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Impossible cube

The impossible cube, also known as the irrational cube, is a two-dimensional optical illusion representing a three-dimensional cube that defies the laws of Euclidean geometry and cannot be constructed in physical space.^[1] It appears as a perspective drawing of a cube where edges and faces inconsistently shift between foreground and background, creating an undecidable figure that challenges the viewer's perception of depth and structure.^[1] This illusion was invented by Dutch graphic artist M. C. Escher in 1958 as a central element in his lithograph Belvedere, a print depicting an architectural scene incorporating multiple impossible objects.^[2] In Belvedere, the impossible cube is shown as a model held by a figure in the foreground, set within a larger impossible tower that blends realistic shading with paradoxical geometry.^[3] Escher's work built on earlier explorations of impossible figures, such as the impossible triangle by Oscar Reutersvärd in 1934, but the cube variant specifically exploits ambiguities in cubic projection similar to the reversible Necker cube illusion.^[4] The impossible cube has become a seminal example in the study of visual perception, cognitive psychology, and the philosophy of impossible objects, illustrating how the human brain interprets ambiguous two-dimensional lines as coherent three-dimensional forms despite inherent contradictions.^[1] It influences fields from art and design to computer graphics, where it demonstrates limitations in rendering realistic 3D models from 2D perspectives.^[1]

Definition and Description

Visual Representation

The impossible cube is commonly rendered as a two-dimensional wireframe diagram composed of twelve straight line segments representing the edges of a cube, arranged to outline the front, side, and top faces while suggesting spatial depth through converging lines.^[5] These edges form a closed structure with vertices where three lines typically meet, creating the visual impression of interconnected surfaces in a three-dimensional form.^[6] In standard depictions, the drawing emulates isometric projection, with non-parallel edges of the cube oriented at equal angles—often approximately 120 degrees—to parallel axes, fostering an illusion of uniform depth without a single vanishing point.^[7] This style avoids true perspective distortion, keeping all faces equally foreshortened for a balanced, symmetrical appearance that enhances the cube's apparent solidity.^[6] Variations often incorporate shading or coloring to amplify the three-dimensional effect, such as subtle gradients on faces to simulate light incidence or distinct solid colors to delineate surfaces and imply occlusion.^[6] The classic wireframe version, frequently executed in bold black ink against a white background, maximizes line contrast to emphasize the geometric framework without additional embellishments.^[5] One notable example appears in M.C. Escher's 1958 lithograph Belvedere, where the cube is portrayed as a tangible polyhedron assembled from beams, maintaining the wireframe essence amid a surreal architectural scene.^[2]

Key Features of the Illusion

The impossible cube illusion deceives the visual system through ambiguous depth cues, particularly monocular indicators like interposition and Y-junctions at edge overlaps, which prompt the brain to infer a three-dimensional structure from the two-dimensional line drawing. These cues lead to conflicting interpretations of relative distances among the cube's bars, as the visual processing integrates local geometric signals into a global form that cannot consistently exist in Euclidean space. Studies on infant perception demonstrate that even young viewers detect these inconsistencies by relying on such cues to discriminate impossible from possible cubes, highlighting the early emergence of depth processing mechanisms.^[8] A key element is the role of figure-ground organization, where the cube's edges create a reversible figure that allows faces to alternate in perceived orientation, exploiting Gestalt principles such as Prägnanz to favor simpler, closed 3D interpretations over fragmented 2D alternatives. This reversibility arises from ambiguous overlaps, where parts of the structure appear simultaneously as figure and ground, preventing stable segregation and perpetuating perceptual instability without a preferred resolution. The brain's tendency to impose continuity on these edges reinforces the illusion by suppressing local anomalies in favor of an overall coherent object hypothesis.^[9] The perceptual effects unfold in stages: an initial impression conveys a valid cube seemingly rotating in depth, as the visual system rapidly constructs a plausible 3D model from familiar line configurations. Upon sustained inspection, however, the inconsistency becomes apparent, revealing edges that cannot connect without violation, which underscores the cognitive bias toward hypothesis-driven perception over direct sensory verification. This delayed detection illustrates how low-level feature integration precedes higher-level consistency checks in visual processing.^[10] The illusion's potency is modulated by viewing angle, as minor shifts in observer position—such as head tilt—alter the apparent convergence of parallel lines, thereby changing the relative salience of depth cues and the overall coherence of the perceived form. These variations can weaken or intensify the deceptive quality without eliminating the core impossibility, since the drawing's fixed inconsistencies persist across perspectives. This sensitivity to viewpoint echoes the critical angles observed in three-dimensional realizations of impossible objects, where alignment enhances the paradoxical effect.^[10]

Historical Development

Origins in Art and Geometry

The foundations of the impossible cube trace back to Renaissance experiments in perspective, where artists sought to represent three-dimensional forms like cubes on a two-dimensional plane. Albrecht Dürer, in his 1525 treatise Underweysung der Messung mit dem Zirckel und Richtscheyt, systematically explored geometric projections of solids, including cubes, to achieve realistic depth through linear perspective techniques.^[11] These efforts highlighted the challenges of translating Euclidean geometry into visual art, as inconsistent viewpoints could lead to perceptual ambiguities in cube renderings, laying groundwork for later paradoxical figures.^[12] In the 18th and 19th centuries, advancements in descriptive geometry further illuminated the limitations of such projections. Gaspard Monge, often credited as the inventor of descriptive geometry, developed methods in the 1790s to accurately depict three-dimensional objects using orthogonal projections onto two planes, as detailed in his lectures at the École Polytechnique. While intended for precise engineering and artistic representation, Monge's system inadvertently underscored the difficulties in maintaining consistent spatial relations for complex forms like cubes, where multiple projection views could reveal inconsistencies not apparent in single perspectives.^[13] Precursors to explicit impossible figures appeared in artistic critiques of perspective errors; for instance, William Hogarth's 1754 engraving Satire on False Perspective deliberately incorporated paradoxical architectural elements and lines, to mock flawed geometric depictions in art.^[14] By the late 19th century, optical illusion studies began to formalize these perceptual discrepancies, with early cube-like figures emerging in psychological research. Louis Albert Necker's 1832 rhomboid drawing, later interpreted as a cube, demonstrated how line-based representations could flip between two stable 3D interpretations, revealing the brain's role in resolving ambiguous projections—a direct antecedent to impossible cubes that defy any single valid interpretation.^[15] Johann Joseph Oppel's 1855 work on geometrical-optical illusions, including distortions in perceived lengths and angles within grid-like structures, extended these ideas by coining the term and experimenting with shapes that challenged Euclidean expectations, predating formal impossible object recognition.^[16] Impossible figures, including cube variants, arose culturally from the tension between artistic pursuits of illusionistic depth—rooted in Renaissance techniques—and the inherent constraints of Euclidean geometry, which assumes consistent spatial rules ill-suited to paradoxical 2D renderings.^[17] This intersection fostered informal sketches in 19th-century illusion studies, where researchers like Oppel explored how perspective could produce undecidable forms, influencing later artistic explorations without yet achieving the deliberate impossibilities of the 20th century.^[18]

Evolution and Popularization

The concept of impossible figures originated earlier with Swedish artist Oscar Reutersvärd's invention of the impossible triangle in 1934.^[19] The mid-20th-century popularization of impossible figures occurred in 1958 when psychiatrist Lionel S. Penrose and mathematician Roger Penrose published their seminal paper "Impossible objects: A special type of visual illusion" in the British Journal of Psychology. Their work focused on the impossible triangle and endless staircase but extended the framework to other inconsistent three-dimensional representations in subsequent psychological studies of visual perception and cognitive processing.^[20] In the same year, Dutch artist M.C. Escher independently introduced the impossible cube in his lithograph Belvedere, depicting a paradoxical architectural structure that highlighted geometric ambiguities and inspired numerous cube-based variants explored by artists and researchers in the 1960s. The impossible cube's popularization accelerated in the 1970s and 1980s through its integration into educational materials and emerging visual technologies. It featured prominently in textbooks on perceptual psychology, notably Richard L. Gregory's Eye and Brain: The Psychology of Seeing (first edition 1966; revised editions through 1990), where it exemplified how the brain interprets contradictory depth cues, influencing curricula in cognitive science and art education. Concurrently, the figure appeared in graphic design applications, including experimental logos and early computer graphics, where vector line drawings facilitated demonstrations of projection and rendering in systems like those developed at Xerox PARC and academic labs during the vector graphics era. In the modern digital era, the impossible cube proliferated via accessible design software, such as Adobe Illustrator (introduced 1987), which streamlined the creation of scalable vector variants for print and web use, democratizing its application in digital art and animation. Its spread intensified online around 1995 with the advent of graphical web browsers, as early illusion websites and forums shared interactive and animated versions, leading to viral dissemination in educational and entertainment contexts.^[1]

Mathematical Explanation

Perspective and Projection Errors

Linear perspective is a technique for creating the illusion of depth in two-dimensional representations of three-dimensional objects, achieved by drawing parallel lines in the object as converging toward one or more vanishing points on a horizon line.^[21] This convergence mimics how the human eye perceives distance, where objects farther away appear smaller and parallel features, such as the edges of a road or building, seem to meet at a distant point.^[22] In depictions of cubes, linear perspective ensures that all sets of parallel edges in the 3D form share the same vanishing point when projected onto the 2D plane, maintaining spatial coherence.^[23] Relevant projection types for rendering cubes include orthographic and isometric projections, both of which differ from linear perspective by avoiding convergence altogether. Orthographic projection represents objects as if viewed from an infinite distance, keeping all parallel lines parallel in the drawing to preserve true dimensions without distortion from depth.^[24] Isometric projection, a subset of orthographic (specifically axonometric), displays three faces of a cube equally, with axes at 120-degree angles and no vanishing points, allowing accurate measurement but lacking the depth illusion of perspective.^[25] The impossible cube disrupts these principles by inconsistently blending elements: some edges remain parallel as in orthographic or isometric views, while others converge toward vanishing points as in linear perspective, resulting in a hybrid that defies consistent spatial interpretation.^[23] Specific errors arise in the impossible cube's structure, where the front face may align with a standard perspective projection using a single vanishing point for its receding edges, but the adjacent side and top faces employ incompatible vanishing points or parallel lines that fail to align.^[23] This mismatch causes non-Euclidean overlaps, such as edges that appear to intersect impossibly or faces that shift between foreground and background positions.^[1] To demonstrate visually, trace one edge from the front face rearward; it should connect to a side edge converging to the same vanishing point, but instead leads to a parallel or misaligned segment on the top face, creating a loop where the path returns to the starting point in a contradictory orientation that cannot exist in three-dimensional space.^[1] Continuing the trace around the figure reveals further inconsistencies, such as a bar appearing as the front edge from one viewpoint but the rear from another, underscoring the projection's fundamental incompatibility.^[23]

Geometric Inconsistencies

The impossibility of realizing the impossible cube in three-dimensional Euclidean space can be demonstrated through a coordinate geometry approach. Attempting to assign 3D coordinates to the eight vertices of the figure—typically labeled A through H based on the 2D line drawing—reveals fundamental inconsistencies in the z-depth values required to connect all edges as depicted. For example, placing vertex A at (0,0,0) and propagating coordinates along one set of edges (e.g., A to B to C to D) yields a specific position for an opposite vertex like H, but tracing an alternative path (e.g., A to E to F to G to H) results in a conflicting location for H, as the z-coordinates cannot simultaneously satisfy both routes without violating the projected 2D positions. This depth contradiction emerges from the propagation of local depth information across the structure's components, where beams or edges that appear connected in the 2D projection are physically separated in 3D, leading to incompatible global positioning.^[26] A topological analysis further underscores the issue, as the edges of the impossible cube form a graph whose embedding in 3D space is impossible without self-intersections or distortions that contradict the figure's apparent connectivity. The cube's skeletal structure represents a specific configuration of cycles and links that violates embedding theorems in Euclidean 3D; for instance, the arrangement requires disjoint components to intersect or link in ways prohibited by properties of spatial graphs. This aligns with intrinsic linking theory, where theorems such as the Conway-Gordon-Sachs result demonstrate that certain point sets in 3D inevitably produce linked cycles, but the impossible cube's topology cannot be realized without forcing such prohibited intersections in its projection.^[27] These inconsistencies manifest quantitatively through distance calculations between vertices. The Euclidean distance formula,

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2},

applied to pairs of vertices assumed to form equal-length edges (as in a regular cube), yields unequal values when coordinates are constrained to match the 2D outline. Edges that should measure the same length—say, unit length 1—compute to disparate distances (e.g., one path giving d \approx 1 while another gives d > 1) due to the unresolved z-depth conflicts, proving no uniform scaling or positioning satisfies all constraints simultaneously.^[26] Efforts to resolve these issues in 3D modeling software, such as Blender or Maya, fail to generate a coherent model from the 2D outline without introducing distortions, self-intersections, or fragmentation. Standard rendering pipelines, relying on consistent perspective or orthographic projections, cannot accommodate the ambiguous depth cues of the impossible cube, as the linear system for vertex placement becomes overconstrained and unsolvable (e.g., the affine transformation matrix for projection has insufficient rank to fit the 3D points). To approximate the illusion, software requires manual decomposition into separate 3D parts—each locally valid but globally disconnected—followed by view-dependent compositing during rendering, which simulates connectivity only from specific angles but breaks under rotation or alternative viewpoints.^[28]

Applications and Interpretations

Usage in Visual Art

The impossible cube serves as a powerful motif in surrealist visual art, where artists integrate it to evoke dreamlike and paradoxical realities that challenge perceptual norms. Following M.C. Escher's seminal 1958 lithograph Belvedere, which features the cube as a central architectural element in an impossible structure, subsequent works in the 1960s drew inspiration from this form to amplify surreal effects, blending geometric precision with irrational spatial logic.^[29] These pieces, often incorporating impossible geometries for perceptual disruption, appeared in paintings and prints that mirrored the era's fascination with altered consciousness and multidimensional ambiguity.^[30] In the 1970s, Dutch artist and mathematician Bruno Ernst extended the impossible cube into sculptural forms, creating physical approximations that rely on perspective shifts and mirrors to manifest the illusion in three dimensions. Ernst's stainless steel models, such as those depicting the cube alongside other impossible figures like the triangle, allow viewers to observe the paradox from dual angles—one revealing the seamless impossibility and the other exposing structural breaks—thus emphasizing the interplay between illusion and reality in tangible art.^[31] His adaptations underscore a creative intent to democratize Escher's concepts through accessible, interactive installations.^[32] The motif also permeates graphic design, notably in 1980s psychedelic album covers, where the impossible cube symbolizes perceptual paradox and mind-expanding themes. Designers like those at Hipgnosis incorporated Escher-inspired optical distortions into photomontages for rock albums, enhancing the era's fusion of visual psychedelia with musical exploration.^[33] Since the 2010s, contemporary digital art has revitalized the impossible cube through algorithmic generation and NFTs, enabling dynamic variations in distortion for interactive experiences. Artists leverage computational tools to produce evolving versions of the cube, often in generative formats that explore infinite permutations of the illusion, as seen in AI-assisted creations that render and manipulate physically impossible geometries.^[34] This evolution highlights the cube's adaptability in virtual realms, where code-driven aesthetics amplify its paradoxical essence.^[35]

Role in Optical Illusions and Design

The impossible cube plays a significant role in psychological research on visual cognition, particularly since the 1960s, where it exemplifies how the brain constructs three-dimensional interpretations from inconsistent two-dimensional cues. Early studies, such as those by Richard Gregory in 1973, utilized impossible figures like the cube to investigate perceptual organization, demonstrating that viewers rely on Gestalt principles—such as closure and continuity—to interpret the figure as a coherent object despite inherent geometric contradictions.^[9]^[36] These experiments highlight the brain's tendency to prioritize local depth cues like interposition over global inconsistencies, often leading to prolonged fixation as the visual system resolves or ignores paradoxes.^[9] These mechanisms involve neural processing that combines learned and inferred approaches to depth perception in impossible figures. Neuroimaging research has extended these findings, revealing that impossible cubes engage similar early visual cortical areas as possible objects, with event-related potentials indicating initial processing without immediate detection of impossibilities, though cognitive conflicts emerge in higher-order areas.^[37] For instance, a 2011 fMRI study on analogous impossible figures (such as the two-pronged trident) showed no significant differences in early neural activation between possible and impossible conditions, suggesting the visual system builds illusory depth before inconsistency detection, which can introduce subtle processing delays in interpretive stages.^[38] Developmental studies since the late 2000s have also incorporated the impossible cube, finding that even 4-month-old infants exhibit preferences for and discriminate between possible and impossible cubes, evidencing innate sensitivities to geometric violations in visual cognition.^[8] In education, the impossible cube functions as a hands-on tool for illustrating perspective and geometric principles, appearing in museum exhibits and curricula to foster spatial reasoning. This encourages exploration of projection errors, aligning with broader perceptual education goals. In formal settings, impossible figures like the cube are integrated into geometry textbooks and lessons; for example, Australian curricula from the early 2000s, such as Queensland's Mathematics A Senior Syllabus (2000), emphasize 3D-to-2D representations using such illusions to address student difficulties in spatial visualization.^[39] Classroom applications have proliferated since the 2010s, with high school programs in regions like Northern Nevada (2010) and Italy (2012–2019) employing the impossible cube alongside M. C. Escher's artworks to teach perspective drawing and inconsistency detection through activities like creating personal impossible objects.^[39]^[17] These methods, rooted in earlier examples like the Penrose triangle (1958), help students grasp Euclidean geometry violations, improving engagement and metacognitive skills in visualizing non-Euclidean forms.^[17] In computer graphics, the impossible cube demonstrates challenges in rendering 3D models from 2D perspectives, influencing techniques in video games and virtual reality to create perceptual effects. Recent tools, such as MIT's Mescher (as of August 2025), allow visualization and editing of physically impossible objects, advancing research in digital illusion generation.^[35] Commercially, the impossible cube inspires branding and design elements that leverage optical intrigue to convey innovation and complexity. Abstract variants appear in tech interfaces and logos, where isometric illusions mimic the cube's paradoxical structure to draw viewer attention; for instance, vector-based designs from the 1990s onward in software graphics evoke endless depth for user interfaces.^[40] Such motifs are common in modern emblem creation, as seen in royalty-free collections emphasizing retro 3D effects for halftone labels and emblems, enhancing brand memorability through perceptual ambiguity.^[41] In therapeutic contexts, the impossible cube contributes to art therapy programs focused on perception paradoxes, particularly from the 2000s, by prompting clients to confront and reinterpret visual ambiguities. Documented initiatives, such as mindfulness-based exercises inspired by Op Art illusions, use cube-like figures to build cognitive flexibility and reduce anxiety, as participants draw or assemble models to navigate shifting viewpoints.^[42] These activities, integrated into sessions since the early 2000s, draw on perceptual psychology to explore self-perception, with therapists noting improved emotional regulation through engagement with geometric inconsistencies.^[43]

Other Impossible Figures

The Penrose triangle, also known as the impossible triangle or tribar, is considered the foundational impossible figure, originally created by Swedish artist Oscar Reutersvärd in 1934 and popularized by Lionel S. Penrose and Roger Penrose in their 1958 paper on visual illusions.^[20] This two-dimensional polygonal form depicts three bars meeting at right angles to form a closed triangular loop, creating a paradox through inconsistent perspective that cannot exist in three-dimensional space.^[44] Unlike the impossible cube, which projects a three-dimensional polyhedron with volumetric enclosure, the Penrose triangle relies on a simpler planar ambiguity, emphasizing a flat, infinite loop rather than depth and solidity.^[45] The impossible trident, or impossible fork, emerged in the 1960s as a variant of impossible figures, first described by D. H. Schuster in 1964 as a "three-stick clevis."^[46] It illustrates an object with three cylindrical prongs at one end that inexplicably merge into two rectangular prongs at the other, exploiting inconsistencies in line connections to defy logical assembly.^[47] Similar to the impossible cube in its use of edge ambiguity to mislead depth perception, the trident focuses on manipulative object features like prong divergence rather than the cube's enclosed spatial paradox, resulting in a more localized perceptual flip at the junction points.^[14] The blivet, another 1960s impossible figure synonymous with the impossible trident or devil's pitchfork, gained its name through a 1967 article in Worm Runner's Digest by Harold Baldwin, which popularized the term for such ambiguous forks.^[46] This figure features an apparent proliferation of holes or prongs that multiply illogically from three to two, inducing a perceptual reversal akin to other impossible objects but without the pronounced spatial depth illusion of the cube.^[48] In contrast to the cube's emphasis on volumetric inconsistency across an entire structure, the blivet highlights surface-level topological anomalies, such as impossible tunneling through solid forms.^[49] These impossible figures, including the cube, share common construction methods rooted in ambiguous line junctions, where edges appear to connect locally in a consistent manner but fail globally when viewed as a whole.^[6] Techniques often involve axonometric projections and stylized edges—such as flat, convex, or reflex stylizations—to resolve apparent dihedral angles at junctions, creating local plausibility while ensuring overall impossibility.^[6] However, the impossible cube uniquely stresses a volumetric paradox through its polyhedral projection, differentiating it from the more linear or prong-based ambiguities in the Penrose triangle, trident, and blivet.^[6]

Connections to Modern Geometry

The impossible cube, as a projection that defies Euclidean realization, finds parallels in non-Euclidean geometries where such structures can emerge naturally. In hyperbolic spaces, modeled by the Poincaré disk from the early 1900s, curved geometries allow for tilings and embeddings that appear "impossible" under Euclidean rules, such as cube complexes with branching factors exceeding those possible in flat space. These models demonstrate how the cube's inconsistent projections might correspond to valid configurations in negatively curved manifolds, bridging optical illusions to foundational non-Euclidean concepts developed by Henri Poincaré.^[50] In topology and graph theory, impossible figures like the cube relate to challenges in interpreting 2D projections as consistent 3D embeddings, connecting to broader topological invariants such as those explored by Roger Penrose using cohomology to formalize such illusions mathematically.^[51] Computational geometry addresses the impossible cube through algorithms for validating 3D model consistency in CAD systems, pioneered by Kokichi Sugihara since the 1980s. Sugihara's work on anomalous pictures developed linear-time methods to determine if line drawings like the impossible cube can be realized as polyhedra, using constraint satisfaction to detect geometric inconsistencies in projections. These algorithms, integrated into CAD software for 3D modeling validation, ensure designs avoid impossible configurations by solving inverse problems in polyhedral reconstruction.^[52]^[53] The impossible cube influences virtual reality design by testing rendering techniques and immersion limits since the 2010s, where non-Euclidean engines simulate such figures to create disorienting yet engaging environments. Systems for prototyping impossible objects in VR, such as those using interactive geometry processing, allow users to navigate apparent paradoxes, enhancing spatial perception studies and narrative depth in immersive simulations. This application extends the cube's theoretical inconsistencies to practical tools for exploring perceptual boundaries in digital spaces.^[54]^[55]

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The Impossible Trident is an impossible figure (or impossible object or undecidable figure): it depicts an object which could not possibly exist.
[50]
4. Forks - Cameron Browne. Impossible fractals
The Devil's fork, also known as the impossible fork, Devil's pitchfork, Devil's tuning fork or the blivet [1], is another famous impossible object.
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Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes ...
Oct 2, 2012 · Thirty years ago, the mathematician William Thurston articulated a grand vision: a taxonomy of all possible finite three-dimensional shapes.
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[PDF] Math 228: Kuratowski's Theorem
Theorem 1 (Kuratowski's Theorem, layman's terms). We know two nonplanar graphs, they are K3,3 and K5. So of course any graph containing those is not planar.Missing: cube | Show results with:cube
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Three-dimensional realization of anomalous pictures—An ...
Anomalous pictures are naively regarded as pictures of impossible objects, but some of them are realizable as three-dimensional polyhedral objects.
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Kokichi Sugihara's English Homepage
My impossible objects are exhibited at National Palace Museum in Taipei from September 21, 2018 to February 23, 2020. Please see here. IMA Publid Lecture on " ...Missing: computational CAD
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Prototyping impossible objects with VR - ACM Digital Library
Nov 28, 2018 · The purpose of the present study is to develop a system for prototyping impossible objects with VR, which can be used for prototyping impossible ...
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[PDF] Illustrations of non-Euclidean geometry in virtual reality - arXiv
Aug 3, 2020 · Weeks, which makes use of state-of-the-art computer graphics algorithms to render several non-Euclidean geometries. (Weeks 2002). The ...