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4D

Four-dimensional space, commonly abbreviated as 4D, is a mathematical construct extending three-dimensional Euclidean space by incorporating an additional perpendicular dimension, allowing points to be described by four coordinates rather than three.^[1] This fourth spatial dimension, often denoted as w alongside the familiar x, y, and z, enables geometric objects and transformations that transcend human visual intuition, with volumes (or hypervolumes) scaling as the fourth power of the linear dimension.^[2] In pure mathematics, 4D geometry employs the "hyper-" prefix for analogs of lower-dimensional figures, such as the hypercube (or tesseract), a bounded 4D polytope composed of eight cubic cells, sixteen square faces, and thirty-two edges.^[1] Key features of 4D space include its role in advanced topology and differential geometry, where it exhibits unique properties like the existence of exotic structures not generalizable to arbitrary dimensions, and facilitates projections onto 3D for partial visualization, such as stereographic or cross-sectional methods.^[1] Historically developed in the 19th century as part of higher-dimensional analysis, 4D concepts underpin modern fields like string theory and computer graphics, though direct empirical observation remains impossible due to perceptual limits confined to three spatial dimensions.^[3] In physics, a distinct but related formulation treats spacetime as a four-dimensional continuum, integrating three spatial dimensions with time under Minkowski geometry in Einstein's special relativity, fundamentally altering causal structures and predictions for phenomena like time dilation.^[4] These frameworks highlight 4D's abstract yet indispensable utility in modeling complex systems, from quantum field theories to multidimensional data visualization.^[5]

Mathematics and Physics

Four-dimensional space

Four-dimensional Euclidean space extends three-dimensional space by adding a fourth orthogonal coordinate axis, typically denoted as w, resulting in points represented by coordinates (x, y, z, w). The distance between two points P = (x_1, y_1, z_1, w_1) and Q = (x_2, y_2, z_2, w_2) is given by the [formula d](/page/Formula_D) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (w_2 - w_1)^2}, which generalizes the Pythagorean theorem to four dimensions by summing squared differences along each axis.^[6] This metric preserves Euclidean properties such as isotropy and the triangle inequality, allowing constructions analogous to those in lower dimensions, including hyperspheres defined by the equation x^2 + y^2 + z^2 + w^2 = r^2 and hypercubes known as tesseracts. The tesseract, a regular four-dimensional polytope, consists of 8 cubic cells, 24 square faces, 32 edges, and 16 vertices, with its hypervolume (four-dimensional analog of volume) calculated as V = s^4, where s is the edge length; for a unit tesseract (s=1), V=1.^[7] First-principles constructions build from lower dimensions: a 0D point extrudes to a 1D line segment, to a 2D square, to a 3D cube, and to the 4D tesseract by further extrusion along the w-axis. These objects underpin n-dimensional generalizations in linear algebra and topology, where four-dimensional manifolds serve as models for smooth, locally Euclidean spaces without boundary. Historical development traces to the mid-19th century, with Bernhard Riemann's 1854 habilitation lecture formalizing n-dimensional Riemannian manifolds through metric tensors describing infinitesimal distances, laying groundwork for higher-dimensional geometry independent of embedding in Euclidean space.^[8] Earlier intuitions appeared in works like Edwin A. Abbott's Flatland (1884), which employed the analogy of two-dimensional beings encountering three-dimensional intrusions—such as a sphere appearing as a growing and shrinking circle—to convey perceptual challenges of higher dimensions.^[9] Human perception, constrained by three-dimensional sensory input, precludes direct visualization of four-dimensional space, necessitating indirect methods like orthogonal projections onto three or two dimensions or analogies such as cross-sections (slicing a 4D object to reveal 3D volumes) and net unfoldings (analogous to cube nets). Schlegel diagrams provide a perspective projection of a 4D polytope's boundary into three dimensions, with one cell at the center and others surrounding it, further projectable to two dimensions for rendering.^[10] In topology, four-dimensional spaces reveal unique phenomena, such as the possibility of exotic smooth structures on \mathbb{R}^4, differing from lower dimensions. Applications include modeling quasicrystals in crystallography, where certain quasiperiodic structures emerge as projections of periodic lattices from four-dimensional space, influencing mechanical and topological properties observable in three dimensions.^[11] In computational simulations of protein folding, configuration spaces are embedded in high-dimensional Euclidean spaces (including four-dimensional approximations) to capture the vast degrees of freedom in polypeptide chains, facilitating analysis of folding landscapes and energy minima via dimensionality reduction techniques.^[12]

Spacetime in special and general relativity

In special relativity, spacetime constitutes a four-dimensional continuum merging three spatial dimensions with time, formalized as Minkowski space to preserve the invariance of the speed of light c across inertial frames. Hermann Minkowski presented this geometric interpretation in 1908, building on Einstein's 1905 postulates that reject an absolute luminiferous ether and enforce Lorentz transformations.^[13] The line element ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 defines the proper interval between events, invariant under Lorentz boosts, with the negative sign for the time component yielding a Lorentzian (indefinite) signature that distinguishes timelike paths (possible for massive particles) from spacelike separations (acausal). Light cones emerge from null intervals (ds^2 = 0), bounding causal domains: events within the future light cone can influence an observer, while those outside cannot, grounding causality in the metric's hyperbolic geometry rather than Euclidean equivalence of dimensions. This framework derives from empirical null results like the 1887 Michelson-Morley experiment, which measured no directional variation in light speed, contradicting ether drag predictions and supporting light's frame-independence.^[14] Time dilation and length contraction follow directly, validated in particle accelerators where muon lifetimes extend by factors matching \gamma = 1/\sqrt{1 - v^2/c^2}. In applications, global positioning system (GPS) satellites require daily corrections of approximately 38 microseconds for net clock gain from special relativistic velocity effects (slowing) and weaker gravitational potential (speeding), ensuring positional accuracy within meters; uncorrected, errors would accumulate to kilometers daily.^[15] General relativity extends Minkowski spacetime to a pseudo-Riemannian manifold curved by mass-energy, as Einstein finalized in November 1915 via field equations G_{\mu\nu} = (8\pi G/c^4) T_{\mu\nu}, where geometry encodes gravitational causation.^[16] Mercury's perihelion advances 43 arcseconds per century beyond Newtonian mechanics, precisely matching observed residuals after planetary perturbations, a prediction unexplained by alternatives like modified Newtonian dynamics until GR's tensorial curvature.^[17] Direct tests include the 2015 LIGO detection of gravitational waves from binary black hole merger GW150914, with strain h \approx 10^{-21} rippling flat spacetime approximations, confirming quadrupole radiation formulas over scalar or vector gravity theories lacking empirical support.^[18] Misinterpretations equating time fully to space ignore the metric's signature, which precludes closed timelike curves in asymptotically flat regions and upholds causal structure against acausal artifacts in Euclidean analogies.

Extra dimensions and theoretical models

Kaluza-Klein theory, proposed by Theodor Kaluza in 1919 and refined by Oskar Klein in 1926, extends general relativity to five dimensions, interpreting electromagnetism as geometric effects arising from the fifth dimension compactified into a small circle of radius on the order of the Planck length.^[19] This framework aimed to unify gravity and electromagnetism without introducing new fields, deriving Maxwell's equations from the five-dimensional Einstein equations under the cylinder condition that fields are independent of the extra coordinate.^[20] Despite its elegance, the theory lacks direct experimental verification, as no observable deviations from four-dimensional physics have confirmed the extra dimension, and quantum inconsistencies persist without modern extensions.^[21] Contemporary theoretical models build on this idea, positing multiple compactified extra dimensions to reconcile quantum mechanics and gravity. String theory, formalized in the 1980s, requires 10 spacetime dimensions—nine spatial and one temporal—with six spatial dimensions curled up via structures like Calabi-Yau manifolds to evade detection at macroscopic scales.^[22] M-theory, introduced in 1995 by Edward Witten, encompasses string theories within an 11-dimensional framework, where the extra dimension emerges dynamically through dualities and brane configurations.^[23] These models promise unification of all fundamental forces, with extra dimensions enabling vibrational modes of strings to correspond to particles and interactions.^[24] However, extra-dimensional theories face substantial empirical challenges. Large Hadron Collider experiments through 2025 have yielded null results for supersymmetric particles, a cornerstone prediction in many string-inspired models, excluding gluinos and squarks up to masses exceeding 2 TeV and first-generation sleptons to nearly 400 GeV.^[25] The theory's vast landscape of approximately $10^{500} possible vacua, arising from different compactification geometries and fluxes, undermines predictive power by allowing virtually any low-energy observation to fit some configuration, rendering the framework difficult to falsify.^[26] Absent confirmatory evidence, such as microscopic black hole production or deviations in gravitational force laws at sub-millimeter scales, these hypotheses prioritize mathematical consistency over causal mechanisms grounded in observable four-dimensional spacetime.^[27] Laboratory simulations of higher dimensions provide analogs but no fundamental validation. In 2023, researchers at the University of Missouri developed metamaterials exploiting synthetic dimensions—internal degrees of freedom like momentum or frequency—to manipulate energy waves in an effective fourth dimension, enabling precise control of wave paths unattainable in three-dimensional space.^[28] These engineered systems, which map physical parameters to higher-dimensional lattices, demonstrate wave phenomena mimicking extra-dimensional topology but rely on artificial constructs rather than intrinsic spacetime structure, highlighting technological utility over evidence for unseen spatial extents.^[29] Such approaches underscore the distinction between simulated higher-dimensional effects and verifiable extensions of physical reality.

Technology and Computing

4D printing and adaptive materials

4D printing refers to the additive manufacturing of three-dimensional structures embedded with stimuli-responsive materials that undergo programmed transformations over time in response to external triggers such as temperature, moisture, light, or magnetic fields.^[30]^[31] The term was coined by Skylar Tibbits of MIT Self-Assembly Lab in 2013, building on 3D printing by incorporating a fourth dimension—time—to enable dynamic reconfiguration of printed objects.^[30] These materials, often shape-memory polymers or composites, exploit differential expansion or contraction; for instance, hydrophilic hydrogels swell upon water absorption to actuate self-assembling hinges or biomedical implants like expandable stents.^[32]^[33] Early prototypes demonstrated water-responsive mechanisms, with hydrogels expanding over 200% in volume due to moisture absorption, as explored in foundational research on programmable matter. More recent advancements include Khalifa University's 2024 vat photopolymerization of photochromic Fresnel lenses that dynamically alter focal properties and block UV light under exposure.^[34] In 2025, magnetically responsive composites have advanced soft robotics applications, enabling shape changes via external fields for precise actuation in biomedical and robotic systems, as reviewed in NIH-funded studies.^[35] The technology has shown empirical shape recovery rates exceeding 91% in controlled tests, with some shape-memory polymer composites achieving 100% fixity and recovery in initial cycles under thermal stimuli.^[36]^[37] Applications span aerospace, where 4D-printed composites enable morphing wings that adapt to flight conditions for improved efficiency, as demonstrated in Concordia University's drone wing prototypes and NASA-MIT collaborations on shape-shifting structures.^[38]^[39] In pharmaceuticals, 4D printing facilitates personalized drug delivery systems that release payloads in response to physiological cues like pH or temperature, enhancing targeted therapy compliance.^[40] Market projections reflect growing interest, with the global 4D printing sector valued at US$213.76 million in 2024 and forecasted to reach US$3,313.32 million by 2033 at a CAGR of approximately 35%.^[41] Despite these advances, 4D printing faces scalability hurdles, including slow actuation times—such as hours required for hydrogel drying—and high material costs that limit production volumes compared to conventional 3D printing.^[42] Empirical durability tests reveal reduced mechanical strength and fatigue resistance in adaptive structures versus static prints, with shape-memory alloys prone to degradation over repeated cycles.^[43] These limitations have tempered early hype, as real-world pilots, such as manufacturing explorations by HDI Global in 2025, highlight the need for cost reductions and faster responses to achieve broader viability.^[44]^[45]

4D visualization and computational modeling

Visualization of four-dimensional (4D) objects requires projecting higher-dimensional geometry onto lower-dimensional spaces accessible to human perception and current hardware, typically through 3D or 2D representations. Common techniques include orthographic and perspective projections from 4D to 3D, followed by further projection to 2D screens, which preserve structural analogies like wireframes or slices. Ray tracing methods extended to 4D treat the fourth dimension as an additional axis for intersection calculations, enabling rendering of solid volumes by simulating light paths in hyperspace, as demonstrated in early implementations for 4D fractals and objects.^[46]^[47]^[48] Topology-based approaches leverage manifold properties to interpret 4D structures without full geometric rendering, focusing on invariant features like connectivity and homotopy to "slice" or embed 4D spaces into comprehensible forms, such as through cross-sections or unfolding analogies. Software tools support these methods indirectly; for instance, Mathematica facilitates 4D coordinate plotting in xyzw systems for hypercube projections, while custom extensions to ray tracers like POV-Ray handle exported scenes for enhanced lighting and depth cues. Empirical validations, such as Duke University's 2017 explorations of 4D plotting, confirm that coordinate-based projections accurately depict tesseract distortions under rotation, aligning with mathematical expectations despite perceptual limitations.^[49]^[50] In applications, computational modeling simulates 4D dynamics for physics and graphics, including animations of tesseract rotations where dual cubes morph via 4D transformations, rendered through iterative projection matrices to mimic hyperspace motion. Integration with virtual reality (VR) and augmented reality (AR) provides pseudo-4D immersion by mapping 4D slices to stereoscopic 3D views, as in interactive hypersphere applications that allow users to navigate manifold embeddings for educational topology insights. Recent developments incorporate AI for hypersurface reconstruction, aiding rendering of complex 4D data by optimizing neural approximations of multidimensional manifolds, particularly useful for reducing high-dimensional datasets to analyzable 4D forms in scientific visualization.^[51]^[52]^[53] Despite advances, 4D rendering faces severe hardware constraints, with computational complexity scaling poorly—often approaching O(n^4) for volume intersections in grid-based methods due to the exponential growth in hypersurface elements—necessitating approximations like sparse sampling or hierarchical bounding volumes to achieve feasible runtimes. These techniques yield projections that convey topological fidelity but do not enable direct perceptual access to 4D geometry, as human cognition remains anchored in three spatial dimensions, underscoring the reliance on mathematical rigor over intuitive visualization.^[54]^[51]

4D as a database system

4D is a relational database management system (RDBMS) integrated with an application development environment, enabling the creation of custom business software through a combination of data storage, querying, and graphical interface design tools. Developed by Laurent Ribardière in 1984 as "4th Dimension," it originated from early efforts to provide a graphical relational database for professional application building on platforms like the Macintosh.^[55] The system evolved under ACI US (later 4D Inc.), incorporating support for SQL standards alongside a proprietary 4D language that allows developers to define data models, business logic, and user interfaces within a unified IDE.^[55] Central to 4D's design is its built-in form and menu designer for constructing graphical user interfaces without external tools, alongside client-server architecture that supports multi-user deployments for distributed applications.^[56] The Object Relational Data Access (ORDA) feature, enhanced in versions like v19 released around 2021, provides an object-oriented layer over relational data for more efficient querying and manipulation, reducing boilerplate code in application development.^[57] These elements make 4D suitable for building standalone or web-connected apps handling structured data, such as inventory tracking or CRM systems, where integrated development accelerates deployment compared to assembling separate database and frontend components.^[58] With over 40 years of continuous refinement since its inception, 4D demonstrates empirical stability in maintaining backward compatibility and core functionality across operating systems, appealing to small and medium-sized enterprises (SMEs) that prioritize reliable, low-maintenance solutions over cutting-edge scalability.^[55] Its rapid prototyping capabilities—stemming from the all-in-one environment—offer practical advantages for iterative business app development, where developers can prototype, test, and refine without switching between disparate tools, contrasting with pure SQL systems that demand additional frameworks for UI and logic integration.^[56] Market adoption persists in niche sectors requiring customized, on-premise or hybrid deployments, underscoring its utility for non-enterprise-scale operations.^[56] To address limitations in handling very large datasets, where independent evaluations highlight slower query performance relative to optimized open-source RDBMS like PostgreSQL in high-volume scenarios, recent updates emphasize cloud-compatible integrations and hybrid architectures for improved scalability.^[59] These adaptations, including support for REST APIs and cloud hosting options, enable 4D applications to leverage external services for data-intensive tasks while retaining core strengths in application logic.^[56]

Arts and Entertainment

Fourth dimension in visual arts

The concept of the fourth dimension entered visual arts in the early 20th century as artists sought to depict multiple perspectives and non-Euclidean spatial relationships, drawing from mathematical ideas of hyperspace rather than temporal or mystical interpretations.^[60] Maurice Princet, a mathematician frequenting Pablo Picasso's circle in Paris around 1903, introduced concepts from Henri Poincaré's work and Esprit Jouffret's 1903 book Traité élémentaire de géométrie à quatre dimensions, influencing Cubist experiments in simultaneous viewpoints.^[61] ^[62] This period, spanning Picasso's proto-Cubist works from 1907 to analytic Cubism by 1914, used fragmented forms to evoke multidimensional perception, as seen in Picasso's Les Demoiselles d'Avignon (1907), where angular distortions simulate rotations in higher dimensions.^[63] Critics have debated the depth of this mathematical influence, with some art historians labeling claims of direct fourth-dimensional inspiration on Cubism a "myth" propagated by later interpretations, arguing that Picasso denied overt reliance on such geometry and that stylistic shifts stemmed more from perceptual innovation than precise hyperspace modeling.^[64] ^[65] Nonetheless, these efforts expanded viewer cognition by challenging Euclidean representation, prioritizing empirical analogies to projection and slicing techniques over occult or pseudoscientific notions of ethereal realms, which lacked verifiable geometric basis.^[66] In mid-20th-century Surrealism, Salvador Dalí explicitly incorporated fourth-dimensional forms, as in his 1954 oil painting Crucifixion (Corpus Hypercubus), featuring an unfolded tesseract (hypercube) as the cross to symbolize divine hyperspatial transcendence while grounding it in mathematical unfolding.^[67] ^[68] Dalí collaborated with mathematicians like Thomas Banchoff to visualize such projections, using stereo effects for perceptual depth beyond three dimensions. Later artists like Tony Robbin advanced these ideas through sculptures and paintings of hypercube tessellations, beginning in the late 1970s after consulting with Banchoff on 4D rotations and distortions.^[69] Robbin's works, such as projections in his 1992 book fourfield: Computers, Art & the 4th Dimension, employ wireframes and color to render 4D polytopes in 3D space, fostering interactive viewer engagement with non-intuitive geometries. Contemporary math-art intersections, documented in proceedings like the 2016 Bridges Conference, continue this tradition via computational projections that analogize hyperspace slicing for perceptual experiments.^[60] These approaches achieve cognitive expansion by verifiable projection methods, distinguishing empirical art from unsubstantiated mystical claims.

4D cinema and immersive experiences

4D cinema refers to motion picture presentations that combine three-dimensional visuals with synchronized sensory effects, such as seat vibrations, wind bursts, water sprays, scents, and strobe lighting, to simulate physical immersion in the film's action. This format evolved from early prototypes like Morton Heilig's Sensorama, a multi-sensory simulator developed starting in 1957 and prototyped by 1962, which integrated stereoscopic 3D imagery, motion simulation, audio, wind, and odors to evoke environmental realism in a booth accommodating up to three viewers.^[70]^[71] Unlike static cinema, 4D effects are algorithmically timed to on-screen events, often using pneumatic or hydraulic actuators in theater seats to replicate movements like tilts, jolts, or rotations.^[72] Modern 4D systems, exemplified by CJ 4DPLEX's 4DX technology introduced in 2009, employ proprietary motion control algorithms that parse film data to generate per-seat effects via networked actuators, enabling effects like leg ticklers or back pokers for precise haptic feedback.^[73] The first 4DX theater opened in Seoul, South Korea, in 2009, with effects calibrated to avoid overpowering the narrative while enhancing causal links between visual cues and physical sensations, such as wind during flight scenes or scents for environmental context.^[74] Patents like US7934773B2 detail synchronized motion induction in seats relative to video playback, using transducers to translate digital signals into multi-axis movements without requiring full simulator gimbals.^[72] Technical challenges include synchronizing effects across rows via "master-slave" drive mechanisms, as patented by MediaMation in 2013, to minimize latency and ensure uniform audience response.^[74] Key milestones include the 2010 release of Avatar in 4DX format in South Korea and Hong Kong, where augmented effects like explosive scents and seat shakes drew premium ticket sales at $15.80 versus standard $6.90, marking an early commercial success for sensory cinema.^[75] Adoption expanded globally, with the first U.S. 4DX venue at Regal in 2014, European rollout via Pathé in 2017, and partnerships like Cinépolis in Mexico since 2011 leading to over 50 theaters by 2025.^[76] In the 2020s, 4D integrations grew in theme parks, such as SimEx-Iwerks' ultra-accessible 4D platforms at Morgan's Wonderland in 2025, featuring wheelchair-compatible motion bases for inclusive sensory rides.^[77] Audience reception data indicates heightened sensory engagement, with effects fostering multisensory causality that aligns physical responses to plot events, though empirical studies reveal trade-offs. A 2019 experimental analysis found correlations between 4D motion intensity and viewer discomfort, including elevated nausea scores linked to vestibular mismatches, particularly in susceptible individuals.^[78] Related research on 3D-augmented formats shows symptoms like oculomotor strain and disorientation increasing by up to 20-30% in motion-heavy sequences, with women reporting higher incidence due to visual-vestibular sensitivities.^[79] Proponents argue these elements boost immersion by 20-30% in subjective engagement metrics from pilot studies, yet critics note potential narrative distractions, as effects can prioritize spectacle over storytelling coherence. High implementation costs—seats exceeding $10,000 each plus effect licensing—limit accessibility to premium multiplexes, confining 4D to about 700 global screens as of 2025.^[76] Despite marketing hype, disinterested evaluations emphasize verifiable synchronization patents over unproven psychological benefits, with nausea risks underscoring the need for viewer warnings and adjustable intensity controls.^[78]

4D in literature, music, and philosophy

In literature, H.G. Wells' The Time Machine (1895) popularized the concept of time as a fourth dimension traversable like spatial ones, with the Time Traveller asserting that "there are really four dimensions, three which we call the three planes of Space, and a fourth, Time."^[80] This narrative blends spatial navigation with temporal progression, portraying the fourth dimension as a medium for human agency amid evolutionary decline. Earlier, Edwin A. Abbott's Flatland: A Romance of Many Dimensions (1884) used a two-dimensional world to analogize higher dimensions, inspiring extensions in fiction where three-dimensional beings encounter four-dimensional intrusions, challenging perceptual limits without direct temporal fusion.^[81] Modern science fiction, such as Rudy Rucker's Spaceland (2002), critiques dimensionality constraints by depicting protagonists navigating four-spatial-dimensional realms, highlighting ontological puzzles like non-Euclidean geometries and identity fragmentation across extents.^[82] References to four-dimensional concepts in music remain scarce and largely symbolic, often evoking hyperspatial abstraction through structural innovation rather than literal representation. For instance, Shorty Rogers' album The Fourth Dimension in Sound (1961) employs jazz arrangements with spatial layering to suggest multidimensional depth, though its inspiration draws more from perceptual expansion than mathematical hyperspace. Experimental works like Keith Lay's Four Dimensions (2012) integrate orchestral elements with electronic synthesis to simulate temporal-spatial interplay, using an electronic wind instrument to bridge auditory "dimensions" in a seven-minute composition.^[83] Such efforts prioritize evoking perceptual transcendence over rigorous geometric mapping, reflecting the challenge of translating four-dimensional ontology into sequential sound. In philosophy, four-dimensionalism, also termed perdurantism, posits that objects persist as extended "worms" through spacetime, comprising temporal parts analogous to spatial ones, thereby resolving puzzles of change via part-whole relations.^[84] Theodore Sider formalized this in Four-Dimensionalism: An Ontology of Persistence and Time (2001), arguing that everyday persistence language accommodates temporal parts, such as a person's infancy stage as a proper part of their full four-dimensional self, supported by linguistic evidence like "is shaped like a statue" applying to wholes over times.^[85] This view contrasts with presentism, which denies past and future existence, leading to debates over empirical persistence: perdurantism aligns with causal continuity in relativity but faces critiques for diluting personal identity, as the "you" now is merely a temporal slice, not the enduring whole, potentially undermining intuitive self-unity.^[86] Proponents credit it with advancing causal realism by modeling events as spacetime intersections, yet detractors highlight unfalsifiable metaphysics, lacking direct empirical disconfirmation beyond theoretical coherence with physics.^[87] These ontological tensions persist, with perdurantism offering explanatory power for diachronic relations at the cost of presentist intuitions about temporal immediacy.^[88]

Commercial and Miscellaneous Uses

In construction management, 4D Building Information Modeling (BIM) integrates time as a fourth dimension into 3D models to simulate construction sequences, enabling project teams to visualize progress, detect scheduling conflicts, and optimize resource allocation.^[89] This approach has been applied commercially since the early 2010s, with firms using it to reduce delays by up to 20% through early identification of logistical issues, such as equipment clashes or labor overlaps.^[90] For instance, in federal and commercial building projects, 4D BIM facilitates safer site operations by forecasting high-risk phases and supports cost savings via precise material delivery timing.^[91]^[92] Precision metrology employs 4D interferometry for non-contact surface measurements in manufacturing, where the fourth dimension accounts for dynamic environmental factors like vibration to ensure accurate defect detection on optics and components.^[93] 4D Technology Corporation, founded in 2001, commercializes products such as the AccuFiz laser interferometer and the handheld 4D InSpec gauge, which measure surface flaws down to micrometer scales in industries including aerospace, automotive, and semiconductors.^[94]^[95] These tools are deployed in production environments, enabling real-time quality control for applications like telescope mirrors and engine parts, with the InSpec model specifically designed for shop-floor portability since its 2017 release.^[96]^[97] Miscellaneous applications include geophysical modeling, where 4D geometric computations analyze volumetric curvatures of subsurface features for oil exploration and seismic risk assessment, aiding commercial resource extraction decisions.^[98] In parameter optimization for engineering design, such as cell phone antenna tuning, 4D spaces represent multi-variable trade-offs to minimize trial-and-error in product development.^[99] These uses leverage abstract 4D mathematics for practical efficiency gains without physical instantiation of higher-dimensional objects.

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4D Printing: A Review on Recent Progresses - PMC - NIH
4D printing was originally defined with the formula of “4D printing = 3D printing + time”, which refers to changes in the shape, structure, or function of 3D ...
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Development of 4D printed smart fresnel lenses
Researchers at Khalifa University have developed switchable 4D printed Fresnel lenses with dynamic photochromic and UV-blocking properties using vat ...
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4D Printing of Magnetically Responsive Materials and Their ... - NIH
Sep 22, 2025 · Magnetically responsive materials are stimulated by magnetic fields to change their shapes, properties, or functions.Missing: NASA | Show results with:NASA<|separator|>
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Controlling Morphing Behavior in 4D Printing: A Review About ...
All printed specimens exhibited a shape recovery rate of >91%. However, in this work, it was found that the shape recovery rates of the lowest samples occurred ...
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4D composite printing can improve the wings of drones, according to ...
May 10, 2022 · 4D composite printing can improve the wings of drones, according to new Concordia research. Suong Hoa's manufacturing technique could make small aircraft ...
[39]
Understanding the Promise of Smart Materials and 4D Printing
In aerospace, MIT and NASA have already developed a shape-shifting wing that adjusts mid-flight to reduce fuel burn, emissions, and maintenance costs. In health ...
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Revolutionizing Personalized Medicine with 4D Printing in Drug ...
Apr 18, 2025 · Different from conventional technologies, 4D printed systems can control the administration rate of drugs depending on the internal environment ...
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Global 4D Printing Market to Worth Over US ... - GlobeNewswire
Jun 13, 2025 · Chicago, June 13, 2025 (GLOBE NEWSWIRE) -- The global 4D printing market was valued at US$ 213.76 million in 2024 and is expected to reach ...
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New Trends in 4D Printing: A Critical Review - MDPI
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4D Printing: The Development of Responsive Materials Using 3D ...
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4D printing: Fundamentals, materials, applications and challenges
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Four-Space Visualization of 4D Objects - Steve Hollasch on GitHub
This research covers the visualization of four dimensional objects through wireframe methods with extended visualization cues, and through raytracing methods.
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https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1813&context=cstech
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Raytracing 4D fractals, visualizing the four dimensional properties of ...
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The goal of the new algorithm is to minimize the rendering time by improving the average time of the ray-object intersection tests. The algorithm focuses on ...
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The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces.
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About us - 4D
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4D: Fast development platform for business apps
4D provides integrated platforms that simplify and speed up the development and deployment of Web, mobile, desktop, and client-server applications.
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4D
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Cubism and the Fourth Dimension: a Myth in Modern Criticism
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A New Facet of Cubism: 'The Fourth Dimension' and 'Non-Euclidean ...
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Salvador Dalí often used mathematical objects and ideas in his paintings and he enjoyed sharing ideas with mathematicians and scientists.
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Tony Robbin Hypercubes - Brown Math
These ideas, for which he is best known as an artist and author, transformed from interstes to plans when Robbin met with Thomas Banchoff in 1979. Banchoff ...
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The First Virtual Reality | The Saturday Evening Post
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The present document describes an actuated chair for inducing motion with respect to the ground as a function of motion signals synchronized with a video ...
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CJ 4DPLEX and Cinépolis Expand Partnership
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Experimental Study on the Effect of Motion Sickness on 4D Movie ...
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Are there side effects to watching 3D movies? A prospective ...
Seeing 3D movies can increase rating of symptoms of nausea, oculomotor and disorientation, especially in women with susceptible visual-vestibular system.
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The Time Machine — Ch 1
There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction ...
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The Fourth Dimension - Rudy Rucker
FLATLAND, first published in 1884, is the story of a square who takes a trip into higher dimensions. A century has passed and people are still talking about it.
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Sci-fi book where a 3D man travels into 4D world and meets some ...
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The Fourth Dimension in Sound - Wikipedia
The Fourth Dimension in Sound is an album by bandleader and arranger Shorty Rogers recorded in late 1961 and released on the Warner Bros. label.<|separator|>
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Four Dimensions - Keith Lay, composer
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Four-Dimensionalism - Theodore Sider - Oxford University Press
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Persistence in Time | Internet Encyclopedia of Philosophy
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Mauro Dorato, Presentism/Eternalism and Endurantism/Perdurantism
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What does 4D BIM Mean for Your Construction Business? - TrueCADD
Feb 18, 2025 · Let's discuss three potential benefits of 4D BIM · Improved planning and management of construction activities · Safer working site · Savings in ...
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What are 4D and 5D Modeling? | GPRS
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Optimize Federal Builds: 4D Construction Scheduling Services
Rating 5.0 (10) Oct 29, 2024 · Achieve timely federal project success! United-BIM's 4D BIM scheduling empowers you to minimize delays, manage risks, and meet your project ...
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How 4D BIM with AI Supports Commercial Construction Projects?
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4D Technology - 4D Technology
4D Technology's Optical Metrology Products assure the quality of precision optics and optical systems. Our Dynamic Interferometry assures the surface and ...Optical Metrology Products · Contact us · About · Careers at 4D Technology
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Optical Metrology Products - 4D Technology
4D custom interferometers and laser interferometers overcome difficult environments, complex setups and one-of-a-kind measurement problems.
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4D Technology Accufiz Interferometers - QD UK
Flexible, proven and affordable, AccuFiz Fizeau laser interferometers fit well in confined lab spaces and offer high stability in any environment.<|separator|>
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4D Technology Announces 4D InSpec: The First Handheld, High ...
4D metrology products utilize state-of-the-art technology that is deployed in production environments, even in the presence of severe vibration and turbulence.
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Industries - 4D Technology
4D gauges can be found in industries from aircraft maintenance to furniture production. They measure anywhere in the factory, and anywhere on a component.
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What are the applications of four-dimensional objects? - Quora
May 11, 2023 · The geometry of four dimensions is applied (in particular) in Geophysics when computing the volumetric curvatures of geological features.
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Are there any practical applications for higher dimensional geometry?
Mar 29, 2016 · But higher dimensional spaces are practical in other ways. One of my favorite examples is parameter space. Suppose you were designing a new cell ...