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The Perpendicular Paradox:
A Model-Induced Conflict Between Euclidean Geometry and Non-Euclidean Reality
Abstract
In Euclid’s Elements (Book I, Definitions 10 and 23, Propositions 12 and 31), perpendicularity and parallelism form the bedrock of spatial reasoning, epitomized by the square’s right-angled adjacent sides and parallel opposite sides (see Figure 1, illustrating a square’s Euclidean structure with perpendicular horizontal and vertical lines). This framework assumes a plane with zero curvature where perpendicular lines support parallel structures. Yet, a paradox arises when applied to physical reality: under the non-Euclidean model of Earth, vertical lines—aligned with gravitational pull, as in the perpendicles of architectural practice—converge toward the planet’s centre, violating Euclid’s parallel postulate. This Perpendicular Paradox, and its specific horizontal-vertical instantiation the #HorizontalParadox, is evidenced by reciprocal zenith angle measurements (see Figure 2, where plumb lines at two stations yield angles summing to greater than 180°, indicating non-parallelism due to Earth’s geometry). This tension reveals a model-induced conflict: Euclidean assumptions, rooted in Elements’ axioms, falter in a non-Euclidean world where geodesics intersect. We analyse how perpendicularity, as a Euclidean ideal, presupposes an uncurved space, only to be disrupted by the non-Euclidean model. Non-Euclidean geometries resolve the paradox by accommodating converging geodesics, while Euclidean methods remain valid for local approximations, as in construction. Bridging historical geometry, philosophy of science, and applied mathematics, this study elucidates the epistemic limits of Euclidean reasoning, offering a framework for understanding model-driven paradoxes and their implications for geometric thought.
Keywords: Euclidean geometry, non-Euclidean geometry, Perpendicular Paradox, geodesics, model-induced paradox
The Perpendicular Paradox:
1. Introduction
Euclidean geometry, as formalized in Euclid’s Elements, has long served as the cornerstone of human spatial reasoning, providing a framework of perpendicularity and parallelism that shapes our understanding of the world. In Elements (Book I, Definitions 10 and 23, Propositions 12 and 31), Euclid defines perpendicular lines as those meeting at right angles and parallel lines as those that never intersect in a plane, axioms that underpin the structure of fundamental shapes like the square (Euclid, 300 BCE/1956). The square, with its right-angled adjacent sides and parallel opposite sides, epitomizes Euclidean order, embodying a symmetry and structural clarity that resonates deeply with human cognition. Its perpendicularity—adjacent sides meeting at 90°—and parallelism—opposite sides never converging—rely on the assumption of a Euclidean plane, a space with zero curvature where these properties hold universally.
Yet, a paradox emerges when this Euclidean framework is applied to physical reality, revealing a tension that sharpens scientific inquiry. As Hans Christian von Baeyer notes, “Paradox is the sharpest scalpel in the satchel of science. Nothing concentrates the mind as effectively, regardless of whether it pits two competing theories against each other, or theory against observation, or a compelling mathematical deduction against ordinary common sense” (von Baeyer, 2003, Ch. 23). Under the widely accepted non-Euclidean model of Earth, vertical lines—aligned with gravitational pull, as in the perpendicles used in architectural practice—converge toward the planet’s centre, violating Euclid’s parallel postulate. This Perpendicular Paradox, or its specific horizontal-vertical instantiation the #HorizontalParadox, exposes a model-induced conflict: the Euclidean assumption of an uncurved plane, implicit in perpendicular relationships, clashes with the non-Euclidean geometry of Earth’s surface. For instance, in surveying, reciprocal zenith angle measurements between two stations demonstrate that plumb lines are not parallel, as their zenith angles sum to greater than 180° (see Figure 2). This paper explores how perpendicularity, as a Euclidean ideal exemplified by the square (Figure 1), presupposes an uncurved space, only to falter in a non-Euclidean reality. A Euclidean proof further underscores this contradiction, showing that lines cannot simultaneously be horizontal and vertical in the presence of a circle (Figure 3), highlighting the paradox’s roots in Euclidean axioms.
We argue that non-Euclidean geometries resolve the paradox by accommodating intersecting geodesics, while Euclidean methods remain valid for local approximations, as in construction. Bridging historical geometry, philosophy of science, and applied mathematics, this study elucidates the epistemic limits of Euclidean reasoning, offering a framework for understanding model-driven paradoxes and their implications for geometric thought. Moreover, the paradox invites philosophical reflection, particularly in the realm of epistemology, as it challenges the reliability of Euclidean geometry as a model for knowing physical reality, raising questions about how our mathematical frameworks shape—and sometimes distort—our understanding of the world. The Perpendicular Paradox thus serves as a scholarly tool to interrogate the interplay of mathematical theory, physical reality, cognitive modelling, and philosophical inquiry, bridging Euclid’s legacy with modern geometric and epistemological insights.
The Perpendicular Paradox:
2. Euclidean Geometry and the Square
Euclidean geometry, as articulated in Elements, provides a systematic framework for understanding spatial relationships through axioms and propositions. Central to this system are the concepts of perpendicularity and parallelism. Euclid defines a right angle as one formed when a straight line standing on another makes adjacent angles equal (Book I, Definition 10), and perpendicular lines as those meeting at such angles (Book I, Proposition 12). Parallel lines, meanwhile, are straight lines in a plane that do not meet, no matter how far extended (Book I, Definition 23; Proposition 31 constructs parallels through a point). In three dimensions, Euclid extends these principles: Proposition 6 of Book XI states that if two straight lines are perpendicular to the same plane, they are parallel to each other (Euclid, 300 BCE/1956). These principles are vividly embodied in the square, a shape whose four equal sides and four right angles make it a paragon of Euclidean order.
The square’s structure is a direct consequence of Euclidean axioms: its adjacent sides are perpendicular, forming 90° angles, and its opposite sides are parallel, maintaining a constant distance (see Figure 1, where a square is depicted with clearly labeled perpendicular and parallel sides in an enhanced diagram). This configuration assumes a Euclidean plane, a two-dimensional space with zero curvature, as Euclid’s geometry does not account for non-Euclidean geometries. The square’s significance in Euclidean geometry is further underscored by its connection to other geometric constructions, such as the Vesica Piscis (see Figure 4), a shape formed by the intersection of two circles of equal radius, where each circle’s center lies on the other’s circumference. The Vesica Piscis can be used to construct a square: by connecting key points of the overlapping region, one can form a right-angled triangle or a square, a process that aligns with Euclid’s Proposition 47, Book I—the Pythagorean Theorem—which states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Euclid, 300 BCE/1956). This theorem reinforces the square’s role as a symbol of Euclidean order, where perpendicularity and parallelism are foundational, and geometric constructions like the Vesica Piscis highlight the interconnectedness of Euclidean principles.
When extended to three dimensions, as in architectural applications, the square’s perpendicularity informs the use of vertical lines—such as perpendicles—that are expected to be perpendicular to horizontal planes, per Proposition 6, Book XI. The square’s appeal lies not only in its mathematical elegance but also in its practical utility—perpendicles, tools used in construction to ensure vertical alignment, rely on the same perpendicularity that defines the square’s form. As such, the square is often regarded as the “most Euclidean shape,” a symbol of the uncurved geometry that Euclid codified, further enriched by its geometric ties to constructions like the Vesica Pisces and foundational results like the Pythagorean Theorem.
However, this Euclidean ideal encounters a challenge when confronted with physical reality. A Euclidean proof illustrates the inherent contradiction when applying these axioms to a non-Euclidean context: if horizontal and vertical lines are perpendicular to each other and intersect a circle (representing Earth’s geometry), they cannot consistently maintain their horizontal and vertical properties simultaneously (see Figure 3, where lines A, B, and C are horizontal lines parallel to each other, D, E, and F are vertical lines perpendicular to A, B, and C, yet their perpendicularity leads to an absurdity with the circle). This proof, rooted in Euclid’s own definitions, foreshadows the Perpendicular Paradox: the very axioms that make the square a Euclidean exemplar also render it incompatible with a non-Euclidean world, setting the stage for the conflict between model and reality.
The Perpendicular Paradox:
3. The Perpendicular Paradox Defined
The Perpendicular Paradox arises from a fundamental tension between the idealized principles of Euclidean geometry and the physical reality of Earth as a non-Euclidean manifold, a conflict that becomes evident when examining the behaviour of perpendicular and parallel lines in both contexts. In Euclidean geometry, perpendicularity and parallelism are defined with precision: perpendicular lines meet at right angles (Euclid, Elements, Book I, Definition 10), and parallel lines never intersect in a Euclidean plane (Book I, Definition 23). In three dimensions, Euclid’s Proposition 6, Book XI, further asserts that lines perpendicular to the same plane are parallel to each other (Euclid, 300 BCE/1956). These properties are exemplified in the square, where horizontal and vertical lines form right angles, and opposite sides remain parallel, maintaining a constant distance (see Figure 1). A key expectation in this framework is that horizontal lines—such as those used in differential leveling—are parallel in an uncurved Euclidean plane, and vertical lines (e.g., plumb lines) are perpendicular to these horizontals. Consequently, verticals perpendicular to the same horizontal plane should be parallel, per Proposition 6, Book XI.
However, this expectation is disrupted when we apply Euclidean principles to the physical world under the non-Euclidean model of Earth. On a non-Euclidean manifold, vertical lines—defined as those aligned with the local gravitational pull (i.e., plumb lines)—are perpendicular to level surfaces (non-Euclidean horizontals) at points of equal elevation, but these verticals do not remain parallel; instead, they converge toward the planet’s center, directly contradicting the parallelism expected in a Euclidean context. This convergence is empirically demonstrated through reciprocal zenith angle measurements, a technique used in surveying to determine the relative orientation of plumb lines at two stations. As shown in Figure 2, two stations, A and B, on the Earth’s surface have plumb lines that form angles with a line of sight between them. In a Euclidean plane, these angles would each be 90°, summing to 180°, indicating that the plumb lines (perpendicular to parallel horizontal planes) are parallel, as expected from Euclid’s axioms. Yet, on a non-Euclidean manifold, the measured angles are greater than 90° (94°03'00" at A and 94°04'05" at B), summing to more than 180° (188°07'05"), which confirms that the plumb lines are not parallel but converge toward the Earth’s center due to the manifold’s curvature (Woolfson, 2012). This deviation from Euclidean expectations forms the crux of the Perpendicular Paradox: the perpendicular relationship between horizontal and vertical lines, which implies parallelism among verticals in a Euclidean plane, fails in a non-Euclidean reality.
The paradox is further illuminated by a Euclidean proof that exposes the contradiction inherent in applying Euclidean axioms to a non-Euclidean context. As depicted in Figure 3, lines A, B, and C are horizontal lines parallel to each other, while lines D, E, and F are vertical lines perpendicular to A, B, and C. These lines intersect a circle representing the Earth’s geometry, with angles marked at 90°. The proof demonstrates that if horizontal and vertical lines maintain their Euclidean properties (perpendicularity and parallelism) while intersecting a circle, an absurdity arises: for instance, line A cannot simultaneously be horizontal (parallel to B and C) and serve as a radius extending past the circle’s circumference, contradicting Euclidean definitions (Sarugaki, personal communication). This contradiction underscores the model-induced nature of the paradox: the non-Euclidean model of Earth forces a reevaluation of Euclidean assumptions, as the uncurved plane required for the square’s perpendicular and parallel properties—and the parallelism of verticals perpendicular to horizontal planes—does not align with the non-Euclidean reality of Earth’s surface as a manifold with positive curvature. The Perpendicular Paradox, or its specific horizontal-vertical instantiation the #HorizontalParadox, thus emerges as a conflict between the idealized geometry of Euclid and the physical world, highlighting the limits of applying Euclidean principles universally.
The Perpendicular Paradox:
4. Reality’s Non-Euclidean Nature
The non-Euclidean nature of physical reality, particularly under the non-Euclidean model of Earth, is a well-established scientific fact that fundamentally challenges the Euclidean framework central to the Perpendicular Paradox. Early Greek philosophers like Pythagoras (c. 570–495 BCE) are often credited with proposing that the Earth might not be flat, and by around 240 BCE, Eratosthenes measured its circumference, suggesting a three-dimensional shape consistent with the Euclidean geometry of the time (Russo, 2004). However, their understanding would have been rooted in Euclidean terms—interpreting the Earth as a three-dimensional sphere embedded in Euclidean space, not as a non-Euclidean manifold. It is only through modern scientific advancements, such as satellite imagery, GPS technology, and gravitational data, that we recognize Earth’s surface as a two-dimensional non-Euclidean manifold, exhibiting positive curvature—meaning that the geometry of the surface lacks parallel lines, and all geodesics intersect. This geometry directly impacts the behavior of geometric constructs like vertical lines: because gravity pulls toward the Earth’s center, plumb lines—used as practical verticals in applications like perpendicles—converge at the planet’s core rather than remaining parallel, as they would in a Euclidean plane.
This geometry aligns with non-Euclidean principles, specifically those of a geometry with positive curvature, where the parallel postulate does not hold in the Euclidean sense. In such a non-Euclidean geometry, there are no parallel lines; all geodesics—the intrinsic equivalents of straight lines on the surface of the manifold—intersect. For example, on Earth’s surface as a non-Euclidean manifold, meridians (lines of longitude) are geodesics lying in the azimuthal plane, orthogonal to lines of latitude, and they intersect at the poles, a direct violation of Euclid’s parallel postulate (Greenberg, 2007). More pertinently, if we consider “vertical” lines as those aligned with gravity—such as plumb lines extending radially from the Earth’s surface to its center—these radial lines converge at the center when extended inward. On the two-dimensional surface itself, each plumb line intersects the manifold at a single point, effectively becoming a point on that surface. These points can be grouped into level surfaces defined by equal elevation, often referred to as non-Euclidean horizontals, which are non-Euclidean parallel in the sense that surfaces at different elevations do not meet, forming concentric surfaces in the non-Euclidean geometry. The directions of these plumb lines at those points, if projected as paths on the surface, would follow geodesics—the intrinsic “straight” paths in non-Euclidean geometry—that intersect, rather than remaining parallel as they would in a Euclidean plane where straight lines maintain parallelism (Euclid, 300 BCE/1956). The plumb lines themselves, being perpendicular to these level surfaces, converge at the Earth’s center due to the manifold’s curvature, a behavior empirically shown in the zenith angle measurements of Figure 2: the sum of angles exceeding 180° indicates that plumb lines, despite being perpendicular to non-Euclidean horizontals, are not parallel, consistent with the geometry of the non-Euclidean manifold. This non-Euclidean behavior extends beyond Earth to cosmic scales, as described by Einstein’s general relativity, where space-time itself exhibits non-Euclidean properties due to mass and energy (Einstein, 1916).
The historical development of non-Euclidean geometry provides further context for understanding this reality. In the 19th century, mathematicians like Carl Friedrich Gauss, Nikolai Lobachevsky, and János Bolyai independently developed non-Euclidean geometries by modifying Euclid’s parallel postulate. Gauss, for instance, conducted surveys to test the geometry of physical space by measuring the angles of a large triangle formed by three mountain peaks in Germany, aiming to determine whether the angle sum deviated from 180°, which would indicate non-Euclidean curvature (Gauss, 1828). Although his measurements were inconclusive due to the small scale of the triangle relative to Earth’s radius and the precision of instruments at the time, Gauss’s theoretical work on curved surfaces—particularly his development of Gaussian curvature—demonstrated that in a geometry of positive curvature, such as on a sphere, the angle sum of a triangle would exceed 180°. These insights laid the groundwork for modern geometry, which recognizes that Euclidean principles are an approximation valid only in spaces with negligible curvature, such as small-scale local contexts. In architectural practice, for example, the use of perpendicles to ensure vertical alignment assumes a Euclidean plane locally, where Earth’s geometry is imperceptible. However, over global scales—such as in geodesy or astronomy—the non-Euclidean nature of reality becomes undeniable, forcing a reconciliation with the Perpendicular Paradox. The square, while a perfect embodiment of Euclidean geometry in an uncurved plane, thus serves as a point of departure for exploring how physical reality diverges from Euclidean ideals, necessitating a shift to non-Euclidean frameworks to fully understand the world.
Figures
Figure 1: The Euclidean Square
An enhanced diagram of a square with labelled horizontal and vertical sides, showing adjacent sides meeting at 90° (perpendicular) and opposite sides marked as parallel. This figure illustrates the Euclidean ideal of a plane with zero curvature, where perpendicularity between horizontal and vertical lines supports parallelism, as assumed in Euclid’s Elements. It serves as a visual representation of the “most Euclidean shape,” highlighting the structure that the Perpendicular Paradox challenges when applied to Earth as an elliptic manifold.
Figure 2: Reciprocal Zenith Angle Measurements
A detailed diagram depicting two stations (A and B) on Earth as an elliptic manifold, with plumb lines (verticals) aligned with gravity. The horizontal line at each station forms a 90° angle with its plumb line, but the line of sight between A and B intersects the plumb lines at angles greater than 90° (94°03'00" at A, 94°04'05" at B). The sum of these angles exceeds 180°, demonstrating that the plumb lines converge at the Earth’s centre, consistent with elliptic geometry. This figure, based on surveying principles, visually confirms the Perpendicular Paradox: Euclidean expectations of parallel verticals fail in a non-Euclidean reality.
Figure 3: Euclidean Proof of Contradiction
A diagram showing lines A, B, and C as horizontal lines parallel to each other, and lines D, E, and F as vertical lines perpendicular to A, B, and C. These lines intersect a circle (representing Earth’s geometry), with angles marked at 90°. The accompanying proof demonstrates that if horizontal and vertical lines are perpendicular and intersect a circle, they cannot consistently maintain their horizontal and vertical properties simultaneously (e.g., line A cannot be both horizontal and a radius extending past the circle’s circumference). This contradiction, rooted in Euclid’s definitions, underscores the Perpendicular Paradox’s origin in Euclidean axioms applied to a non-Euclidean context (Sarugaki, personal communication).
Figure 4: Vesica Pisces and the Square
A diagram of the Vesica Pisces, formed by the intersection of two circles of equal radius, where each circle’s canter lies on the other’s circumference, creating an overlapping almond-shaped region. This geometric construction can be used to form a square or a right-angled triangle, illustrating Euclid’s Proposition 47, Book I—the Pythagorean Theorem—where the square of the hypotenuse equals the sum of the squares of the other two sides. This figure reinforces the square’s role as a fundamental Euclidean shape, highlighting the interconnectedness of Euclidean geometry’s principles of perpendicularity and parallelism.
References
Einstein, A. (1916). The Foundation of the General Theory of Relativity. Annalen der Physik, 354(7), 769–822.
Euclid. (300 BCE/1956). The Thirteen Books of Euclid’s Elements (T. L. Heath, Trans.). Dover Publications
Gauss, C. F. (1828). Disquisitiones Generales Circa Superficies Curvas. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores.
Greenberg, M. J. (2007). Euclidean and Non-Euclidean Geometries: Development and History (4th ed.). W. H. Freeman.
Isaacson, W. (2007). Einstein: His Life and Universe. Simon & Schuster.
Kuhn, T. S. (1962). The Structure of Scientific Revolutions. University of Chicago Press.
Russo, L. (2004). The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn. Springer.
von Baeyer, H. C. (2003). Information: The New Language of Science. Harvard University Press
Woolfson, M. M. (2012). Surveying and Geodesy: Principles and Applications. Oxford University Press.
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THE BLUE MARBLE INDUCED HORIZONTAL PARADOX
Obviously horizontal planes exist in reality.
So why do some people get so upset at hearing that level is a straight line on a horizontal plane?
Finding these people is what led us to realise that simplifying everything down to it's most basic form and building up from there, is what was needed to fully understand.
Taking the first principles approach to even the most simple task is what we call ADVANCED SIMPLICITY and it's how we assure ourselves that WE fully understand what it is that we are speaking of.
The Horizontal Paradox is an ideal introduction to Advanced Simplicity and it goes like this;
A plane is a geometric concept that is 2D, that means it has only 2 dimensions, length and breadth, it has no edges and therefore is infinite.
A plane is well defined and understood as the base for understanding geometry within the foundations our learning process, but have we ever really considered such simple things at the basis of our own thought processes?
There are proofs within geometry that can give you an understanding of these basics, but what's important is that you know they are there for you to find. The picture to the right is A page from a book called 'The Elements' it is a series of proofs and theorems that guide to find truth.
This one proposition (along with all the others preceding it) lets us know that all perpendiculars to a plane will be parallel, knowing this may cause cognitive pain at some point if a contradiction is pointed out.
Here's the contradiction; for some people, placing this plane in the horizontal position becomes a problem for the reason stated above - verticals being parallel is a big problem for the dogmatic '''globe''' believers.
The Horizontal paradox is only a paradox when a model is introduced, and once introduced how do you resolve the contradiction? How would you resolve a contradiction? How can you resolve the contradiction?
Welcome to Advanced Simplicity and On The Level Thinking.