and ending with ocean basin closure; Explain how the ...
[53]
Weathering and Soils - Tulane University
Sep 10, 2015 · Physical Weathering - disintegration of rocks and minerals by a physical or mechanical process. Chemical Weathering - chemical alteration or ...
[54]
Global Carbon and other Biogeochemical Cycles and Feedbacks
This chapter assesses how physical and biogeochemical processes of the carbon and nitrogen cycles affect the variability and trends of GHGs in the atmosphere
[55]
Soil and the intensification of agriculture for global food security
Anthropogenic changes in the use of soils, including for the intensification of agricultural production in order to achieve food security, modifies the ability ...
[56]
The Human Impact on All Soil-Forming Factors during the ...
Sep 13, 2021 · Human activity is altering the time scale of soil formation. It has been argued that the anthropogenic factor is orders of magnitude faster than ...
[57]
[PDF] Climate change and urban forest soils [Chapter 10]
The purpose of this chapter is to explore the array of modifications of soil from anthropogenic activities in urban areas, particularly within the context of ...
[58]
Generalized Musical Intervals and Transformations - Oxford Academic
May 3, 2007 · This book is recognized as the seminal work paving the way for current studies in mathematical and systematic approaches to music analysis.
[59]
[PDF] Style and Idea - Monoskop
Composition with twelve tones has no other aim than comprehensibility. In view of certain events in recent musical history, this might seem astonishing, for ...
[60]
[PDF] Timbral Movements in Electronic Music Composition
Transformation methods include ring-modulation, transposition, filtering, reverberation, enveloping, distortion by overloading, and tape-splicing. In ...
[61]
Understanding the Leitmotif
The musical leitmotif, having reached a point of particular forcefulness in the music of Richard Wagner, has remained a popular compositional device up to ...Missing: URL | Show results with:URL
[62]
MIDI Tools — Ableton Reference Manual Version 12
In order to transform notes or generate notes using a MIDI Tool, open the Transform or Generate panel, select a tool from the Transformation/Generator Selector ...Missing: composition software
[63]
Opusmodus
Opusmodus is a comprehensive computer-aided environment for the whole work of music composition, a virtual space where a composer can develop ideas and ...
[64]
SCROOGE'S CHARACTER DEVELOPMENT IN CHARLES ...
This paper deals with Scrooge's characterization in Dickens' A Christmas Carol who is a rich, cold-hearted, miserly, misanthropy, and an anti Christmas old man.
[65]
[PDF] Analysis of the Metamorphosis - Digital Commons @ Butler University
The first transformation is a physical one. Gregor wakes up first thing in the morning and realizes that he has been transformed into a giant insect (Kafka 89).
[66]
Shakespeare and Ovid's Metamorphoses | Blogs & features
Sep 22, 2021 · Ovid's Metamorphoses has a strong claim to being Shakespeare's favourite book, surfacing throughout his career as a narrative source, a spring of thematic ...
[67]
The Fly — New Buzz on an Old Theme - American Cinematographer
Oct 1, 2024 · Mark Irwin, CSC details his experience working with frequent collaborator David Cronenberg on this horror masterpiece.<|separator|>
[68]
How Visual Effects Made Modern Superhero Movies Feel Real - IGN
Mar 15, 2021 · From super speed to scrawny Steve Rogers, this is how visual effects made modern superhero movies feel real.
[69]
Does Ovid Still Matter Today? - Electric Literature
Nov 8, 2022 · Ovid's Metamorphoses has been hugely instrumental in the formulation of various later canons within the Western tradition. Yet many have rightly ...Missing: thematic | Show results with:thematic
[70]
Intimate Citizenship, Trans Activism, and the Gender Identity Law in ...
Dec 15, 2022 · Lelio's film brought the trans lived experience to a wider public, providing an important visual portrayal of the complexities of living as a ...
[71]
Media Portrayals of Trans and Gender Diverse People
Feb 27, 2024 · The present study investigated how media representations of TGD people in news headlines varied across three European countries with different levels of legal ...
[72]
Enduring Ideas: The 7-S Framework - McKinsey
Mar 1, 2008 · Peters and Robert H. Waterman, the framework maps a constellation of interrelated factors that influence an organization's ability to change.
[73]
McKinsey 7S Model - Overview, Structure and Application, Example
The McKinsey 7S Model refers to a tool that analyzes a company's “organizational design.” The goal of the model is to depict how effectiveness can be.
[74]
What is business transformation? - McKinsey
Apr 17, 2023 · Business transformations are designed to boost overall performance through increased revenue, lower operating costs, and better customer satisfaction and ...
[75]
Reinventing your business by transforming the core - McKinsey
Feb 17, 2017 · Netflix's evolution from a company that rented DVDs to a company that streams entertainment for a monthly subscription to one that now creates ...<|control11|><|separator|>
[76]
Agility in the time of COVID-19: Changing your operating model in ...
Nov 18, 2020 · Our research shows that agile companies have outperformed others in adapting to COVID-19. By examining their priorities and using practices such as objectives ...Missing: mergers | Show results with:mergers
[77]
The Pandemic Created a Renewed Urgency for Agile Transformation
Mar 1, 2022 · Dispersed workforces, unpredictable supply chains, and rapidly changing market dynamics caused agile ways of working to shift overnight from ...Missing: restructuring 2020
[78]
Changing change management | McKinsey
Jul 1, 2015 · 70 percent of change programs fail to achieve their goals, largely due to employee resistance and lack of management support.Missing: ROI | Show results with:ROI
[79]
Mapping Digital Transformation Value – Metrics that Matter - Deloitte
Nov 17, 2023 · The most commonly cited challenge, reported by 3 in 4 leaders, is the inability to define exact impacts or metrics—the solution to which begins ...
[80]
Waking Up IBM: How a Gang of Unlikely Rebels Transformed Big Blue
IBM was a case study in complacency. Insulated from the real world by layer upon layer of dutiful managers and obsequious staff.
[81]
Checkpoint: Sustainable supply chain transformation - Deloitte
Jan 9, 2025 · Case study. Transforming ESG compliance and supply chain transparency for a global fashion brand. As a prominent Consumer Products brand, this ...
[82]
[PDF] A Deep Dive into Transformational Grammar
Deep Structure vs. Surface Structure: The Two. Sides of a Sentence. Chomsky distinguished between the deep structure and the surface structure of a sentence.
[83]
Natural Language Processing and Computational Linguistics
Dec 23, 2021 · Linguistic structures, with which NLP technologies such as parsing have previously been concerned, play less important roles than we initially ...NLP, CL, and Related... · Grammar Formalisms and... · Text Mining for Biomedicine
[84]
[PDF] Chapter 7 Morphology: the structure of words - Geert Booij's Page
Feb 24, 2014 · Word formation deals with the creation of new (complex) words by various morphological mechanisms such as compounding, affixation, truncation, ...<|control11|><|separator|>
[85]
[PDF] Sound changes from Proto-Indo-European to Early Modern English
The survey of sound changes from Proto-Indo-European to Early Modern English pre- sented in this chapter is based on a selection of the most important ...
[86]
[PDF] Head-Driven Statistical Models for Natural Language Parsing
This article describes three statistical models for natural language parsing. The models extend methods from probabilistic context-free grammars to ...
[87]
Cognitive Development - StatPearls - NCBI Bookshelf - NIH
Apr 23, 2023 · Children in this age group learn to imitate and pretend to play. This stage is characterized by egocentrism, i.e., being unable to perceive that ...Definition/Introduction · Issues of Concern · Clinical Significance
[88]
10.1 Piaget's Sensorimotor and Preoperational Stages
Egocentrism in early childhood refers to the tendency of young children not to be able to take the perspective of others, and instead the child thinks that ...
[89]
Cognitive Behavior Therapy - StatPearls - NCBI Bookshelf - NIH
Examples of CBT in practice: Anxiety: CBT often focuses on replacing negative automatic thoughts that can occur in generalized anxiety disorder and may be ...Missing: seminal | Show results with:seminal
[90]
Cognitive Behavioral Therapy for Anxiety Disorders (Part One)
Jan 6, 2022 · The therapist explained that exposure gives Kayla evidence with which to challenge her negative thoughts and gain confidence in her ability ...<|separator|>
[91]
A. H. Maslow (1943) A Theory of Human Motivation
The present paper is an attempt to formulate a positive theory of motivation which will satisfy these theoretical demands and at the same time conform to the ...
[92]
Toward neuroscientific understanding in posttraumatic growth
This comprehensive review critically examined the limited neural PTG research, to identify electrophysiological training targets for future research.
[93]
[PDF] USING MOTIVATIONAL INTERVIEWING IN - Advisory 35
(Miller & Rollnick, 2013). This Advisory is based on the Substance Abuse and Mental Health Services Administration's. (SAMHSA) Treatment Improvement Protocol ...
[94]
Motivational interviewing in treating addictions. - APA PsycNet
Motivational interviewing (MI) began as a tool for treating alcohol problems and quickly found applications in addressing other drug and gambling problems ...<|control11|><|separator|>
[95]
[PDF] Hegel-Phenomenology-of-Spirit.pdf
... dialectical reasoning. Dialectic is, in fact, a richer and more supple form ofthought,.advance than mathematical inference, for while the latter proceeds on ...
[96]
Hegel's Dialectics - Stanford Encyclopedia of Philosophy
Jun 3, 2016 · “Dialectics” is a term used to describe a method of philosophical argument that involves some sort of contradictory process between opposing sides.Applying Hegel's dialectical... · Why does Hegel use dialectics? · Bibliography
[97]
[PDF] being-and-nothingness.pdf - Antilogicalism
... Nothingness. 21. Chapter Two. Bad Faith. 47. I. Bad Faith and Falsehood. 47. II.. Patterns of Bad Faith. 55. III. The "Faith" of Bad Faith. 67. PART 1WO. BEING- ...<|separator|>
[98]
Jean Paul Sartre: Existentialism - Internet Encyclopedia of Philosophy
Insofar as bad faith is self-deceit, it raises the problem of accounting for contradictory beliefs. The examples of bad faith which Sartre gives, serve to ...
[99]
14. The Conversion of Saul (Acts 9:1-31) - Bible.org
Aug 13, 2004 · Saul was not “seeking God.” Saul was saved in spite of himself. This Paul knew and testified to. God chose Saul and had his destiny planned, ...
[100]
What happened on the road to Damascus? - Got Questions
Jan 30, 2023 · The events that happened on the road to Damascus relate not only to the apostle Paul, whose dramatic conversion occurred there, but they also provide a clear ...
[101]
Japanese Zen Buddhist Philosophy
Jun 28, 2006 · Through za-zen the Zen practitioner attempts to embody non-discriminatory wisdom vis-à-vis the meditational experience known as “satori.” A ...
[102]
The Noble Eightfold Path - Vipassana Fellowship
The Noble Eightfold Path: The Way to the End of Suffering. by Bhikkhu Bodhi. Contents. Preface, Abbreviations, Copyright; The Way to the End of Suffering
[103]
Transhumanism and Transcendence (Chapter 8)
Nov 20, 2024 · This chapter examines transhumanism as the use of technological means to enhance human biological function.
[104]
[PDF] Transhumanism, Technological Evolution, and Theosis - Principles
Apr 10, 2025 · 4 This essay examines Transhumanism's fundamental principle that humans have a responsibility to correct the limits of biological evolution ...