From Number to Geometry

In the Greek tradition, and especially in Timaeus, geometry occupies an intermediate position between the intelligible and the visible. Plato describes the cosmos itself as being ordered through number and form, the visible world shaped according to intelligible patterns that cannot themselves be directly seen. He distinguishes between the eternal pattern, which does not exist in time, and the created cosmos, which comes into being together with time:

Time, then, and the heaven came into being at the same instant in order that, having been created together, if ever there was to be a dissolution of them, they might be dissolved together. It was framed after the pattern of the eternal nature, that it might resemble this as far as was possible; for the pattern exists from eternity, and the created heaven has been, and is, and will be, in all time. (1)

Within this framework, geometry can be understood in two ways, one true to its ideal form, abstract, beyond time, and the other, in the world, in time, finite and perceptible. This idea of finitude, of being within time is central here. Certain ratios or quantities, such as √2, √3, φ, and π, cannot be fully expressed within any finite numerical system. Within our finite world, in time, they resist completion; they extend indefinitely. In that sense they could be interpreted as belonging to infinity, beyond time, and if we take Plato’s ideas to heart, to the pattern used by the creator. Yet the geometric constructions to which they belong, the square, the equilateral triangle, the circle, for example, can be constructed in this finite world, though in a necessarily approximate way. Thinking about the relationship between the side of a square and its diagonal, or the diameter of a circle and its radius, brings us closer to this world beyond time, as these ratios, which can only ever be approximate in this world, may pertain to another. 

At Giza, we can interpret the dimensions of the plateau and the pyramids as numerical expressions of astronomical cycles. Lengths correspond to periods; distances become durations; the inch itself operates as a mediating unit between space and time. The Great Giza Rectangle, in particular, is interpreted in this study as a numerical framework within which planetary cycles, lunar periods and long durations such as precession can be brought into relation.

Yet the significance of the system extends beyond number alone. Geometry introduces transformation. A length becomes a perimeter; a perimeter generates a circle; a circle becomes a square; a diagonal produces a new figure; an area extends a cycle into a surface; a height becomes a radius. At every stage the underlying quantity is preserved while its form changes. Number remains constant, but its geometrical expression is continually renewed. It is this succession of transformations that gives the system its generative character. The Great Giza Rectangle generates the pyramids; the pyramids introduce further geometrical relationships; those relationships, in turn, become the basis for new constructions. Geometry is therefore not simply descriptive but productive: it creates a network of equivalent forms through which one reality is translated into another.

Seen in this light, the geometry is not merely an intellectual exercise. It offers a way of thinking about permanence and change simultaneously. The same mathematical relationships persist while continually assuming new forms, be it line, square, circle, pyramid, Earth, or astronomical cycle. What changes is not the underlying order but the way in which that order is embodied. Geometry becomes a language of transformation, capable of expressing unity within diversity and continuity within becoming.

Human existence unfolds in precisely this way. Our bodies change, generations succeed one another, landscapes are cultivated and transformed, kingdoms rise and disappear, while the heavens continue their cycles overhead. Ancient sacred architecture may therefore have been intended not simply to represent the cosmos but to participate in it. By translating the same numerical relationships between architecture, astronomy and the Earth itself, monuments such as Giza suggest that change need not imply disorder. Transformation becomes the very means by which enduring order is revealed. In this sense the geometry is neither abstract mathematics nor mere symbolism. It is a way of contemplating the relationship between time and eternity, between the visible world and the invisible patterns that give it coherence. 

The geometry explored here may therefore represent more than mathematical ingenuity. It offers a way of reconciling permanence with change. Heraclitus famously observed that no one steps into the same river twice, because both the river and the person are continually changing. Yet change itself presupposes continuity. The river remains recognisable despite the perpetual movement of its waters. Geometry performs an analogous function. Numerical relationships remain invariant while their embodiments continually change. A length becomes a perimeter; a perimeter becomes a circle; a circle becomes a square; a square becomes a pyramid; architecture becomes astronomy; spatial measure becomes time. Identity is preserved, not by resisting transformation, but through it. 

We can read the elements of the Giza plateau as a sequence of operations, underpinned by geometric constants, which stand at the threshold between the measurable and the immeasurable. Yet this tension is not confined to irrational quantities such as√2, √3, ϕ, or π; it lies within geometry itself. Euclid begins the Elements with the deceptively simple definition of a point as “that which has no part,” followed by the definition of a line as “breadthless length.” These definitions open onto a profound problem. A geometrical point has position but no extension: it has no length, breadth or depth, and cannot be subdivided into anything smaller. No physical point can therefore correspond perfectly to it. When a dot is drawn on paper,  on papyrus, however small, it has extension within the world. So it’s not straightforward to explain how a line could be composed from such points: however many things without length are assembled, they do not simply acquire length by accumulation. Such questions generated debate in antiquity and continued through medieval philosophy, touching upon the nature of continuity, divisibility and the relationship between mathematical objects and the physical world. The geometrical point is perfectly intelligible but physically unrealisable; the physical mark is perfectly real but geometrically imperfect.

Geometry therefore confronts us, from its most elementary objects onward, with entities that can be conceived exactly but never perfectly embodied. The point exists with absolute precision in thought, in definition and in construction, but, if it is to be indivisible, never quite as a material object; the same is true of the perfect line, circle, or square. There is the figure that can be drawn, measured or built, and there is the exact figure according to which it is conceived. The distinction recalls Plato’s separation between sensible things and their intelligible patterns, but it is not dependent upon Plato alone: it arises from the practice and foundations of geometry itself. To work with√2, √3, ϕ, or π intensifies the same tension. These quantities can be defined and constructed exactly, while resisting complete expression as finite numerical measures. If we take Plato’s formulation, what is constructed participates in a pattern without ever achieving the perfection of that pattern.

In this sense, geometry offers a way of thinking about the human condition itself: situated within limits, yet continually oriented toward what exceeds them. We inhabit a world in which there are no dimensionless points, breadthless lines or perfect circles, yet we can conceive them, reason from them and use them to impose intelligible order upon matter. The Giza complex can be approached as a system in which astronomical time is translated into spatial form through a sequence of geometric operations: an encounter between ideal relations and their finite embodiment in stone. The yantra illustrates an idea that recurs throughout this chapter: geometry can itself become an object of contemplation. It does not depict mountains, people or stars, but relationships. Squares, triangles and circles become vehicles through which abstract truths are explored. In this respect, geometry is no longer simply descriptive but participatory, inviting the observer to move intellectually between the finite figure before the eye and the ideal forms it represents. 

Figure 1. Bronze yantra meditation plaque, India, 1801-1900, Wikimedia Commons. Abstract geometric constructions, like drawing the diagonal of a square, or the diameter of a circle, or indeed squaring the circle, embody ideal forms and mathematical truths that are not possible in the physical world.

To work with √2, √3, φ, or π is to engage with quantities that can be constructed, but if we are to take Plato’s idea, which I think would have been similar to ideas formulated in ancient Egypt, constructed after a pattern but never achieving the perfection of the pattern. How does the infinite or indivisible relate to the finite and divisible? How does an intelligible form become matter? A pyramid built from millions of imperfect stones can instantiate a geometrically exact pyramid that, strictly speaking, exists nowhere in the stones themselves. Its apex, as a mathematical point, has zero dimensions; its edges as mathematical lines have zero breadth; its faces as mathematical planes have zero thickness. None of those things physically exists in the monument, and yet without them we cannot even say what geometrical object the monument is. Geometry mediates between what can be conceived perfectly and what can only be embodied approximately.

For Plato, the ideal world of forms is considered separate from the world of physical things, and is more perfect from the world of experience, though just as real. In his view, objects of pure geometry, such as straight lines, circles and squares, are only ever approximated in the finite world of our existence. Precise mathematical truths and geometrical objects exist only in a separate world, an ideal world of concepts. The only way we can access this other world is via the intellect. 

More recently, Roger Penrose's framework of the three worlds is a philosophical and conceptual model that aims to explain the relationship between mathematics, the physical universe, and human consciousness. The three worlds Penrose identifies are the physical world, comprising everything that exists physically, including all matter and energy, from subatomic particles to galaxies, the mental world, which is the realm of human consciousness, thoughts, perceptions, and mental experiences, and the Platonic world, which is the abstract realm of mathematical forms, concepts, and truths. The Platonic world includes numbers, geometrical shapes, and all mathematical structures that exist independently of human thought and the physical universe, whereas the mental world is about subjective experiences, including emotions, creativity, and the sense of self. Human consciousness allows access to the Platonic world of mathematical truths, and mathematics provides a precise language to describe the physical universe. Penrose's three worlds framework emphasises the profound interconnectedness between mathematics, physical reality, and human consciousness.

Just as ancient astronomy distinguishes between the enduring framework of the heavens and the cyclical motions unfolding within it, the geometry of Giza appears to distinguish between invariant relationships and the transformations through which they are expressed. This suggests that geometry and astronomy may have been understood as complementary expressions of a single cosmic order: one revealing permanence, the other revealing ordered becoming - even though, for Plato, all eight motions, the seven planets and the background stars are all set in motion together by the Demiurge. Like the ordered motions of the heavens, the geometry at Giza appears to preserve invariant relationships through continual transformation. Whether expressed in celestial cycles or in geometrical constructions, order is maintained not by remaining static, but by unfolding according to intelligible laws 

Figure 2. Gaia, Zeugma museum, Wikimedia Commons. The square, and the octagon made up of two squares, are associated with life and death. Many baptistries and mausoleums use the octagon as a central feature. It is possible that one square symbolises what Mother Earth gives and the other what she takes away. Here the octagon mediates between square and circle. Architecturally it functions as a transitional form, which may explain why octagons appear so frequently in baptisteries, mausoleums and sacred buildings. If the square represents earthly existence and the circle the heavens, then the octagon becomes a geometry of passage. 

Circle and Square Symbolism

The relationship between circle and square is highly symbolic. Across many cultures, the circle is associated with the heavens, the infinite, and, obviously, the cyclical, whereas the square represents the earth, the finite, and the ordered.

The earliest indications of the heavens being symbolised by a circle and the earth by a square come from diverse cultures such as ancient China, Egypt, Mesopotamia, Greece, India, and Native American traditions. These symbols reflect their understanding of cosmology and the natural order, with the circle representing the infinite and divine nature of the heavens, and the square representing the finite and orderly nature of the earth. If the circle represented the infinite, it is possible that the infinite nature of pi as a number was known.

 In ancient Chinese cosmology, the principle of heaven round, earth square expresses this distinction directly. Circular temples and square enclosures embody a cosmological structure in architectural form. Similar patterns appear in mandalas and yantras in India, where circles and squares are combined to represent the universe. In ancient China, there were two opposing concepts, of Heavenly Roundness and Earthly Squareness. In ancient Chinese cosmology, the concept of "Tian yuan di fang" (天圆地方) literally translates to "heavenly round, earthly square." This idea is evident in texts dating back to at least the Zhou Dynasty (1046–256 BCE). The circle symbolises the heavens, which are seen as boundless and infinite, while the square represents the earth, which is perceived as finite and orderly. This philosophy has been incorporated into traditional architecture. For example in Beijing there are two temples, the Temple of Heaven and the Temple of Earth. The Temple of Heaven is round, symbolising the heavens and sky, and the Temple of Earth has a square base and many square walls and altars.

In the contemporary world, the principle of the square and circle representing the earthly and the divine remains important, for example in China. For the 2008 Olympic Games, the Water Cube, based on the square, and the Bird Nest Stadium, based on the circle, were built side by side.

Figure 3. Water Cube – National Aquatics Center and Bird's Nest Stadium, China. More than two thousand years later, the same symbolic vocabulary was consciously revived for the Beijing Olympic Games, demonstrating the remarkable persistence of these geometrical ideas. 

Figure 4. The classic Shri Yantra (1800s), Wikimedia Commons, left; and Lakota Medicine Wheel, a Native American sacred site and National Historic Landmark in Wyoming U.S. Forest Service Photo, Wikimedia Commons, right.

In Native American traditions, the medicine wheel uses the circle to represent cycles and continuity, while the four directions introduce a square framework. These symbolic systems reflect a shared intuition: that the world can be understood through the interplay of circular and rectilinear forms. Although separated by continents and cultures, both the sri yantra and the medicine wheel employ a remarkably similar geometric grammar. The yantra begins with nested squares, triangles and circles that guide contemplation toward the centre, while the medicine wheel organises sacred space around the circle, the centre and the four directions. Neither is simply decorative. Both express the conviction that geometry reveals an underlying order within nature and human existence. These examples span different continents and more than two millennia, yet all employ the same elementary geometrical vocabulary of circle, square and centre. Whatever their historical connections, they suggest that certain geometrical forms repeatedly became vehicles for expressing cosmological ideas.  

The Receptacle and the Problem of Exactness

The question of approximation, which in a modern context is often treated as a technical limitation, may be understood more fundamentally as a philosophical condition. In the Timaeus, Plato distinguishes between three kinds of being: that which is eternal and unchanging, that which comes into being and passes away, and a third kind, more difficult to grasp, which he describes as the receptacle or “nurse of all generation.”

The distinction is crucial. The forms themselves, number, proportion, ratio, belong to the intelligible order. They are stable, exact, and not subject to change. The visible world, however, is not composed of these forms directly, but of their appearances within a medium that is itself without fixed character. The receptacle receives, transmits, and transforms, but does not preserve any form in a perfectly stable way.

Geometry occupies precisely this intermediate position. It is not identical with pure number, nor is it reducible to material form. It operates within the space where intelligible relations are brought into contact with the conditions of the visible world. In this passage, exactness is not lost; it is translated. The problem of approximation arises from this translation.

Quantities such as π, √2, √3, or √5 cannot be fully expressed within any finite measurement. They belong, in Plato’s terms, to the order of that which “is always the same.” Yet when they are constructed geometrically, or embodied within architecture, they must appear within the receptacle: within matter, measurement, and space. They are therefore necessarily rendered as approximations.

This does not diminish their significance. On the contrary, it defines the conditions under which they can appear at all. A circle drawn in stone is not π, but it participates in π. A diagonal constructed within a square is not √2 as an abstract quantity, but it gives access to it. The relation is preserved, even if its expression is finite.

Plato illustrates this point through the example of gold:

The underlying substance remains, while the forms imposed upon it change. In the same way, the numerical relations explored in this study persist across multiple geometric constructions, even as their specific expressions vary. The relations described throughout this chapter are therefore not exact in the sense required by modern mathematical proof. They involve small discrepancies, variations within the limits of measurement, and the use of constants that cannot be expressed as finite numbers. This is not a weakness of the system, but a consequence of its operation within the receptacle. What matters is not exact equivalence, but coherence: the consistent reappearance of the same relations across different constructions. At Giza, specific values recur through a sequence of transformations. A length becomes a perimeter; a perimeter becomes a circle; a diagonal produces a new figure; a height becomes a radius. At each stage, the same numerical identity is preserved within the limits imposed by material realisation. The importance of this distinction becomes apparent when we turn to Giza. The geometrical relationships explored in the following sections are not proposed as exact mathematical identities existing independently of measurement. Rather, they are understood as ideal relations translated into stone through the necessary medium of approximation. 

Squaring the Circle 

One of the most persistent geometric themes, both at Giza and in mathematical history, is the relationship between the circle and the square. Squaring the circle, in ancient cultures, may have been linked to an exploration of the universe's structure, of human understanding, and spiritual aspirations. Knowledge of the irrationality of pi, if it existed, might have meant such an exercise was about uniting the two worlds, the ineffable divine (the circle), and the material world (the square), knowing that it was impossible to truly understand the area of a circle in terms of the area of a square in the material world, in an applied way. Yet in the world of the intellect, of thought, constructing perfect shapes, a circle, a square, and by extension, a square with the same area as a circle, is not problematic. By integrating divine symbolism with practical measurement, ancient civilisations interested in the idea of a square and a circle of the same area sought not only to quantify physical and temporal dimensions but also to explore deeper philosophical truths about the nature of existence and the cosmos. Mathematics and geometry become tools for exploring the mysteries of nature, and expressing profound insights through measurement.

The earliest known person to have attempted to square a circle is Anaxagoras, the Greek philosopher, circa 500 BC. Plato (circa 428-348 BCE) and his Academy placed great emphasis on geometric problems, including squaring the circle, which in turn inspired many others to study the problem. Traditionally, the problem of “squaring the circle” is defined as the construction of a square equal in area to a given circle using only a compass and straightedge.The usual formulation is that this is impossible, due to the transcendental nature of π. Yet there is something slightly misleading in this way of framing the problem. What exactly are we unable to construct: the transformation of the circle into an equivalent square, or the perfect circle itself? A compass appears to solve the latter problem effortlessly: fix one point, choose a radius, and turn. But the circle traced by an actual compass is no more a mathematical circle than a dot made by a pencil is a Euclidean point. The pencil line has thickness; the compass point has width; and no physical circumference can stand in the exact relation C = 2πr to its radius, and it is with a radius that the compass starts to trace a circle. Already the logical tension is within the moving parts of the compass, before ever anything has been drawn. In the squaring of the circle by compass and straightedge problem, the circle is already the first hurdle. Long before we attempt to square it, we have crossed from an exact object existing in mathematical thought to an approximate object embodied in matter. The celebrated impossibility of squaring the circle concerns something more specific: the impossibility of constructing the required exact length from a given ideal circle in finitely many compass-and-straightedge operations. But physically, every geometrical construction presents the same deeper problem. There is necessarily some degree of approximation in all applied geometry. As Petrie put it:

Approximation, in this sense, is not an error, but a condition of manifestation. It is the means by which relations that are, in themselves, exact and unbounded, are brought into the finite world.

Geometry mediates between what can be conceived perfectly and what can only be embodied approximately. The circle and the square therefore dramatise a tension already present in the point and the line: between ideal form and material realisation, between the exact and the measurable. 

There is therefore a danger in allowing the modern formulation of the problem to obscure the more interesting questions that lie behind it. “Squaring the circle” is now commonly presented as a problem with a definitive answer: an exact square equal in area to a given circle cannot be constructed in a finite number of steps using only an unmarked straightedge and compass, because π is transcendental. Attempts to answer this by proposing ever more accurate approximations to π consequently appear simply to have missed the point. In the strict mathematical sense, they have: approximation cannot solve a problem that demands exact construction.

Yet this is not the end of the matter. In another sense, the modern formulation risks missing a different point. The moment geometry is transferred from the ideal realm of mathematical definition onto paper, stone, board or screen, approximation has already begun. The point acquires size; the line acquires breadth; the perfect circle becomes a visible trace. An irrational or transcendental ratio presents no obstacle to seeing such a figure: a diagonal corresponding to √2​, or a circumference involving π, can plainly be drawn. What cannot be physically drawn is its infinite numerical expansion, and neither can any physical construction reproduce an ideal geometrical object with absolute exactness.

This does not make the ancient problem of squaring the circle meaningless. Quite the opposite: it makes it interesting on several different levels. At the practical level, constructing relationships between circles and squares with the deliberately restricted tools of compass and unmarked straightedge is a genuine geometrical challenge. Approximate solutions may be mathematically inexact while remaining ingenious and highly effective as constructions. At the level of pure geometry, meanwhile, the relationship itself reveals important transformations involving π, √π​, and the conversion of one kind of magnitude into another. And at a symbolic level, the encounter between circle and square carries an obvious potential significance: two fundamentally different forms, long associated in many traditions with heaven and earth, the celestial and terrestrial, the boundless and the bounded, are brought into correspondence. The problem of squaring the circle should therefore be situated within a more fundamental distinction between pure and applied geometry. 

Irrational quantities make this distinction particularly vivid. We tend today to encounter √2 or π as numbers followed by an endless sequence of decimal places, but geometrically they are first and foremost relationships. A square of unit side immediately generates √2​ as its diagonal; an equilateral triangle generates √3​ through its altitude; the circle embodies the relation expressed by π. Such quantities can therefore be understood, constructed and manipulated geometrically without ever being exhausted numerically. Their decimal expansions are infinite, but their geometrical meanings are immediate.

This is particularly important when considering ancient applied geometry. A builder did not need to know √2​ to ten decimal places in order to work with it. Once a square had been constructed, its diagonal provided the magnitude directly: it could be measured, transferred, doubled, divided or used to generate another figure. The resulting physical length would necessarily be approximate, as every constructed length is, but the relationship being expressed could nevertheless be exact in conception. Geometry thus provides a remarkable mediation between the finite and the infinite: an irrational magnitude that cannot be completely written as a finite number can nevertheless be made visible as a finite line.

It is within this distinction that the ancient problem of squaring the circle becomes most interesting. The familiar problem is not merely one of finding a sufficiently accurate numerical value for π, nor should the history of attempts to relate square and circle be dismissed simply because an exact material construction is unattainable. The deeper question is one of transformation: how can the relationship embodied by one ideal form be translated into another, and how can that relationship then be realised in finite matter? Whether equality is sought through area or perimeter, the exercise brings the circle and square, two fundamentally different geometrical forms, into correspondence. At the level of pure geometry this introduces relations involving π, π/2, and √π​; at the level of applied geometry it becomes a problem of construction and approximation; and at the symbolic level it brings together forms that have repeatedly acquired cosmological associations with heaven and earth.

There is not one relationship between the square and the circle, but several. A circle may be transformed into a square by preserving area, or by preserving perimeter. These are mathematically different operations. The first introduces √π​; the second introduces π/2. At Giza, this relationship appears repeatedly, though never in a literal or explicit form. There are no circles constructed in stone. Instead, circular geometry is implied through ratios, perimeters, and derived values. The base of the Great Pyramid, for example, relates to its height through π, such that the perimeter corresponds closely to the circumference of a circle whose radius is the height. 

In this sense, squaring the circle should not be understood as an impossible task, but as a method. It is a way of expressing relationships between domains that are otherwise incommensurable. It allows cyclical time to be written into spatial form, and abstract proportion to be realised in architecture. The importance of approximation follows directly from this. Approximation is not a failure of precision, but the necessary condition of any geometry that seeks to operate in the world. It is through approximation that ideal relations become physically meaningful.

What emerges, then, is not a system striving for exactness, but one organised around coherence. The same relationships reappear across different constructions, in different forms, and at different scales. This consistency suggests not accident, but design—not necessarily in the narrow sense of intention, but in the broader sense of a structured way of thinking.

Geometry, in this context, becomes a language capable of moving between worlds. It does not resolve the tension between the ideal and the material; it articulates it.

Squaring the circle reflects the human desire to understand and embody universal truths in material forms. This interplay invites philosophical inquiry into the nature of reality, the limitations of human knowledge, and a balance between the ideal and the empirical. It highlights the tensions between the infinite and the finite, the divine and the human. Yet it also seems to suggest that while they are separate, they can exist one within the other, and that there is in infinitely small gap between the human and the truly divine, which is beyond our comprehension. Pi can be approximated, and must be in any geometry which uses pencil and paper, in architecture, in engineering. But ultimately, pi remains a mystery, a constant which is impossible to completely define or understand.

If we do concede that it is possible to construct a circle with a compass, then it is possible to construct a square with the same area as a circle, as long as we agree at the outset that these constructions will be approximate. This has been done by outstanding mathematicians, such as Ramanujan, with pi as 355 / 113. By using a specific rational value (355/113) for π, he provides a clear and precise framework for practical constructions, ensuring that the limitations of the approximation are understood and accounted for. This approach contrasts with the implicit approximations often made when drawing geometric shapes, where the inaccuracies are not explicitly quantified. Ramanujan’s method demonstrates mathematical honesty and rigour by working within the known limits of approximation and providing a highly accurate solution within those bounds.

If we accept that drawing a true circle is possible in the ideal, non-material world (as per Plato and Penrose), then squaring the circle is also possible in this realm. Both are ideal geometric constructs that exist perfectly in the world of forms. Requiring π to be exact in physical constructions is inherently impossible due to its irrational and transcendental nature. Both drawing a circle and squaring the circle in the material world are approximate activities. However, in the ideal world of ideas, or Platonic forms, exact constructions are possible. Therefore, if it is possible to draw a true circle in the ideal realm, it is also possible to square a circle in that same realm. In practical terms, accepting the approximate nature of geometric constructions means that squaring the circle is as feasible as drawing a circle, both being approximate representations of their ideal forms. Understanding the distinction between the material and ideal realms clarifies why certain mathematical constructs are seen as impossible in practice but possible in theory.

Squaring the circle could symbolise a process of travelling between the material world, and this other realm, in which perfect concepts such as pi can exist, and as a result, where perfect circles, and squares can exist, and within which squaring a circle is possible. The paradox is that it is within this perfect world of forms, that the bridge between the circle, representing this perfect, or divine world, and the square, representing the material world, can happen. If the circle symbolises the divine, the infinite, and the perfect, it can also be said to represent some perfect world of forms, within which pi exists, in a way which cannot be fully grasped or measured by finite means. On the other hand, the square represents the material world, the finite, and the measurable, and perhaps the more tangible aspects of human existence.

In the realm of forms (perhaps we could also call it the divine realm), exact squaring of the circle is possible because this realm contains perfect concepts and entities. In this realm, π is not just an irrational number but a fundamental and exact ratio that defines the circle perfectly. Thus, squaring the circle here symbolises the unity and harmony of divine principles. In the material world, any attempt to square the circle is inherently approximate due to the limitations of physical reality and our tools. This symbolises the human striving to comprehend and bridge the divine with the material, acknowledging our limitations.

The paradox lies in the fact that the act of squaring the circle, a bridge between the divine (circle) and the material (square), is itself bound by the nature of the realm in which it is performed. If done exactly, it belongs to the divine realm, yet it represents a connection to the material world. If done approximately, it acknowledges the material world's limitations while striving towards the divine. This paradox symbolises the human quest to understand and reach the the world of concepts, the ideal, the infinite, the divine. While we can conceptualise perfect truths and entities, our physical realisation of these truths is always approximate. Human endeavours to bridge this gap, even if never fully successful, are meaningful and symbolic of our connection to the divine. If the concepts of π and the perfect circle suggest that there is a realm where these ideals are real and exact, it is a realm which is also a foundation for the material world, influencing and giving meaning to our attempts to understand and replicate divine principles. It would make sense to use the squaring of a circle as a basis for measurement, and perhaps also music and architecture, inviting us to appreciate the symbolic meaning of mathematical and geometrical pursuits as reflections of deeper philosophical and metaphysical ideas.

Ultimately, at the level of the individual, the square and the circle represent something of human nature, and at the level of society, the square and the circle create a space within which to live within a sense of order, derived from the workings of the universe. The squared circle is the basis for a cosmic temple. As John Michell put it in City of Revelation:

Yet although from the human point of view the universe is irrational, it nevertheless continues to function in a most satisfactory manner, and must thus be supposed capable of providing the answer to the problem that besets each individual throughout his life: how to reconcile the conflicting element of different sides of his nature, symbolised by the square and the circle. The square is solid matter; it perimeter is precisely and rationally four times the length of its side. The circle represents spirit, and the measure of its perimeter, which is pi times its diameter, can never be defined on account of the irrational nature of pi. Square and circle are therefore incommensurable for there is no way of showing that the perimeter of a circle is exactly equal to that of a given square. Yet, the geometer who sets out to create the true image of the cosmos must combine square and circle of equal perimeters in one scheme of proportion. If he succeeds in the task, he obtains the greatest possible regard for himself and the community - the plan of the cosmic temple. (5)

John Michell understood the problem of squaring the circle in a broader philosophical sense. Rather than treating it simply as a mathematical impossibility, he regarded it as a symbolic expression of the meeting between two fundamentally different orders of reality. The square represented the measurable, rational and material world; the circle the immeasurable, spiritual and cosmic. The significance of squaring the circle therefore lay not in producing an exact numerical solution, but in seeking a harmonious correspondence between these two domains. Michell described the union of circle and square as the basis of a “cosmic temple.” In this view, geometry becomes a language through which the structure of the universe is expressed in built form. In Michell's words: 

The circle squared, a familiar cipher in the alchemical language, is an obvious symbol of the union of two incommensurable elements, and thus an image of both man and cosmos... The square is solid matter; its perimeter is precisely and rationally four times the length of its side. The circle represents spirit, and the measure of its perimeter, which is π times its diameter, can never be defined on account of the irrational nature of π. Square and circle are therefore incommensurable... Yet the geometer who sets out to create the true image of the cosmos must combine square and circle of equal perimeters in one scheme of proportion. If he succeeds in the task, he obtains the greatest possible reward for himself and the community – the plan of the cosmic temple. (6)

Rectangling the Circle

If the squaring of the circle expresses a relation between area and form, a second, less familiar transformation appears throughout the geometries considered here: the rectangling of the circle. By this I mean a transformation in which a circular quantity is expressed by means of a rectangle. 

This rectangle, defined by the outer limits of the three pyramids, establishes the primary framework of the site. Its dimensions, as measured by Petrie, are 29 227.2 inches in width and 35 713.2 inches in length. From these, the perimeter can be derived as 2 × (29 227.2 + 35 713.2) = 129 880.8 inches. If this perimeter is treated as the circumference of a circle, a new figure emerges. Dividing this circumference by π gives a diameter of approximately 41 342.3 inches. This value corresponds closely to 1400 × 29.53059 inches, or 1400 synodic lunar months. The circle is not drawn, but it is implied. The rectangular boundary becomes a linear expression of a circular cycle. The equivalence is approximate, but precise within the limits of measurement. More importantly, it is consistent with the broader behaviour of the system. The number fourteen, as half of twenty-eight, connects the construction to the numerical framework discussed earlier. The rectangle and the circle, geometry and astronomy are thus brought into relation at the level of the entire plateau. Within this circling of the rectangle, the seven planets, the background stars, the sun, the moon,  Venus, and the number 8 (80 years, in the width, and 7+1 circles in the length), present in the rectangle, are brought into relation with the moon and the number 14 (or at least 14 x 100), present in the circle. 

A similar principle can be observed in the King’s Chamber. The floor of the chamber is a rectangle (a double square). Its perimeter, expressed in metres, closely approximates 10π, that is, a circle with a diameter of 10 metres. Here again, a rectilinear figure encodes a circular relation. The chamber does not contain a circle, but its dimensions carry one within them. The circle is present implicitly, as a measure rather than a form.

Within the Great Pyramid a similar principle applies, but in this case a circle becomes a square - more on that later in the chapter. This repetition across scales is characteristic of the system. The transformations are not isolated. They recur, linking different parts of the site through shared geometric procedures. The rectangle generates the pyramids; the pyramids introduce new relations; and these relations reflect back onto the rectangle.

The rectangling of the circle is an example of how the system operates: by taking a given figure and subjecting it to a transformation that reveals a new form while preserving an underlying numerical identity. In this way, geometry becomes a method of translation, allowing the same quantities to move between different spatial expressions.

The interest lies less in the final value than in the sequence of transformations themselves. The principle of rectangling the circle is not confined to Giza. A striking parallel appears at the ancient city of Khorsabad, in modern day Irak, where the geometry of the city enclosure reveals a similar transformation between circular and rectilinear measures. The city wall forms a large rectangle. The diagonal of this rectangle is of particular interest.  The construction proceeds in three stages. We begin with the measured rectangle of the city wall. Its diagonal generates the diameter of an implied circle. That circle is then transformed into a square of equal area. When converted into metres, it can be interpreted as 19 × 28 × 8 /π x √π. Geometrically, this implies a circle with a circumference of 19 x 28 x 8 x √π metres. This circle in turn implies a square with sides of 19 x 28 x 8 metres. A circle with a radius of 1 has an area of pi. A square with an area of pi will have sides of √π. From the rectangle emerges a circle, in relation to perimeter and circumference. From the circle emerges a square, in relation to area. The numbers 19, 28 and 8 evoke three well-known time cycles: the Metonic cycle of 19 years, the 28-year solar cycle in which calendar dates repeat on the same weekday, and the 8-year octaëteris linking lunar and solar time, as well as Venus’s 8 year cycle. In fact, the 8 year cycle is also present in the width of the GGR, in inches, multiplied by 10. And the numbers 28 and 19 are present in the length of the GGR, though differently: this length is close to 1 000 000 / 28 inches, and this number contains, approximately, the cycles of the 7 planets, precession and the Metonic cycle of 19 years. 

The square with sides of 19 x 28 x 8 metres has a perimeter four times the side. Multiplying the Metonic cycle of 19 years by 4 reduces error in reconciling the cycles of the sun and moon, and is called the Callipic cycle.  The cycle of 28 x 19 years is used in the Christian world to calculate Easter. If the side of the square is interpreted in inches, then it is 19 x 28 x 8 x 10 000 / 254 inches. 19 / 254 is the number of sidereal lunar months in a year, 0.0748. So in fact bringing in the number 254 is about the moon in relation to the sun, scaled up by 10 000. This is the role of the metre at Khorsabad. There, as at Giza, the circle is not drawn but encoded. The diagonal of the city enclosure becomes a linear expression of a circular construction rooted in time cycles. The rectangle does not replace the circle; it carries it. 

The purpose of the following construction is not simply to produce another numerical correspondence, but to illustrate that the same sequence of geometric transformations, rectangle, circle and square, appears outside Egypt as well. 

.

 The Great Pyramid: Square Base, Circular Perimeter

The first and most fundamental geometric relation at Giza appears in the Great Pyramid itself. At ground level, the monument is defined by a square. Its four equal sides establish a stable, bounded figure, oriented precisely to the cardinal directions. This square base anchors the structure within the terrestrial domain: it is measurable, finite, and aligned with the surface of the Earth. A second geometry is about the circle. Using Petrie’s measurements, the mean base side of the Great Pyramid is 9068.8 inches, giving a total perimeter of 4 × 9068.8 = 36 275.2 inches. The height is 5776 inches. If we take twice the height and multiply by π, the result is 2 × 5776 × π ≈ 36 275.2 inches. In other words, the perimeter of the square base is essentially equal to the circumference of a circle whose radius is the height of the pyramid. The height functions as a radius; the base perimeter functions as a circumference. A square and a circle are brought into relation, by perimeter. The square base and the vertical axis are linked through a constant that belongs to the circle. 

This quite remarkable connection was first made in modern times by John Taylor. In The Great Pyramid: Why Was It Built? And Who Built It?, published in 1859, Taylor argued that the pyramid’s height and base perimeter expressed the relationship between a circle’s radius and circumference. Charles Piazzi Smyth came across Taylor’s book shortly afterwards, and brought this idea into his own work, crediting Taylor’s 1859 work as the source of the “very new” ideas.  He then investigated through earlier measurements and, later, through his own survey at Giza. Piazzi Smyth published Our Inheritance in the Great Pyramid in 1864, travelled to Egypt in 1865, and produced a much more detailed set of measurements and interpretations. 

Jomard didn’t notice this pi ratio, on Napoleon's expedition (1798–1801). Neither did Greaves, a century and a half before that, but as their measurements were not true to the actual structure, they would not have been in a position to find it.  

One of the most interesting aspects of Jomard's analysis lies not in his conclusions, but in the way he thinks about the geometry of the pyramid. He distinguishes carefully between dimensions that could have been measured directly by the builders and those that exist only as geometrical constructions. The base and the sloping height (the apothem) are, for him, genuine architectural dimensions. The vertical height, by contrast, is "only a geometrical line, impossible to obtain except by calculation" (« La hauteur perpendiculaire n'était qu'une ligne géométrique, impossible à atteindre autrement que par le calcul ») (7). It is therefore the relationship between the base and the sloping height that most interests him, since these are the quantities that could be laid out physically during construction.

This perspective helps explain why Jomard never recognised the celebrated π relationship between the perimeter and the vertical height. Although he reconstructed the original perpendicular height mathematically, his principal concern was with measurable architectural dimensions rather than derived geometrical ones. In a striking irony, he even describes the vertical height as "incommensurable" with the side of the base. Here he is not using the term in Euclid's technical sense of irrational magnitudes, but simply to indicate that the two lengths do not share a direct common measure within the architecture itself. His emphasis therefore falls upon integer relationships between physically measurable lines rather than upon geometrical constructions generated from them. The distinction is an important one, for it illustrates two different ways of reading the Great Pyramid: as a monument defined by measurable dimensions, or as one whose hidden geometry may deliberately include calculated lines such as diagonals, radii and perpendicular heights.

Using Petrie’s values for the height and side, the pi ration is approximate, but intriguingly there is a better correspondence using the height as derived from the equilateral triangle through √3, so a height of 10 000 / √3 = 5773.5 inches. Indeed, that gives a near perfect match to Petrie's figure for the side. Petrie: 9068.8 inches. Calculated figure: 10 000 / √3 x 2π / 4 = 9068.997.

In geometric terms, the pyramid can therefore be understood as a transformation between forms. The square defines the base; the height introduces a radial dimension; and the perimeter completes the relation by invoking the circle. 

This interpretation also extends the earlier reading of the inch as a unit of time. When the height is expressed in inches, it can be understood as a duration, just as the length of the Great Giza Rectangle was read as a long temporal cycle. The transformation from height to radius then becomes a transformation from duration to circumference: a temporal quantity generates a spatial curve. In this way, the pyramid mediates between different orders of measure. The vertical axis, associated with elevation and structure, is brought into relation with circular motion, associated with cycles and return.

Figure 6: The relationship between the Great Pyramid's height and base can be understood as a sequence of geometric transformations. Beginning with the height, interpreted as approximately 10 000 / √3​ inches, a circle is constructed using this value as its radius. The circumference of this circle is then equal to the perimeter of a square whose sides measure approximately 9069 inches, remarkably close to Petrie's measured mean side of 9068.8 inches. The pyramid may therefore be viewed as a geometric translation from radius to circumference, and from circumference to square. Rather than existing as isolated dimensions, the height and base participate in a single geometric construction. 

If we remember that the height can be interpreted as 10 000 / √3 inches, then this adds another dimension. We can begin with a vesica piscis, two circles of radius 1, to generate the √3 ratio in the lens created by the intersection of the two circles. If instead, we take the lens to be 1, then the radii of the circles becomes 1 / √3. Scaling up, if we take the lens to be 10 000 inches, then the radii of the two circles will be 10 00 / √3 inches, the height of the Great Pyramid. The circumference of each of the two circles will then be 10 000 / √3 x 2π, and in inches, this is the perimeter of the square base of the Great Pyramid. (10 000 / √3 x 2π = 36 275.987, Petrie gives a base perimeter of 9068.8 x 4 = 36 275.2, and this is within less than an inch of the calculated value). All that remains then is to transform this circle into a square, equal in circumference / perimeter. Each side of the square will be a quarter of the circle’s circumference in length.  Beginning with the height of the pyramid, interpreted geometrically as 10 000/√3 inches, a circle can be constructed whose radius equals that height. The circumference of this circle is almost exactly equal to the perimeter of the square base. Nothing new has been introduced; the geometry simply allows one figure to become another. The pyramid therefore appears as a transformation between circular and square forms. 

Figure 7: The familiar vesica establishes √3. We begin with the familiar vesica piscis. Two circles of radius one intersect to produce a lens whose height is √3. The construction therefore provides a natural geometric origin for the √3 relationship. 

Figure 8: The construction can be viewed in reverse. If the lens is assigned a value of one, the radius of each circle becomes 1/√3. Rather than generating √3, the vesica now generates its reciprocal. 

Figure 9: Scaling the lens to 10 000 inches gives circles whose radii become 10 000/√3 inches. This is almost exactly the height of the Great Pyramid expressed in inches. Each circle therefore possesses a circumference of 2π×10 000/32\pi \times 10\,000/\sqrt32π×10000/3​ inches. The circular geometry has now generated a new linear quantity through π. 

Figure 10:  The circle is next transformed into a square by equating circumference with perimeter. Each side of the square is therefore one quarter of the circle's circumference. 

Figure 11: Finally, this square becomes the base of the Great Pyramid, while the original radius becomes its height. The familiar relation between the pyramid's base perimeter and height therefore emerges naturally from the successive transformations of the vesica piscis. 

The figures explored here can be understood as elements within a geometric grammar, a small set of transformations capable of generating a remarkably rich family of forms. If the interpretation developed in this book is correct, this grammar was not simply a mathematical convenience, but part of a wider cosmological language through which architecture, astronomy and measure were brought into relation. 

π / √3 as a Recurrent Grammar

As the geometric transformations at Giza are followed across different figures, a recurring relation begins to appear with increasing clarity: the combination of π and √3. This pairing is not introduced in a single location, but emerges across multiple constructions, linking lengths, areas, and implied circles through a shared proportional structure. And it links back to the vesica pisicis and squaring of the circle seen in the basic ratios of the Great Pyramid. Rather than treating each appearance of π/√3 as an isolated curiosity, we can understand it as one of the generative rules of the system. 

One of the most direct instances occurs in the base of the Great Pyramid. The mean side, measured by Petrie as 9068.8 inches, can be expressed with close accuracy as:

5000 × (π / √3) ≈ 9068.9968 inches.

The agreement is within a fraction of an inch. Here, the side of the square base is generated through a relation that combines circular and triangular geometry: π, the constant of the circle, and √3, which defines the proportions of the equilateral triangle. The same relation appears at the level of the Great Giza Rectangle. The north–south length of 35 713.2 inches can be closely approached through a construction that again involves π / √3, combined with the synodic month and simple rational factors. The recurrence of this ratio across distinct dimensions suggests that it is not incidental, but part of the underlying structure of the system.

Geometrically, the presence of √3 points toward the equilateral triangle. If a triangle has sides of length 2, its height is √3. This figure therefore establishes a relation between linear extension and vertical projection. When combined with π, the ratio between diameter and circumference, the result is a link between triangle and circle: between straight edges and curvature.

The repeated appearance of π / √3 signals a point of intersection between two fundamental geometries. It provides a means of moving between them, allowing linear figures to generate circular relations, and circular quantities to be expressed within rectilinear forms. This can be understood as a kind of geometric grammar. Just as a small set of rules can generate a wide range of expressions in language, a small set of proportional relations can generate a wide range of figures in geometry. The recurrence of π / √3 across different constructions suggests that it functions as one of these generative elements: a relation through which diverse forms can be derived from a common structure.

The significance of this becomes clearer when viewed alongside the transformations already observed. The height of the Great Pyramid acts as a radius; the perimeter of its base corresponds to a circumference; the rectangle can be converted into a circle through its perimeter; and now, within these relations, a consistent proportional link appears, tying them together. In this context, π / √3 is part of the mechanism by which the system operates. It allows different geometries to be brought into relation, ensuring that the transformations from line to circle, from rectangle to circumference, and from base to height remain internally coherent.

The presence of this ratio across multiple scales reinforces the impression that the geometry at Giza is not assembled from isolated correspondences, but generated from a small number of underlying relations. These relations do not fix the form in advance, but guide its development, allowing variation within a consistent framework.

In what follows, this interplay between geometry and number will be extended further. The vertical dimension, already linked to the circle through the height of the pyramid, will be brought into relation with long-duration cycles, introducing √3 in a new context and extending the system from geometry into time once again.

A related expression of this triangular geometry can be seen if the base of the Great Pyramid is treated as the side of an equilateral triangle. Taking a value of 9069.13 inches, close to Petrie’s 9068.8, the triangle generates two associated circles: an inscribed circle and a circumscribed circle. The radius of the inscribed circle is approximately 2618.03 inches, while the radius of the circumscribed circle is approximately 5236.06 inches. These values correspond closely to φ² × 1000 and φ² × 2000 respectively, introducing the golden ratio into the same geometric framework that already contains π and √3. The larger value, 5236.06, also corresponds to 10 000 Egyptian cubits when expressed in metres, linking the construction to known systems of measure. What is striking here is not only the numerical agreement, but the continuity of transformation: the same base length generates a triangle, which in turn generates two circles, each preserving a proportional identity across different geometric forms. The inch, once again, functions as the mediating unit through which these relations become visible, extending the system outward toward larger measures, including the mile.

This diagram presents a proportional relationship between solar and lunar time cycles, showing how the ratio of the civil year and draconic year to the lunar year and synodic month approaches a simple expression involving π / √3. The result approximates the number of lunations per year, suggesting that this irrational constant may emerge naturally from the interaction of independent astronomical cycles rather than being imposed externally. 

Figure 12

Figure 13

This construction illustrates how the relationship π/√3 can arise naturally from the synodic month of 29.53059 days. The lunar month first defines the height of an equilateral triangle. The side of this triangle then becomes the diameter of a circle, introducing π through the circle's circumference. Dividing that circumference into three equal parts generates a second equilateral triangle, bringing the construction back to √3. The resulting expression, 29.53059 × 2π / (3√3), is therefore not imposed externally but emerges from successive geometric transformations linking the lunar cycle, the equilateral triangle and the circle.

The previous figures showed how π and √3 emerge through geometry. This final diagram asks whether the same proportional relationship also appears independently within astronomical cycles. Using the civil year, draconic year, tropical year and synodic month, the resulting ratio approaches the familiar expression involving π/√3. If so, the significance of this constant would not be confined to geometry alone. It would represent a point at which independent systems, geometry, architecture and astronomy, begin to converge upon the same proportional relationship. 

The Great Pyramid: Two Triangles (And Another at Stonehenge)

The side of the Great Pyramid can be said to be based on the inch. The diagram below shows that a circle with a radius of 1000 x Phi squared inches fits inside an equilateral triangle with sides equal to the GP. The equilateral triangle generates √3. 

The equilateral triangle also provides a geometrical setting for the draconic month. Taking the perimeter of the triangle as approximately 27.2122 units—the draconic month in days—gives sides of approximately 9.0707. The radius of the inscribed circle is then approximately 2.6185, very close to ϕ² . The point is not that the draconic month literally “contains” the golden ratio, but that an astronomical period, once translated into an equilateral triangle, generates a familiar proportional constant through the triangle’s inradius. Time is again converted into geometry. 

Figure: 14

Figure 15

The value of the circumradius, here in inches, 5236, is equivalent to the value of an Egyptian royal cubit in metres.

The same circle–triangle relationship appears at Stonehenge on a smaller scale. Taking the measured diameter of the Sarsen circle, an equilateral triangle inscribed within it produces a height of approximately 906.9 inches. Multiplied by ten, this is very close to Petrie’s mean side of the Great Pyramid, 9068.8 inches. The comparison does not in itself establish a historical connection between Stonehenge and Giza. It does, however, show that the same elementary construction, a circle containing an equilateral triangle, can generate closely related dimensions at two very different monuments.

A related transformation appears in the Bluestone circle. Its circumference is close to three times the side of the equilateral triangle inscribed within the larger Sarsen circle. The two Stonehenge circles can therefore be read through the same movement between circle, triangle, circumference and linear measure that recurs at Giza. Expressed in inches, the proportional relationship becomes particularly visible 

Figure 16

Figures17 & 18: The height of an equilateral triangle inscribed within the Sarsen circle, enlarged tenfold, closely approaches the side of the Great Pyramid. Stonehenge and Giza are not being presented here as copies of one another, but as different-scale expressions of a shared circle–triangle geometry. 

The Lunar Year

Why would this relationship between the square and the circle be of interest in ancient astronomy? One reason may be that it is relevant to understanding time in terms of cycles. One example of this is the lunar year. If a circle of diameter 400 units is considered, a square of equal area will have sides of √(40 000π). This evaluates to approximately 354.49, which is very close to the length of a lunar year in days (354.36708). A similar construction, using a smaller circle, produces an approximation of the synodic month.

What is notable here is the simplicity of the process. A circular quantity, defined through π, is converted into a linear measure through the side of a square. The result corresponds to a temporal cycle. This suggests that the squaring of the circle may function as a bridge between geometry and astronomy. Cyclical time, represented by the circle, is translated into a measurable length. The lunar year becomes not only a period of time, but a geometric quantity. The larger construction begins with a circle of diameter 400. A square of equal area has sides of 200 200 √π​, or approximately 354.49, close to the lunar year of 354.367 days. The smaller construction repeats the same principle at quarter scale. A circle of diameter 100 produces an equal-area square whose perimeter is close to 354.49; one twelfth of that perimeter is approximately 29.54, close to the synodic month. The year and month are therefore generated through related circle–square transformations at different scales. 

Such constructions do not produce exact values, but they are structured approximations. The relationships are close enough to be meaningful. This balance between precision and approximation is characteristic of applied geometry.

Seen in this way, the problem is not one of impossibility, but of mediation. The circle cannot be perfectly squared in the material world, but it can be meaningfully approximated. This perspective allows us to understand why such constructions might have been of interest. They provide a way of expressing astronomical cycles, from lunar months and years, to much longer cycles, within a geometric framework. The resulting figures are not exact solutions, but they are structured approximations that preserve relationships between the quantities involved.

In this sense, the “squaring of the circle” becomes less a problem to be solved than a method to be used. It is one of the key operations through which the geometry at Giza appears to function.

Figure 19

The Apex

So is there a place where the finite and infinite meet? The geometry explored throughout this chapter continually returns to the boundary between the finite and the infinite, the measurable and the intelligible. Every construction begins with concrete lengths and finite figures, yet each ultimately depends upon quantities that can never be completely embodied within the material world. The square root of two, the square root of three, π, the geometrical point itself: each is perfectly intelligible, yet none can be fully realised in matter. Geometry therefore inhabits a threshold. It belongs neither entirely to thought nor entirely to stone, but continually mediates between them.

Perhaps this is why the problem of squaring the circle exercised such a lasting fascination. It is not simply a question of constructing one figure from another, but of exploring the relationship between two modes of reality: that which may be conceived with perfect clarity, and that which may only ever be approximated in the world of becoming. The impossibility is not a failure of geometry. It is a reminder that the finite continually gestures towards what exceeds it without ever exhausting it.

The Great Pyramid itself may be understood in this light. Its four edges rise inexorably towards a single geometrical point. Yet that point, in Euclid's sense, has no magnitude. It cannot be quarried, measured or set in place as a stone. The monument approaches the point without ever becoming it. The apex is therefore more than an architectural termination. It becomes a symbol of geometry itself: the place where material form tends towards an ideal that it can never completely embody. Whether or not a pyramidion once completed the summit is, in one sense, beside the point. The true apex belongs not only to the monument, but to the imagination. The point is nowhere in the stone, yet without it the stone could never have been shaped. The finite and the infinite can never meet? They are like the hand of Adam and the hand of God almost but never quite touching, on Michaelangelo’s Sistine Chapel ceiling. 

The geometry of Giza, as transformations, becomings, are not merely mathematical devices but meditations upon the relationship between finitude and infinity, matter and thought, the measurable and the immeasurable. Geometry becomes a discipline of attention directed towards that threshold. It reminds us that every finite construction points beyond itself, disclosing a reality that always exceeds complete expression.

While the finite and infinite can never quite meet, there is a way to bridge the gap between the two, at least according to philosophers and theologians, and it is the soul. Plotinus offers an intriguing way of thinking about the relationship between the intelligible and the sensible worlds. Unlike Plato's earlier ethical division of the soul into reason, spirit and appetite, Plotinus is primarily concerned with the soul's orientation. The soul does not simply inhabit the material world. As he famously writes,

The soul therefore belongs, in different ways, to both orders of reality. It acts within the world of becoming while remaining permanently turned towards the intelligible world from which it proceeds. The two worlds are not collapsed into one another, but neither are they wholly separated. The soul itself becomes their mediator.

This mediation is not merely metaphysical but ethical. The question is not only what the soul is, but towards what it is turned. Augustine develops this Plotinian inheritance by shifting the emphasis from cosmology to the inner life. The human soul is continually oriented either towards temporal things or towards God, and the ethical life consists in the continual reorientation of attention towards the highest good. The soul is therefore understood less as a static possession than as a movement of orientation. How we live depends upon the direction in which we turn.

Egyptian conceptions of the person suggest that existence was understood less as a fixed state than as a continual process of transformation. The ba, often depicted as a human-headed bird leaving and returning to the tomb, moved between worlds, the deceased aspired to become an akh, and the maintenance of Ma'at required continual ritual participation. Rather than opposing the temporal and the eternal, Egyptian thought repeatedly explores the ways in which movement, renewal and transformation allow one to participate in enduring cosmic order. Order is not opposed to change; it is realised through perpetual renewal. Perhaps the concept of renewal is even more important, in sacred terms, than becoming or being. 

Perhaps geometry may be understood in a similar way. Rather than acting simply as a collection of figures, it becomes a mediating activity. It belongs simultaneously to thought and to matter, to the ideal and the constructed, preserving intelligible relationships while giving them material expression. Geometry does not abolish the distinction between these domains. Instead, it continually translates between them. In this sense, it performs a role curiously analogous to that of Plotinus' soul: remaining oriented towards one order while acting within another. We could say that to practise geometry, metaphysically, is to attune oneself to one’s soul, and through that process, to the cosmos. 

The Great Pyramid itself seems almost to embody this philosophical intuition. Every stone belongs unmistakably to the earth, yet the whole monument is ordered towards a point that no stone can ever occupy. The apparently missing pyramidion, the stone section of the Great Pyramid capping the structure and ending in a point, has generated much speculation. Where you would expect a pointy top, there is nothing, just a flat platform a little below. Indeed, its four faces and four edges converge upon an imaginary point only. It is a Euclidean point, a point without magnitude, invisible yet perfectly intelligible. But perhaps that is precisely the point. The apex therefore does more than complete the architecture. It draws the eye upwards. It invites orientation. The pyramid behaves, in a sense, as Plotinus imagined the soul behaves: bridging the worlds of the eternal and the temporal, wholly engaged with the material world, yet continually directed beyond it.

The same pattern may perhaps be recognised in the wider geometry of Giza. The Great Giza Rectangle lies horizontally across the plateau, embedded within the landscape and expressed through measurable distances. It is the world of extension, number and architecture. Along its north-south axis, the geometry continually directs attention towards the heavens, in that, as a a distance, as a number, or a value, it seems to embody a super-cycle composed of the cycles of the seven classical planets and the background stars in their precessional motion.  This is turn is potentially the same as the eight circles in Plato’s creation story given in Timaeus.  Yet, this distance is quite flat, horizontal. With our eyes and mind, we can comprehend the cycles of the heavens, and with our human abilities, construct a monument such as the Giza Pyramid complex. With our soul, and through the vertical direction of the height of the Great Pyramid, we can direct ourselves upwards, participating in vast cycles of becoming within time, but also to the eternal and infinite, beyond time. Horizontality and verticality therefore express complementary aspects of reality. One grounds us within time and the measurable world; the other invites contemplation of permanence, intelligibility and eternity.

Figure 20: The upward moving lines of the Great Pyramid converge at a point which has no extension.

Together, geometry, number and astronomy cultivate a particular disposition of the soul: not an escape from the material world, but a way of inhabiting it while remaining consciously oriented towards that which continually transcends it. We may add to this, of course, music.

The north-south axis, the pyramid pointing to its apex, the World Soul turned towards Intellect, Augustine's soul turned towards God, Plato's philosopher turning from becoming towards being: these are all, in different traditions, expressions of the same fundamental act. They are acts of participation and orientation. Sacred geometry is neither the worship of figures nor the search for hidden codes. It is the cultivation of orientation through geometry: a discipline by which human beings learn to inhabit the finite world while remaining consciously directed towards that which transcends it. The pyramids of Giza have not merely survived history, but continually generated new ways of thinking about humanity's relationship to the Earth, the heavens and itself.

Squaring the Circle, Crowning the Pyramid

To square the circle is not simply to solve a geometrical problem. It is to enter a dialogue between forms that can never become identical, yet continually become one another through transformation. The circle remains a circle; the square remains a square; yet each contains the possibility of the other. Identity persists, not by resisting change, but by passing through it.

The missing apex of the Great Pyramid may be understood in a similar way. It is the point towards which every course of masonry converges, yet it is itself dimensionless. Like the mathematical point, it possesses position but no magnitude. It belongs simultaneously to the visible monument and to an ideal geometry beyond it. Whether or not the original capstone survives is almost secondary. The pyramid always points beyond itself, towards an invisible completion.

The question posed by the Great Pyramid may be how permanence arises from continual transformation. How does the visible participate in the invisible? How does time disclose eternity without ever ceasing to be time? The various elements of the Giza complex can be understood as parts of a metaphysical machine, each one the result and the impetus of transformation. Circling, squaring, pyramiding, these are the geometric mechanisms through which heaven and earth, measure and form, thought and stone continually become one another, and are contnually re-generated. The apex is not merely the summit of the pyramid. It is the direction in which the whole monument points: towards the possibility that reality itself is not a collection of fixed things, but an ordered process of continual becoming.

Notes

1. Plato, Timaeus, Translated by Benjamin Jowett

2. Plato, Timaeus, 49–50

3. Plato, Timaeus

4. Petrie, Inductive Metrology, chapter 2

5.  John Michell, City of Revelation

6. Ibid.

7. Jomard, Description de l'Égypte, Antiquités, vol. III, part 1 (Paris: Imprimerie Impériale, 1809), 521.

8. Plotinus. Ennead IV.8: On the Descent of the Soul into Bodies. Translated with Introduction and Commentary by Barrie Fleet. Las Vegas: Parmenides Publishing, 2012.