1.63 Quantum Sphere

Sections 60 (first updated 03.29.2021)

The universal nature of consciousness is not the same as its particular nature. In its particular nature, it remains passive and impartial to the whole, and this passivity maintains the particular forms that it encompasses.[1] For example, she does not demand attention because she is beautiful, but rather, because she demands attention, she takes on the properties of beauty. In this way, the particular is shaped by the direction of attention rather than by an isolated essence.

Yet this finitude itself possesses the universal nature, which is the activity of all things—the rational relation of differences. In this universal sense, consciousness is impartial and external to any particular thing, yet within the particular it remains impartial to the whole of the relation.[2] If a particular consciousness were somehow given universal control, then everything would collapse into its own determinations, as parts of its universal logical relation. Its determinations would define what is taken as universal. However, the fact that such determinations can be contested shows that there is an objectivity beyond mere subjectivity.[3]

Potential, Contradiction, and Universality

In the particular lies potential—the potential for what is otherwise. It cannot both be and not be at once in a simple sense. Yet, for this contradiction not to eliminate itself, there must be a relation that sustains both sides.[4] The inverse—where there is no relation—is itself contradictory, because even the assertion of “nothing” stands in relation to “something.”

This involves universal consciousness, which remains impartial to the particular. In the universal, something must “be,” even if that being is the relation between what is and what is not. The universal is precisely this relation, and even “that” must still be.[5]

Thus, the universal involves both what is and what is not as distinct yet related, whereas the particular involves what is not as either excluded or only partially related. The universal holds contradiction together without collapse; the particular expresses only one side at a time.[6]

Edge and Centre

The edge is, at the same time, whole with the centre. The edge is where the centre conceives the object of the relation. The relation between the edge and the centre is the conception in which the object contains both the edge and the centre.[7]

For example, the perception that the Earth is spherical is taken from a position outside of it, such that the conception contains both the circumference and the area. A conception positioned entirely inside the Earth cannot derive it as spherical. The conception of edge and centre must contain the whole of that relation in order to grasp the object as such.[8]

“What Is Behind Your Eyes”

As Alan Watts suggests in his reflections on consciousness, when we ask “what is behind your eyes?”, we tend to imagine a subject located somewhere inside the head, looking outward at the world.[9] However, this intuition may be misleading. The “observer” is not simply a point inside the body, but is better understood as the field in which experience appears.

In this sense, what is behind your eyes is not another object or image, but the condition for objects to appear at all. The visual field does not have a visible observer within it; rather, the observer is implicit in the entire field of perception. This aligns with the idea that consciousness, in its universal sense, is not localized as a particular thing, but is the relational condition in which all particular things are disclosed.[10]

Thus, the distinction between inside and outside begins to dissolve. The observer is not strictly inside the body, nor is the world strictly outside. Instead, both arise within a unified field of relation. The “edge” of perception—the boundary of what is seen—and the “centre”—the point from which it is seen—are not separate, but are aspects of the same act of consciousness.[11]

The impartiality of consciousness lies in its dual nature: as particular, it remains passive within forms; as universal, it actively constitutes the relations between forms. The edge and the centre, the observer and the observed, the inner and the outer—these are not separate domains, but moments within a single relational structure.

To understand consciousness as the ultimate observer is not to locate it somewhere, but to recognize it as the condition under which all locations, distinctions, and relations appear.[12]

Footnotes

[1] Particular consciousness operates within limits and forms.

[2] Universal consciousness as relational totality.

[3] Objectivity exceeds individual subjectivity.

[4] Classical problem of contradiction and being.

[5] Being as relation between presence and absence.

[6] Universal holds opposites; particular expresses determinate states.

[7] Edge–centre relation as structure of conception.

[8] Perspective determines perceived form (inside vs outside).

[9] Alan Watts critiques the “inner observer” model of self.

[10] Consciousness as field rather than object.

[11] Non-dual interpretation of perception.

[12] Consciousness as condition of possibility for experience.

Consciousness is the centre as the middle point, the point being the duration of a spectrum.[1]

In our explanation that the centre of a sphere is any point on it, a further addition must be made to this idea in order to demonstrate the nature of consciousness. Stating that the centre of the sphere is any point on it does not yet explain how consciousness is the central point of the sphere as any point. Rather, what consciousness means as the centre point of a sphere is that it is the point from which the sphere has a centre point. Consciousness is therefore the “middle point” of the sphere, which explains that the form of consciousness is the activity that remains moderate between extremes. Its being moderate implies that the alteration between extreme points is what constitutes those extremes in the first place.[2] The diameter of the sphere, in this sense, expresses the full span of these extremes.

Satori and Satter

In Buddhism, this meditative state is called Satori. In Arabic, the word satter (also found informally in English) refers to a barrier or borderline that distinguishes two aspects from each other—like a town gate separating the city from the wilderness, or clothing separating the body from the gaze.

Satter and satori may be understood as conceptually related in that both refer to a middle condition: satori as the mental state of consciousness that mediates between two aspects of a relation, and satter as the boundary that distinguishes them.[3] In this way, the idea of a spectrum—where emphasis lies on the middle—is expressed as a conscious state that is aware of both its internal operations and its external happenings, such that the distinction itself belongs to one unified field.

Meditation and the Centre of Experience

If, for example, we enter a meditative state and develop a consciousness that watches the natural functioning of its own mechanisms and the environment as the same activity—breathing, hearing, and so on—the meditative state provides an approximate experience of consciousness as a centre point.[4]

In a closed-eye position, when consciousness focuses on breathing and sound, it is placed between the darkness provided by the eyelids and the darkness “behind” perception. This example shows that the meditative state, in order to be detached from breathing and hearing so as to be aware of them, stands at a middle point. These happenings—including the darkness, the breathing, and the sounds—occur, as it were, around this centre in a spherical manner.

If attention shifts entirely to one element, such as breathing alone, then consciousness becomes identified with that focus, and all other elements organize themselves around it. Thus, whatever is focused on becomes the centre, and the rest of experience becomes its surrounding field. This reflects the insight often emphasized by Alan Watts in discussions of meditation: that awareness is not fixed but dynamically centres itself through attention.[5]

Vision as Radius and Centre

Vision operates in a similar manner. When you look at something, you see the object of focus as the middle point between the farthest and nearest points of vision. When perception selects an object, that object has behind it a farther plane and before it a nearer plane. If you shift focus to the farther point, it too becomes a centre with its own background and foreground.[6]

Thus, vision is always a focusing on a middle point within a field. The individual mind—the point from which perception discloses the perceptual sphere—can be understood as functioning like the circumference of that sphere, while the act of focus is the radius extending toward a centre.

Let dc be the observation disclosing perception (the circumference), and r the line of vision toward the object of focus (the centre). Vision is always the radius toward a centre point, while the diameter expresses the full range between the nearest and farthest limits of that perception. The individual, as the point on the circumference, discloses the entire perceptual sphere through this relation.[7]

Uncertainty and the Structure of Perception

From this perspective, uncertainty is an inherent principle in the thing itself. Because the centre is not fixed but shifts with attention, and because every perceived centre implies further relations beyond it, no single perception can exhaust the object. The structure of experience always contains a degree of indeterminacy.[8]

This insight parallels, in a scientific context, the principle of Heisenberg Uncertainty Principle, where the act of observation is inseparable from what is observed. In both cases, the centre of determination is not independent of the act that establishes it.[9]

Footnotes

[1] Consciousness as temporal and relational midpoint within a spectrum.
[2] The middle defines extremes by mediating between them.
[3] Conceptual relation between boundary (satter) and awareness (satori).
[4] Meditation as experiential access to the structure of consciousness.
[5] Attention dynamically establishes the centre of experience.
[6] Perception always involves foreground–background structure.
[7] Geometric analogy: radius (focus), circumference (observer), diameter (range).
[8] No perception fully captures total reality.
[9] Observation and determination are inseparable in both philosophy and physics.

Where vision ends on a plain, and where the plain continues beyond that vision, constitutes a dimensional limit. When we look into the distance and see a horizon, we often assume that the limit of that plain coincides with the extent of vision; otherwise, the plain is taken to continue indefinitely toward some indefinite point. In this sense, one might say that if vision were perfect, a person could see further and further, as far as the horizon extends.[1]

While this is partially true given the latitude of a plain, an important limit is often overlooked—one that some modern “flat Earth” intuitions mistakenly grasp without fully understanding. The limit of a plain is fundamentally curvature. Even if vision were perfect and could apprehend the full extent of a plain, that plain itself, as a physical reality, possesses curvature as part of its structure.[2]

Thus, what appears flat is not absolutely flat but locally flat within a curved whole. To maintain what is perceived as a flat surface, motion is required—a motion that follows the curvature of that surface. Along this motion, the abstraction of latitude emerges, expressing how flatness is sustained only within a curved continuity. Even a disk, which appears flat, has an extremely acute curvature at its boundary, approaching a limiting angle. This implies that anything with density participates in curvature to some degree.[3]

If you observe an object from a sufficiently long distance, it becomes unclear whether it is moving toward or away from you. This ambiguity is especially evident in phenomena like a mirage. At great distances, perception loses stable reference points, and motion itself becomes indeterminate relative to the observer.[4]

Mirage

A mirage is commonly defined as an optical phenomenon in which the image of an object appears displaced from its true position due to variations in the refractive index of air. More precisely, it involves the bending (refraction) of light as it passes through layers of air at different temperatures.[5]

However, before a mirage is an optical phenomenon, it is first a phenomenon. Optics—our sense organs—do not exist independently of what they perceive. Perception is always receptive to some external phenomenon, whether it represents it accurately or not. The question of accuracy does not eliminate the reality of the phenomenon itself.[6]

Thus, what appears as an “illusion” from a distance may instead reveal something about the conditions under which perception occurs. The distance itself—the “awayness” of the object—contributes to how the object exists within that level of reality. In this sense, a mirage is not merely an error of perception, but an expression of the limits and conditions of observation.[7]

More concretely, mirages occur when hot ground heats the air directly above it, creating layers of air with different temperatures. Light passing through these layers bends, sometimes producing inverted or displaced images. For example, the familiar “water on the road” effect is an inferior mirage, where light from the sky is bent downward and appears as a reflective surface.[8]

Refraction and Temporal Displacement

These refractions or displacements of images can be interpreted as the capture of one among many potential “moments” of an object. Substances like water act as conductors in this sense: they transmit and reflect light, allowing an image to appear outside of the object itself. Mirrors do this with high precision, reflecting light in a way that produces a stable and exact image.[9]

However, when there is significant distance—and therefore time—between the object and the medium reflecting it, the reflection becomes less exact. The image may appear displaced, distorted, or unstable, as if something other than the object is being perceived. This raises the question: if it is a reflection, why is it not exact like a mirror?[10]

The answer lies in the conditions of transmission. Light does not travel through a uniform medium; it passes through varying densities and temperatures, which alter its path. Thus, what is seen is not a perfect reproduction, but a transformation shaped by the medium and the distance involved.[11]

Between Moments and Possibilities

Between any two conceptions of different things—or between two moments in time—there exists a space containing a range of possibilities. This “space” is not empty, but is structured by the relations that connect those moments. In perception, this appears as ambiguity, distortion, or multiplicity of possible interpretations.[12]

The mirage, therefore, reveals something fundamental: that what we perceive is not a fixed image of reality, but a dynamic intersection between the object, the medium, and the observer. The horizon itself, as a limit of vision, is not merely an endpoint, but a boundary where different dimensions of relation—distance, light, and perception—converge.[13]

Footnotes

[1] Horizon as limit of perception, not necessarily of reality.
[2] Earth’s curvature limits any truly infinite flat plane.
[3] Local flatness vs global curvature in geometry and physics.
[4] Loss of depth cues at large distances.
[5] refraction explains bending of light.
[6] Perception depends on phenomena, not independent from them.
[7] Illusion as condition-dependent perception.
[8] Temperature gradients produce inferior and superior mirages.
[9] Reflection as controlled refraction (mirrors vs natural media).
[10] Distance introduces variability and distortion.
[11] Non-uniform media alter light paths.
[12] Indeterminacy between temporal and spatial states.
[13] Horizon as relational boundary, not absolute edge.

In mathematics and physics, intention and action—or the reason why something happens and how it happens—are never entirely separate questions. The concept of gravity, for example, can be understood as describing the physical mechanics of intent, in the sense that it expresses how bodies “tend” toward one another according to lawful relations.[1]

In physics, an inertial frame of reference is defined as a state in which no external forces act upon an object. However, inertia itself must still be understood as a form of motion. More precisely, inertia possesses velocity, since it is the continued state of uniform motion or rest unless otherwise changed. Thus, even in the absence of external forces, there is still a persistence of state, which is itself a kind of directed condition.[2]

Even though inertia describes a situation in which no external force is applied, this does not mean that inertia is the absence of force altogether. As an unchanging state of motion, inertia can be understood as a vector, because remaining in one constant state—whether at rest or in motion—implies an ongoing energetic condition. In other words, to remain the same requires a continuous activity of remaining the same. The object is constantly “changing into itself” in order to persist as what it is. [3]

This idea echoes the insight found in Aristotle, who suggested that even rest must be accounted for as a state that is maintained rather than simply given. Rest, in this sense, is not the absence of motion but a particular mode of it. [4]

Vectors and the Nature of Motion

The nature of change is introduced through the concept of a vector, defined as a quantity having both magnitude and direction. Change itself is therefore not opposed to constancy; rather, it can be a constant state of motion. According to Newton’s First Law of Motion, an object at rest stays at rest, and an object in motion stays in motion unless acted upon by a force.[5]

The implication of this is that inertia and motion are not opposites. Instead, the “acted upon” factor represents the moment where uniform motion and change intersect—an instantaneous point of transformation. This is further elaborated in Newton’s Second Law of Motion, which explains how velocity changes when a force is applied.[6]

However, change should not be understood as arising only from external causes. The difference introduced by change is also inherent within the same phenomenon. The distinction between “no change” and “change” emerges when motion becomes directed in a definite way, making it distinguishable from its prior state. Thus, change is implicit even in inertia itself.[7]

The concept of a radius—the distance between a centre and its circumference—can be interpreted as the relation between change and its origin. Interestingly, the act of drawing a circle suggests that change may precede what we call the “original cause.” One may draw the circumference first and then define the centre, or begin with the centre and extend outward. In either case, the notion of change appears as a necessary condition for identifying origin.[8]

Acceleration and the Derivatives of Change

Acceleration can be understood as the vector composed of the derivatives of change. In basic terms, it describes the transition between no change and the maintenance of change within inertia. Mathematically, acceleration corresponds to the second derivative of position with respect to time, while velocity is the first derivative.[9]

The relationship between velocity and acceleration is often associated with Newton’s Third Law of Motion, which states that for every action there is an equal and opposite reaction. Interpreted philosophically, if inertia represents the persistence of “nothing changing,” then its counterpart is the persistence of “being,” which continuously differentiates itself from nothing. Thus, being remains itself by constantly changing, while nothing remains itself by maintaining its absence.[10]

This dynamic generates acceleration: a condition in which change and constancy are intertwined. What is always changing can still remain the same, insofar as its change preserves its identity.[11]

The direction of velocity implies acceleration because maintaining a direction requires continuous adjustment. Each moment of persistence adds a new instance of change that reinforces the same trajectory. Thus, change implies speed, since the rate at which something changes measures how it maintains its continuity over time.[12]

Inertia, Focus, and Thought

Inertia can be seen as one of the first principles of order. It describes the mechanics of focus necessary for thinking activity. Focus is the mode of reason that allows change in thought while thought remains itself. It is the condition in which variation occurs within a stable framework.[13]

Across disciplines, the concept of focus retains this meaning:

• In geology, it is the point of origin of an earthquake.
• In medicine, it is the site of infection.
• In linguistics, it is the emphasized part of a sentence.
• In perception, it is the point that brings clarity to vision.

In each case, focus organizes activity around a centre. This leads to the idea that gravity itself can be understood as a fundamental force of focus.[14]

Force, Mass, and Meaning

The equation ( F = ma ) expresses that force is proportional to mass and acceleration. The notion of “component” is crucial here, since any motion—even inertia—can be decomposed into components.[15]

The rule of gravity is that force is proportional to mass. However, experiments associated with Galileo Galilei show that in a vacuum, objects of different masses fall at the same rate. This suggests that mass does not determine motion in a simple way; rather, motion reveals the relational structure in which mass participates.[16]

In ordinary language, meaning connects words such that they express an essential fact. Reason, in the sense of aim or purpose, is what “matters”—that is, what has significance. To have a reason is for something to matter. In this way, reason and matter are conceptually linked: what is real is what has significance within a relational structure.[17]

For something to be a reason means that it is experienced as a phenomenon. This connects with insights from Albert Einstein, whose work in relativity shows that observation and physical reality are deeply interconnected.[18]

Infinity and Particularity

In mathematics, the constant ( \pi ) (3.14…) represents an infinite sequence within a finite relation. It expresses how a circle contains an unending series of particular values within a unified form. In this sense, it can be seen as an abstraction of diversity within unity—the unfolding of difference within a consistent structure.[19]

Taken together, these reflections suggest that the laws of physics are not merely descriptions of external phenomena, but also expressions of the structures through which mind apprehends and organizes reality. From this perspective, natural objects are not independent of mind, but are disclosed through the same principles—motion, relation, and form—that govern thought itself.[20]

Footnotes

[1] Gravity as lawful tendency rather than conscious intention.
[2] Inertia defined in classical mechanics.
[3] Persistence requires continuous state maintenance.
[4] Aristotle’s analysis of motion and rest.
[5] Newton’s First Law: inertia.
[6] Newton’s Second Law: force and acceleration.
[7] Change inherent within stable systems.
[8] Circular construction and conceptual priority.
[9] Calculus: derivatives of motion.
[10] Action–reaction interpreted philosophically.
[11] Identity preserved through change.
[12] Speed as rate of sustained change.
[13] Focus as structural stability in cognition.
[14] Cross-disciplinary notion of focus.
[15] Force as product of mass and acceleration.
[16] Galileo’s insights on falling bodies.
[17] Relation between meaning and materiality.
[18] Observer–reality relation in relativity.
[19] π as infinite within finite structure.
[20] Unity of physical law and cognitive structure.

The term “innumerable” means both too many quantities to count and something without a number. A number is a measure in terms of a “unit” that defines a relation of distinct variables that are themselves units of measure. A unit is a measure of itself, or itself is a unit of variables that are units that group together variables of units. A “unit” is therefore defined as a single and complete thing that at the same time forms a part of a greater and more complex system than itself. For example, 2 is a unit of two units of 1, i.e., 1 unit + 1 unit, but 2 is also a unit that is part of 4, which is composed of 2 units of 2, 4 units of 1, and even combinations like 1 unit of 3 plus 1 unit of 1, and other possible arrangements.

The idea is that the process is cumulative: a unit can be a measure of an infinite set of variables that are themselves units. The idea of innumerable as “without a number” is said to result from “too many numbers,” such that the limitation in having too many quantities results in them lacking a number. When there is too much to count, it appears to lack a number. However, it is entirely possible for a quantity to have a number even with the limitation of not being able to count it; just because we do not count to a million does not mean there is no millionth number.

The idea here is not whether something is absent because we lack knowledge of it, but rather that there seems to be an inherent limitation—or an uncertainty principle—built into the very nature of quantity itself. In arithmetic, we see this in the capacity to count toward infinity, or in there being an innumerable amount of numbers without end.[1] In geometry, we see more clearly the limitation being discussed: any shape or figure has within it the incapacity to be fully observed. In other words, no figure is ever completely seen from a single position; any shape is only partially viewed. In a one-dimensional view, a point is only a partial view of a line—it is a face-on view of a line.

A line, on the other hand, is a vector of a point. The “direction” of the vector runs from its tail to its head, and this develops length. The specific geometric definition of position concerns the way a shape is—its orientation or structure, whether it has equal sides or involves curvature—unlike the more general mathematical definition of position as the space occupied by an object relative to another object. If two different positions in space are occupied by two different components of the same figure, or if we abstract two distinct parts from the same figure and assign them different positions, then any single figure—point, line, circle, square—is an abstraction from a more general shape. In other words, they are partial views of a more comprehensive figure.

However, when we say that a line belongs to a figure like a square, it is not limited to that figure. Changing its orientation can derive a more complex figure, and thus a line is universal to all figures. Therefore, a line as a partial view of another figure does not belong to just one figure but to figures in general—it belongs to an innumerable set of figures. A straight line can be understood as a partial view of a curve or an angle within a two-dimensional figure. When a shape changes—when you rotate it—there is always a new part that becomes visible while another part goes out of view. There is always a “blind spot.” You can transform a shape repeatedly, preserving previous and new configurations, and eventually these overlapping transformations produce a dense complexity of angles, sides, and relations.

This is most explicitly seen in a sphere. Ultimately, a sphere is a circle in motion, because if you look at a still conception of a sphere, it appears as a circle. The only way to discern it as a sphere is through rotation, where a tilt in position at the axis reveals that the curvature itself is curved. The reason there is always a blind spot built into the structure of objects is that things are constantly in flux; they are always changing.[2] However, this change is not necessarily the object becoming something else (like a circle turning into a square), but rather a change in the frame of reference—the way the object is presented or perceived.

It is this aspect that is innumerable: the innumerable possibilities of how an object can move and behave. This arises because the figure of the object itself contains an element of uncertainty—a “gap” into which it can infinitely move or be reconfigured. This gap is not a physical empty space located somewhere on the object, but rather the potentiality of spatial and relational change. Space is not merely around or within the object; it is any possible relation the object can enter into, such that any part of it—or the whole—can move. Thus, the innumerable refers to the limitless possibilities for an object to transform its relations.

A curved surface is the same distance from the centre at all its points. This means that all points converge in relation to the same centre, and the place from which they originate is equally distant from that centre. A sphere is defined by this feature: all discernible points are traced back to a single point from which they are equally distant. The “back” of the sphere—what lies behind it spatially—is simply more of its curvature continuing. But it also represents its temporal dimension: its movement. Rotation brings what is behind into the front.

The question then arises: where does this rotation come from? A sphere is often understood as a circle with a vector—thus inherently in motion. Yet we usually attribute this motion to external causes: a planet rotates because of gravity, a ball spins because it is struck. It is also assumed that the whole object is uniformly in motion. However, the motion of a rotating sphere involves an unequal distribution of movement across its surface, producing a specific direction (clockwise or counterclockwise). This directional motion appears to originate from somewhere.

But this origin is not directly observable. What is observed—the roundness—derives from rotation, which includes an unobserved element: the possibility of rotation itself. This possibility extends into an uncertainty point, a blind spot, where there are innumerable ways of being. Thus, when we observe a static sphere, the unseen aspect of it is precisely this movement—the transition of a circle into a sphere. From one dimension it appears static; from another, it is sustained by motion.

Behind the sphere are all the ways it can be positioned and changed while still remaining a sphere. Not everything is possible for everything, however. There are limits: certain possibilities would negate the identity of the thing. For example, ice is not the possibility of fire, because if it were, fire would cease to be fire. However, fire can produce smoke, which emerges simultaneously with it. Thus, each reality has a structured range of possibilities. This is where numbering becomes meaningful: there are distinct variables within an event—smoke, combustion, carbon dioxide—that can be identified and counted.

Even in three-dimensional perception, limitations persist. A 360-degree panoramic view is still limited by distance. Beyond the furthest visible point lies imperceptibility, and curvature hides what lies beyond the horizon. Thus, even in maximal observation, there remains an inherent limit, reinforcing the idea that the structure of reality itself contains an element of the innumerable—both as excess beyond counting and as the intrinsic openness of possibility.[3]

Footnotes

[1] See Infinity and its role in arithmetic and set theory.
[2] Compare with Flux, where reality is understood as constant change.
[3] Related to the Uncertainty Principle, which expresses limits inherent in observation and measurement.Superposition of sphere (qbit)

The concept of a Locus refers to the set of all points that satisfy a particular condition. It is not a single object, but a rule that generates many positions as one unified structure. For example, a circle is the locus of all points that are at a fixed distance from a center. What is important here is that the unity of the figure does not come from any one point, but from the relation that governs all of them. Each individual point expresses the same condition, and therefore the many points together are not merely a collection, but the manifestation of a single governing principle. In this sense, the locus provides a way of understanding how multiplicity can still be one, not by reducing the many to a single element, but by recognizing that each element participates in the same relational structure.

This idea becomes clearer when we consider the example of birds flying together in a flock. Each bird is an individual organism with its own position, velocity, and direction, yet the flock as a whole moves as if it were a single body. There is no single bird that determines the motion of the entire group, and yet the group exhibits coherence, shape, and coordinated movement. The flock, in this sense, is a kind of living locus: each bird occupies a position that satisfies certain relational conditions—distance from neighbors, alignment of direction, responsiveness to motion—and together these conditions produce a unified pattern. The “form” of the flock is not imposed externally, nor is it located in any one bird; it exists in the relations between them.

What this shows is that unity does not require sameness of position, but sameness of relation. Each bird is in a different place, yet each is participating in the same dynamic rule. The locus, therefore, is not static but can also be dynamic: it can describe not only where things are, but how they move together. In the case of the flock, the locus is constantly changing as the birds move, yet the relational condition persists. This is why the flock maintains its form even while every individual within it is in motion. The unity is not a fixed shape, but an ongoing activity.

This connects directly to the idea of superposition. Just as a locus unifies many points under one condition, superposition unifies many possible states into one system. The flock can be understood as a superposition of individual trajectories that together form a single coherent motion. Each bird’s path is distinct, but the totality of these paths constitutes one pattern. In this way, the many bodies become the capability of one body, not by losing their individuality, but by expressing a shared relational structure.

Thus, the example of birds flying together shows how a multiplicity of distinct elements can form a single unity through relation rather than identity. The locus is the conceptual framework that explains this: it is the condition that allows the many to be one without collapsing their differences.

Geometrically, superposition is constructed in three steps:

First, two distinct and inverse points (+) and (−) lead into the same neutral point (0).

The neutral is the vertex of the positive and the negative.

Second, the vertex—or the same point where the two lines meet—discloses the two lines within a circle, as the same figure.

Third, each point (+) and (−) stands outside the disclosure of the circle as a point on the circumference of a sphere.

       +
   (   ÷  > 0   )
       -

This diagram demonstrates the three steps together.

The most important aspect to pay attention to is the point on the circumference of the sphere, which is also the centre of the sphere. Any single potential body contains the relation of which it is only a component. In this way, infinity is disclosed as one finite thing infinitely: the finite point expresses an infinite set of relations within itself.[1]

The principle of superposition states that the many are, as a group, one thing. For equations describing physical phenomena, the superposition principle states that a combination of solutions to a linear equation is also a solution. In other words, having many solutions for one equation is still one solution.[2]

In quantum mechanics, superposition is a state analogous to waves in classical physics: any two or more states can be added together (superposed, or philosophically presupposed), and the result is another valid state. Every quantum state can be represented as a sum of two or more distinct states. If a physical system may exist in many arrangements of particles or fields, then the most general state is a combination of all these possibilities disclosed into one state, often specified by a complex number. This is the equation of variables and their solutions expressed as a unit.

Quantum bits—pieces of information as a unit—illustrate this clearly. A qubit contains all the possible relations of a classical bit within one unit, holding multiple potential states simultaneously.[3]

This relates to the way a point extends to form a line, and a line extends into a circle. Superposition explains how one body forms other bodies, or how a body produces itself into other bodies, because it contains the relation of which it is one part. Implicit within it is the presupposition of itself as not moving in one direction, and simultaneously as moving in every other direction. These must be distinguished as separate entities; thus, when one body moves away, another remains as the counter-position.

If we take one classical bit, it contains within it a qubit as its capacity for motion and change. The qubit discloses all possibilities of the bit—it is the totality of its relations. Unlike classical motion, where an object is in one position and not in another, in the quantum domain the determination of one motion reproduces itself as another possible state.

This leads to quantum entanglement: the entanglement of two inverse determinations means that wherever one determination moves, its counterpart is implicitly disclosed.[4]

Any sum set of relations is disclosed by each single component acting as a part within those relations.

The idea of locus clarifies this structure. A locus is the set of all points that satisfy a given condition.[5] It is not a single point but a rule governing many points as one system. In this sense, the sphere is a locus: it is the set of all points equidistant from a centre. Likewise, the relations of superposition form a locus of possibilities unified by a condition.

A natural analogy is a flock of birds flying together. Each bird is an individual, yet the flock moves as one coherent body. The motion of the flock is not reducible to any single bird; rather, each bird participates in a shared pattern—a locus of motion. The many bodies together become the capability of one body. Just as organs with different functions form one organism, individual elements form a unified system through shared relations.

The conception or mind is itself a superposition of its object, because it discloses all the possibilities of that object.

Sphere and the Inconceivable

How much of a sphere can you see at once? It appears that there is always a “back side” facing away from perception that is inconceivable. First, this is because the sphere is not static but is the substrate of motion. Since it is in constant motion, an aspect of it always escapes perception. Motion, by definition, exceeds its immediate conception.

Second, the inconceivable part is not a fixed object. At any moment during rotation, each part of the sphere is perceivable, but not all parts at once. The “front” and “back” are identical; the distinction depends on the act of perception. Looking at the front excludes the back, and vice versa. This reflects the Law of Non-Contradiction: something cannot be both seen and not seen in the same respect at the same time.[6]

As Charles Sanders Peirce suggests in his theory of signs, perception always involves mediation: what is present implies what is absent.[7]

Thus, the inconceivable is not a defect but a structural feature: the back and front cannot be seen simultaneously, yet they are the same object. The perception of one side produces the rotation away from the other.

In a mirrored sphere, no matter its motion, the facing side appears constant, reflecting the same scene. This suggests a deeper principle: the observed remains stable relative to the observer even while the underlying system changes.

(see Higgs Boson)

One may analogously think of the Higgs field as providing a uniform background that gives consistency to observable properties like mass, much like the constant facing surface of the sphere stabilizes perception.[8]

Point, Sphere, and the Infinite

If the sphere is hollow and all sides are conceivable, there still remains the point of the radius—the centre—which has no measurable magnitude: no angles, no sides. Likewise, if a point is seen from a distance as a sphere, its quantity does not change—only the conception changes. The point can be a plane, a line, or a circle depending on how it is related.

This demonstrates a fractal-like structure: to examine a point is to expand the scope of conception. The inconceivable aspect of the point is its unchanging nature—it is the reference for all change.

The conception is a superposition of itself: all change expresses possibilities of what it is not, and these possibilities become figures that it transcends. The point is, in this sense, an infinite distance from the conception.

The point is the abstract principle of the inconceivable part of a sphere. A sphere in motion is a circle. The point as a distant sphere shows that it is one-sided at a time. The circle expresses the identity of the sphere: a return to itself.

Primary Sphere

(Add to spacetime curvature)

To understand the curvature of spacetime, it is important to see why the sphere is a primary form. The sphere is fundamental because it is the capacity that allows other forms to function within a system. It is a self-enclosing circuit: what extends outward returns inward, forming continuity.[9]

In Spacetime Curvature, gravity is not a force in the classical sense but the curvature of spacetime itself. Objects move along the “straightest” possible paths (geodesics) within this curved structure.[10]

We normally say that the plane, point, and line combine to form a sphere. Yet the sphere, as their totality, is also their presupposition.

A plane is an infinitely extended surface. Its extension is structured by straight lines, the shortest distance between two points. These lines connect in multiple directions, forming shapes like triangles. Eventually, the closure of lines forms a circle, and the motion of the circle produces the sphere.

There are no perfectly straight lines on a sphere—only the straightest possible paths (geodesics). The shortest path between two points on a sphere is part of a circle.

Inscribed, Circumscribed, and Universality

The concept of generality leads to the distinction between inscribed and circumscribed figures.

A circumscribed circle surrounds a polygon, but it is constrained by the vertices it must contain. This relates to the “minimum enclosing circle” problem: the smallest circle that contains all given points.[11] Thus, the circumscribed circle is determined by what it includes.

By contrast, the inscribed circle (incircle) is already minimal. It fits within the figure and is tangent to its sides. Because it is defined by internal relations rather than external bounds, it represents a deeper form of unity.

The inscribed circle defines universality. It is not imposed from outside but arises from within the structure itself. It is shared across different shapes because it expresses the condition of coherence within them.

Thus, universality is not mere generality (covering many things), but the principle that constitutes them together. The inscribed circle represents this: it is the internal condition that allows different forms to exist as parts of one system.

Footnotes

[1] See Infinity as expressed in finite representations.
[2] Superposition Principle.
[3] Qubit.
[4] Quantum Entanglement.
[5] Locus.
[6] Law of Non-Contradiction.
[7] Charles Sanders Peirce, semiotics and mediation.
[8] Higgs Field and mass generation.
[9] Compare with self-relating systems in Alfred North Whitehead.
[10] General Relativity.
[11] Minimum Enclosing Circle Problem.

If we take seriously the quantum mechanical claim that the observer is an invariable part of the phenomenon, then the observer is not divisible from the phenomenon itself. Science, if it is to remain consistent with this claim, cannot treat the observer as merely another fact among facts; rather, it must treat the observer as a principle. This means that the observer must be involved in every proposition about nature. Physical phenomena must therefore always be described in relation to rational notions, or, in other words, the rational principle must always form part of the description of any physical phenomenon. This is evident in quantum mechanics, where the observation of quantum particles introduces physical boundaries and limitations that are subordinate to abstract determinations and principles.

For example, in the case of quantum tunnelling, particles appear to pass through what seem to be solid barriers without the classical requirement of sufficient energy to overcome them. Particles enter through objects and emerge on the other side, not by violating physical law, but by revealing that what we call “solid” is not absolutely impenetrable. This phenomenon suggests that within every seemingly solid object there exists an infinitesimal, inwardly extended dimension—sometimes analogized to concepts such as the Schwarzschild radius or, at larger scales, wormholes. These ideas imply that objects contain within themselves a kind of internal openness or depth, an “entrance” through which motion is possible beyond classical spatial constraints. A related idea appears in the study of blackbody radiation, where energy is absorbed and emitted according to principles that reveal the limits of classical descriptions of matter and light.¹

As we develop this line of thought, we return to the original claim: that physical phenomena must be understood in connection with the role of the observer. The assumption that events occur entirely independently of the observer becomes questionable. This issue is famously captured in Albert Einstein’s phrase “spooky action at a distance,” which refers to the puzzling correlations observed in quantum entanglement.² While Einstein used this phrase critically, it highlights a deeper issue: the reluctance to fully accept the observer as a universal principle. Instead, the observer is often treated as a secondary, subjective feature, rather than something fundamentally embedded in the structure of reality.

A more systematic attempt to integrate the observer into the fabric of nature can be found in the philosophy of Alfred North Whitehead. Whitehead argues that nature is composed not of static particles, but of events or “occasions.” Reality, on this view, is a continuous field of happenings, where each event is defined by its relations to others. Within any such field, the observer is not external but participates as one of these events. The total set of relations within an event constitutes what we call a region of physical space, but this region is always internally related to acts of observation or experience.

From this perspective, the observer’s role in spacetime is not merely local or incidental. Especially when we move beyond classical, macroscopic scales, the observer’s influence takes on a more explicitly structural role. The limits of an observer’s conception—what can be perceived, measured, or thought—can be understood as corresponding to a kind of curvature in spacetime. That is, the boundary of observation is not simply a psychological limit but reflects a structural feature of reality itself. In this way, the observer is not outside the system being observed but is an intrinsic part of the very geometry and dynamics of the world.

What we observe as curvature may be the effects of the observer’s conception on physical phenomena.

Footnotes

1. The concept of blackbody radiation was central to the development of quantum theory; see Max Planck’s work on energy quantization (1900).
2. Albert Einstein, Boris Podolsky, and Nathan Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” (1935), where the phrase “spooky action at a distance” is later attributed to Einstein’s critique of quantum entanglement.

Because of the infinitesimally extended feature of objects, effects beyond the immediate perception of the observer take on a more distinctly quantum-mechanical character. When we deal with objects that are closest to us in everyday experience, we are in fact encountering a highly controlled and finite set of conditions. The environment on Earth is governed by relatively stable parameters—gravity, mass, and force—which constrain how objects behave. These constraints “hide” a vast range of possibilities, making our interactions with objects appear simple, direct, and determinate. As a result, our mental engagement with nearby objects is comparatively rudimentary: the mind seems to passively observe, while the body acts within a stable and predictable framework.

However, as the mind extends its inquiry outward into regions beyond the immediate environment—especially beyond Earth’s atmosphere—it begins to encounter phenomena that appear increasingly abstract. In such domains, physical behavior is no longer governed by the same familiar and tightly bound conditions. Shapes become distorted, spacetime appears warped, and relations between objects seem less rigid and more fluid. These phenomena can appear indeterminate or even unintelligible when judged by everyday standards of perception. One common response is to dismiss such appearances as limitations of our perceptual faculties—as if what we see is merely illusion. Yet this assumption becomes less convincing at larger scales of the universe, where even minute variations can be amplified into significant effects.

This amplification is captured by the scientific concept of the Butterfly Effect, which describes how small changes in initial conditions can lead to vastly different outcomes over time.¹ In this sense, what appears to us as distortion or irregularity in distant regions of space may not simply be a failure of perception, but the visible consequence of complex causal chains extending across vast distances. Thus, when we observe a distortion in space, we face an ambiguity: either we are witnessing the limits of our ability to perceive an effectively infinite complexity, or we are observing the real effects of processes originating far beyond our local frame of reference.

It is therefore conceivable that the perspective of another observer—situated at an immense distance—could, through a chain of interactions, influence the structure of reality as it appears to us. An observer’s frame of reference is not isolated; it extends outward through the propagation of physical processes such as light. In this sense, observation itself participates in the unfolding of reality. We may then ask whether what we perceive is the remnant of another observer’s “conception” of the world, or simply the unfolding of our own. In either case, the effect is physically real: light, as a carrier of information, travels across vast distances and connects distant regions of the universe.

At the quantum level, this interconnectedness is further suggested by the phenomenon of Quantum Entanglement, in which particles exhibit correlated behavior regardless of spatial separation.² Although entanglement does not transmit information in a classical sense, it challenges the idea that distance alone determines the independence of physical events. Thus, when light or other physical influences propagate outward over immense scales, their effects may participate in a network of relations that is not strictly local. What appears in one region may reflect conditions structured elsewhere, not as a simple transmission, but as part of a unified system of relations.

In this way, the universe can be understood not as a collection of isolated objects, but as a continuous field of interaction in which observation, motion, and relation are inseparable. The apparent distinction between local clarity and distant abstraction reflects not merely a limitation of perception, but a transition between regimes of order—between the finite stability of immediate experience and the expansive, relational complexity of the universe at large.

Footnotes

1. The Butterfly Effect is a principle in chaos theory often summarized as: small initial differences in a system can lead to large variations in outcomes over time (popularized by Edward Lorenz).
2. Quantum Entanglement refers to a quantum phenomenon in which the states of two or more particles become correlated such that the state of one cannot be fully described independently of the others, regardless of distance.

Before turning to geometry, we must clarify the problem of objectivity. Objectivity cannot wholly depend on sensation, because sense alone is unable to reconcile opposing principles of change. Objectivity does not merely concern the fact that something exists, but also what that fact is, which may include what it is not. This cannot be derived by sensation alone. From the standpoint of sensation, what an object is not appears only as something external to it—an “other.” Yet the full determination of an object requires grasping both what it is and what it excludes. This raises a deeper question: why is there a distinction between internal and external relations? What maintains the object as distinct from thought, and why are thought and object held apart within consciousness? These questions lead us into the structure of geometry as a model for understanding consciousness.

As is often the case in discussions of polygons, triangles occupy a special position in the study of inscribed and circumscribed figures. Every triangle can be inscribed in a circle (a circumcircle) and can also circumscribe a circle (an incircle). This dual property is unique to triangles and does not hold universally for higher-order polygons.¹ This “universal dual membership” makes the triangle a privileged structure for thinking about the relation between universality and particularity.

Generality can be understood as activity or function, while particularity is result or object. When a circle is inscribed within a triangle, it functions as an internal principle—it touches each side and acts as a unifying activity within the figure. When a circle circumscribes a triangle, it appears as an external completion, enclosing the triangle as a finished object. In this way, the inscribed circle represents activity (function), while the circumscribed circle represents result (object). The inscribed circle reflects an internal organizing principle, whereas the circumscribed circle reflects an external boundary.

The circle, however, is fundamentally always inscribed in a deeper sense. This is suggested by the “minimum enclosing circle” problem in geometry, which shows that any set of points can always be enclosed by a circle, and that smaller circles can often be constructed within given boundaries.² Thus, the internal relation—the inscribed structure—is more fundamental, while the circumscribed relation is limited by the shape it encloses.

This geometric relation provides an analogy for consciousness. Consciousness, when “inscribed,” is the active function within the object—it is the process, the change, the internal relation that gives the object its coherence. Consciousness, when “circumscribed,” is the reflective stance that takes the object as fixed and studies it as something given. These two moments occur simultaneously: consciousness both constitutes the object internally and observes it externally.

In this sense, consciousness as infinity operates as universality. The dialectical sequence can be described as follows: (i) consciousness is identical with itself, like a circle; (ii) it contradicts its self-identity by producing an object, analogous to a triangle within or against the circle; (iii) it then forms the idea of this relation, uniting itself and its object. This is the structure of dialectical development.

More explicitly, consciousness unfolds in three moments:

1. Consciousness produces the object and is initially identical with it.
2. Consciousness critiques or differentiates itself from the object.
3. Consciousness grasps both the object and its critique as a unified whole.

This sequence is not strictly temporal but structural: the critique is already implicit in the generation of the object, and the synthesis is the totality of both.

This dialectical structure can even be seen reflected in physical forms. For example, in the molecular structure of water (H₂O), the distribution of electric charge is uneven, with a concentration around the oxygen atom.³ This asymmetry resembles a kind of “geometric” organization, where relations are structured around a center. Such patterns suggest that what appears as contradiction in thought may correspond to real structural relations in nature.

The notion of touch in geometry further clarifies this relation. A tangent is a line that touches a curve at exactly one point without crossing it. Touch, therefore, is contact without full intersection. The curve represents deviation, while the tangent represents linear direction. The point of tangency is where these meet. The word “intersection,” from the Latin inter (“between”) and secare (“to cut”), indicates not only where lines cross, but where relations are established.⁴ The point is not merely where two lines meet; it is a site of transformation, where line and curve mediate each other.

Touch, in a broader sense, is the mechanism of contact and communication. In physics, contact allows the transmission of forces or signals, such as electric current. Thus, touch is the condition for the transfer of information. Yet the information itself is prior to the mechanism—it is the content that the mechanism conveys.

In trigonometry, these relations are formalized. The tangent function, for example, is defined as the ratio of the side opposite an angle to the adjacent side in a right triangle. The adjacent sides share a common vertex, while the hypotenuse is the longest side opposite the right angle. These relations define the internal structure of the triangle and demonstrate how proportion and relation give rise to measurable form.

Thus, the triangle is not merely a geometric figure but a model of dialectical reasoning. The dialectic is the logical relation of differences, and the triangle is its mathematical expression. It unites internal and external relations, activity and result, function and object. In this way, geometry becomes a language for understanding the structure of consciousness itself.

Footnotes

Etymology of “intersection” from Latin inter (“between”) and secare (“to cut”).

For properties of triangles with incircles and circumcircles, see standard Euclidean geometry; e.g., Elements.

The “minimum enclosing circle” (or smallest enclosing circle) is a well-known problem in computational geometry concerning the smallest circle that contains a set of points.

The polarity of water molecules arises from the electronegativity difference between hydrogen and oxygen; see Chemical Bonding.

Etymology of “intersection” from Latin inter (“between”) and secare (“to cut”).

Adjacent: Having a common vertex (an angular point) and sharing a side with another angle or side in a triangle.

Hypotenuse: The longest side of a right triangle, located opposite the right angle.

In trigonometry, the terms opposite, adjacent, and hypotenuse describe the relations between the sides of a right triangle relative to a given angle. The sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. These ratios express how a fixed relation (the angle) generates different measurable aspects depending on perspective. In this way, trigonometric functions already demonstrate a deeper principle: that a single relation can manifest in multiple forms depending on how it is approached.¹

Universe as Circle

This suggests that the universe, though constituted by the infinite, nevertheless possesses form. The claim that the universe is “spherical” does not merely refer to a physical shape, but to a structural principle: the sphere (or circle) is the form in which every point can relate to every other point through a unified whole. To suppose that the infinite has no shape is, paradoxically, to assign it a direction—yet any direction already presupposes a form within which that direction occurs.

The modern cosmological view, following the Big Bang, describes the universe as expanding outward, even accelerating in that expansion.² This outward motion is often interpreted as a direction into space. However, a deeper interpretation is that this “expansion” reflects an increasing immediacy of becoming—an unfolding of potential into actuality. This unfolding is what we call time. Time is the motion internal to a thing, the process by which it becomes what it is, whereas space is the external relation of things, the field in which they appear to one another.

In this sense, Aristotle describes time as the “number of motion with respect to before and after,” or the affection of motion as it is experienced in change.³ Time belongs to motion as it is internalized within a being, whereas space expresses motion externally.

The Circle as Universal Form

When we say that the circle is the universal form, we are not merely claiming that the universe looks like a circle. Rather, the circle is the condition that enables any form to appear at all. It is the form of total relation. The circle is that shape in which all points are equally related to a center, and thus equally related to one another through that center. In this way, it becomes a symbol of universality.

Yet the circle contains a contradiction. On the one hand, what lies within the circle is limited—it is defined by its relation to other points within the same boundary. On the other hand, what lies outside the circle appears unlimited. However, it is mistaken to think that this “outside” is simply an infinite extension in every direction away from the circle. If the outside were merely external extension, then the circle would not truly be contained within anything and would lose its definition.

Instead, what lies outside the circle must itself be infinite in such a way that it contains the circle. The “void” is not merely absence, but a condition that holds the circle in place. For example, if a laptop floats in space, we still say it exists in space—space is not absent but present as the condition of its existence. Thus, what is outside the circle is not separate from it but is the condition that contains it.

It is equally mistaken to think that something finite, such as a line, could contain the circle. A line can only touch the circle at a point or segment; it cannot encompass the whole. Therefore, the circle, in its completeness, contains the line rather than the reverse. From this, we see that the infinite outside the circle presses upon it in such a way that the circle both limits and expresses that infinity. The circle becomes a contradiction: it is both contained by the infinite and itself contains a determination of that infinite.

Thought, Contradiction, and Actuality

If we reinterpret the infinite “outside” as the principle of reason, we begin to see how thought operates. Every idea exists potentially beyond any given form, but this potentiality presses toward actualization. The circle represents the moment where this potential becomes actual. The infinite possibilities are limited into a determinate form, and that form is the idea as it exists.

Thus, the circle is the transition from the abstract to the concrete. The contradiction between infinite potential and finite determination is not an error but the very condition of reality. This is what later philosophers like Georg Wilhelm Friedrich Hegel describe as the dialectical movement of thought: the unity of opposites within a higher synthesis.⁴

Squaring the Circle

The classical problem of “squaring the circle” illustrates this contradiction. The problem asks whether it is possible to construct a square with the same area as a given circle using only a compass and straightedge. This was proven impossible in modern mathematics because it would require constructing a length proportional to π, which is a transcendental number.⁵

However, beyond its technical impossibility, the paradox reveals something deeper. It is often said that the impossibility lies in the infinite number of steps required to transform one figure into the other. Yet this overlooks the more fundamental issue: the circle itself is the ground of this infinite process. The attempt to square the circle reveals that each figure contains the principle of the other, yet cannot fully become it without ceasing to be itself.

If a square were perfectly transformed into a circle, it would no longer be a square. Likewise, if a circle were fully squared, it would cease to be a circle. This reflects the Law of Non-Contradiction: something cannot be both itself and not itself in the same respect at the same time.⁶ The paradox, therefore, is not merely mathematical but ontological—it expresses the limits of transformation and the necessity of distinction within unity.

The circle, understood in this way, is not merely a geometric figure but a model of the universe and of thought itself. It expresses how the infinite becomes finite, how potential becomes actual, and how contradiction is not a flaw but the very structure of reality. The universe is “circular” not because it is a simple shape, but because it is a system in which all relations return into a unified whole.

Footnotes

1. Basic trigonometric definitions from classical geometry; see Elements.
2. Observations of cosmic expansion and acceleration are associated with modern cosmology; see work related to Edwin Hubble and later dark energy research.
3. Aristotle, Physics, Book IV (discussion of time as the measure of motion).
4. Georg Wilhelm Friedrich Hegel, dialectical method as developed in Science of Logic.
5. The impossibility of squaring the circle was proven in 1882 by Ferdinand von Lindemann, showing π is transcendental.
6. The Law of Non-Contradiction is a foundational principle of classical logic, articulated by Aristotle.

The squaring of the circle is often presented in modern mathematics as an impossibility: the task of constructing, using only a compass and straightedge, a square with the same area as a given circle. This impossibility was rigorously proven only in the 19th century, when it was shown that the number π is transcendental and therefore cannot be constructed by finite geometric means.[1] Yet to interpret this result merely as a “failure” is to miss the deeper philosophical significance that earlier traditions—and later thinkers like Georg Wilhelm Friedrich Hegel—intuitively grasped.

For ancient civilizations, the problem was not simply technical but symbolic. The circle and the square represented fundamentally different orders of being: the circle, continuous and without beginning or end, stood for infinity and unity; the square, composed of straight lines and right angles, represented finitude, structure, and determinacy. The impossibility of perfectly transforming one into the other was not a contradiction to be eliminated, but an expression of a real and necessary tension within reality itself. What appears as a “dead end” in formal reasoning is, in a broader philosophical sense, the very mechanism through which infinity becomes thinkable.[2]

From this perspective, the infinite number of steps required to “square” the circle is not a defect but the point. The fact that no finite procedure can complete the task reveals that the relation between curved and rectilinear forms is inherently open-ended. Every attempt to reconcile them generates further distinctions, further refinements—an endless progression. This is precisely what Hegel identifies as a dialectical movement: the contradiction between forms does not halt thought but propels it forward, generating higher levels of understanding.[3]

Geometrically, this can be seen in the mismatch between curvature and angle. A circle can never be perfectly inscribed into a square without leaving gaps, nor can a square fully conform to the curvature of a circle. There is always a remainder—an excess or deficiency—that prevents total equivalence. But rather than viewing this remainder as failure, it can be understood as the space in which measurement itself operates. It is the “gap” that allows comparison, transformation, and approximation. In this sense, the impossibility of squaring the circle becomes the very condition for the infinite measurability of space.

Historically, many ancient mathematical and architectural traditions employed approximations of this relationship—not to solve it definitively, but to work within its productive tension. The so-called paradox functioned as a generative principle: a way of organizing space, proportion, and symbolic meaning. Temples, mandalas, and city plans often reflect attempts to reconcile circular and square forms, embodying the unity of heaven (circle) and earth (square) in a single structure.[4]

Thus, what modern thought isolates as an abstract impossibility can be reinterpreted as a universal mechanism: the continuous mediation between the infinite and the finite. The circle is not “captured” by the square, nor the square by the circle; instead, their non-coincidence generates an infinite process of approximation. This process is not external to reality—it is the very way reality articulates itself.

In this light, the squaring of the circle is not a failed construction but a conceptual tool: a way of thinking the relation between form and infinity, between determination and indeterminacy. Its “impossibility” is precisely what makes it indispensable.

Footnotes

[1] Ferdinand von Lindemann (1882) proved that π is transcendental, implying the classical construction problem is impossible.
[2] See historical interpretations in Greek mathematics and later Neoplatonic symbolism regarding circle–square relations.
[3] Georg Wilhelm Friedrich Hegel, Science of Logic: contradiction as the engine of dialectical development.
[4] Comparative studies in ancient architecture (e.g., Egyptian, Indian, and Greek traditions) show symbolic uses of circle–square correspondences.

The squaring of the circle phenomenon is seen as one of the oldest principles adapted by ancient civilizations because it is not a contradiction as we see it today, but rather this impossibility that it exhibits is the resolution itself. It is the function or mechanism for capturing infinity, that an infinity of steps cannot make a shape into the other simply means this diversity of possibilities in shape and its relation to area of space. That in the area of space, shape always falls into that area as not perfectly symmetrical with an inverse and opposed shape, yet there is a hierarchy.

The circle can never be captured within an angle, as the angle is straight and the circle is curved. The curve never fills perfectly within that angular corner; there is always a space in which these shapes are contained and not the same. They do not fit by design, and therefore this apparent contradiction is actually the resolution itself according to Georg Wilhelm Friedrich Hegel.[1] For in our hypothetical science it appears as a contradiction, as a dead end in reasoning; however, in nature, outside our abstract constructions, it is the mechanism of measurement—the infinite measurement of all things.[2]

The impossibility is not a failure to reconcile forms, but the very condition under which their relation becomes intelligible. The circle and the square do not coincide, and precisely because they do not, they generate an endless process of approximation. This process is what allows measurement to occur at all: each attempt to equate them produces a new determination, a finer distinction, a further articulation of space. In this way, infinity is not something externally added to geometry, but is already present in the relation between forms that cannot be perfectly unified.[3]

The ancients utilized this paradox as a mechanism of resolution. Rather than eliminating the difference between curved and straight, they preserved it as a productive tension through which form, proportion, and spatial order could be understood. The “failure” to square the circle becomes the very means by which the infinite is grasped within the finite, not as something completed, but as something continuously unfolding.[4]

Footnotes

[1] Georg Wilhelm Friedrich Hegel, Science of Logic: contradiction is not mere error but the driving principle of development.
[2] Classical geometry treats the problem as impossible in finite construction; see the transcendence of π established in 1882 by Lindemann.
[3] The relation between curved and rectilinear magnitudes historically generated the development of limits and calculus.
[4] Ancient mathematical and symbolic traditions often used circle–square relations as expressions of cosmic order rather than solvable problems.

last updated 04.08.2026