The Giza Pyramids
The Giza Pyramids

Most likely responding initially to practical needs—such as restoring landmarks destroyed by the Nile’s annual flooding, estimating the volume of a stone block, or recording of the position of the stars—the ancient Egyptians (like the Sumerians) developed a system of calculations: the predecessor of geometry, Greek for earth measuring.

Although drawn from and referring to material objects—rocks, man-made edifices, stars—these objects and their relative positions were regarded as more or less simple conceptual shapes and more or less simple conceptual relationships: points, volumes, straight lines, triangles. Shifting from objects as physical entities to abstract concepts is, no doubt, a basic function of the human brain, language being its typical, most precious product.

Once codified and rigorously organized in an efficient way, this shift was to become a cornerstone of contemporary civilization; to become the foundation of science as we know it today (which reduces complex natural phenomena into quantifiable entities), and of monetary economics (where everything is measured by the amount of money one needs to acquire it). Mesopotamia and Egypt were probably the first to witness the early steps toward this systematization, manifested in the development both of early mathematics and of writing: cuneiform in Mesopotamia and hieroglyphics in Egypt.

And yet, as we shall see, in the land of the river Nile, the shift from objects as physical entities toward abstract concepts, epitomized by geometry, was associated in the field we are interested in with the worship of matter—literally and figuratively—rendering matter into the very prerequisite for every attempt to rise beyond it.

Early in their history, the Egyptians opted for a simple geometric shape as the most suitable for their most important buildings. In Giza, an area about fifteen kilometers from the center of Cairo and about twenty kilometers north of the then capital city Memphis, pyramids were the dominant features of burial complexes built to accommodate three of the most prominent Pharaohs in the afterlife: they were the deified kings of Egypt, Khufu, Khafre, and Menkaure, until recently known by their Hellenized names Cheops, Chephren, and Mycerinus. The people who labored to build them were more likely “free citizens” than slaves, who were probably charged with a type of corvée, as was often the case in many preindustrial societies; no matter how harsh their working conditions were, they must have taken great pride in their work, as indicated by graffiti found in their communal lodgings.

The first, the pyramid of Khufu, was built within twenty years around 2530 BCE. This period is known as the Old Kingdom—i.e., the first period of unified Egypt, lasting from the twenty-seventh to the twentysecond century BCE (experts disagree on the exact dates of key incidents in Egyptian history). Similar to the other two, its base is oriented with great precision to the four cardinal directions, a fact among many suggesting that designing the pyramid relied partly on the vast, remarkably advanced insights of Egyptian priesthood into astronomy.

Astronomical phenomena were taken seriously into account in the design of monumental buildings, already from earlier on in Mesopotamia, and would continue to do so for many years across the globe. They included both phenomena that were intrinsically important as markers of crucial turning points (such as the summer solstice and the movement of the point of the horizon where the sun rises or sets as the days become longer or shorter), or were attributed rather arbitrarily great importance (such as the appearance in the night sky of a constellation, a group of seemingly neighboring stars billions of kilometers apart). The great rock-cut temple of Abu-Simbel (built around 1265 BCE), for instance, was configured so that twice a year, on significant dates, rays of sun would penetrate the inner sanctum at dawn to shine on all the statues at the far end of the temple except that of Ptah, god of the Underworld. Similar but independently developed astronomical considerations are attested at the Temple of the Sun in Machu-Picchu, Peru, and at the Fatehpur Sikri Jami in India, both built more than 2,500 years later; and Christian churches built even today normally conform to the rule that requires them to be oriented to the east—i.e., the point in the horizon where the sun rises on the equinoxes.

The pyramid of Khufu was built of large stone blocks, and measured 2.5 million cubic meters in total mass and 5.9 million tons in weight. Its height was approximately 146 meters and the length of each side of its square base was 230 meters. Its surfaces were polished and its apex shone under the rays of the sun. Its external and those of the chambers—height to length of side, etc.—conform to special geometric relations. Typically, in a cross section, the proportions of the pyramid of Khufu correspond with what is known as the golden ratio; those of the pyramid of Khafre generate a 3:4:5 Pythagorean triangle, whose properties were known to the Egyptians.

The Egyptians manifested their affection for geometry in almost every edifice of importance they built. They were not the only people to do so. Regular geometrical shapes have often been used in in the course of history, far and wide. From medieval cities in India, such as Madurai, to Renaissance towns in the West, such as Palmanova; from ziggurats in Mesopotamia to the seventeenth-century gardens of Versailles; from the Revolutionary Architecture of E. L. Boullée to the Neo-Rationalism of A. Rossi in the nineteen-eighties, easily recognizable, regular, shapes have been employed to indicate that the form before us is the result of deliberate choices aimed at conveying complex meanings and sophisticated symbolisms and not the outcome of inconsistent, random actions carried out by a large set of active agents.

And yet, it was only during a few periods in their history that the Egyptians opted for elementary geometric shapes to configure their monumental buildings. The earliest such case was the burial monuments of Giza, although experimenting with pyramidal shapes had been going on for some time. But the reign of pyramids as the prominent feature of royal tombs did not last forever.

Already before 2000 BCE—i.e., just five centuries after the Pyramids of Giza—a large burial complex for a pharaoh was built, which was not dominated by a pyramid. It was the mortuary complex of Mentuhotep II, founder of what is known as the Middle Kingdom—the second period during which Egypt was under unified rule, from the twenty-first to the seventeenth century BCE—in Deir el Bahri across from Luxor, ancient The-bes, about 600 kilometers south of Cairo. The complex may have included a pyramid, but a rather small-sized one, if at all, possibly symbolizing the mountaintop where the sun was thought to be reborn every day. The mortuary complexes of some of the later pharaohs assumed the Old Kingdom tradition and had a pyramid as their prominent feature. However, from the eighteenth century on, this practice was gradually abandoned. The last significant pyramid was built by Pharaoh Ahmose, the founder of what is known as the New Kingdom—the period from the sixteenth to the eleventh century BCE—around 1550 BCE and served as his cenotaph. Later mortuary complexes employed different architectural forms to replicate, as did earlier ones, in some idealized way the environment in which the deified king used to live before his death.

Even in the afterlife, the dead king was expected to act for the sake of all his subjects. Assigning a specific architectural form to his last residence constituted the architectural concept; the buildings themselves show that this would change over time. The architectural concept of the three great burial complexes of Giza was the same; their designs varied slightly. By contrast, the concept of the tomb of Mentuhotep II differed considerably; the rock-cut tombs in the Valley of the Kings even more. In the burial complexes of Giza, the focus was on the astronomical/theological symbolisms of the pyramid. In the tomb of Mentuhotep II, half a millennium later, the emphasis was probably placed on providing a setting that would make the ritual performed there every year as majestic as possible. In the rock-cut tombs of the Valley of the Kings, several centuries later, the concern was primarily to hide the burial chamber to prevent desecration; undoubtedly, security had also been a priority in the Giza pyramids. We know that the entrance to the corridors that led to the burial chambers was very well hidden and the obstacles met in gaining access to them were virtually insurmountable.

What remained unchanged was the priesthood’s intention to appropriately accommodate the deceased pharaoh’s afterlife. The person to assume the responsibility for materializing this intention was arguably the priest in charge of the rites, i.e., the High Priest; as a consequence, he also assumed the role of architect. Two thousand years later in Classical Greece, the was in charge of the laborers building the structure. His task was to convert into a physical object the guidelines laid out by the person or the body assigning the project. The range of initiatives that an architect could take, as well as his status and responsibilities, varied considerably from society to society.

About 5,000 years before our time, the Egyptians had evidently grasped the principal act of architecture: to render the product of the mind from whatever “paper” to “stone”; that materializing a concept is the action par excellence through which acquires its identity. The very fact that at some point they chose an elementary geometric shape for their most important buildings not only speaks of their advanced mathematical knowledge; it underlines how clearly they had appreciated that every architectural idea would be in limbo, and it would be outside the scope of architecture if matter was not used as a means of its realization—in the case of pyramids, quite a lot of matter actually.

To them, the immortality of the soul relied on preserving the perishable dead body. Likewise, the Giza pyramids suggest that the validity of a product of the mind relied on its actualization with durable materials. Matter was worshiped—literally in the former and metaphorically in the latter case—as the vehicle for keeping the incorporeal alive. To appreciate how radically different is from drafting the same geometric shape on paper, and from materializing the same concept and the same idea using different material means, let us consider what the pyramids of Giza would be if, rather than 146 meters, they were five meters tall. If architecture differs from other “creative” arts, it is primarily in that it shapes matter to conform to the idea conceived in the human mind.

Nothing corroborates more clearly the Egyptians’ preoccupation with the materialization (in the literal sense of the word) of the respective concept than the pyramid of Pharaoh Djoser, built in Saqqara, close to Memphis probably around 2630–2610 BCE—i.e., a few decades before the pyramid of Khufu.

The mortuary complex was unprecedented in size: at its final stage it measured approximately 150,000 square meters. The main burial monument was shaped initially as a rectangular “mastaba”—a low, truncated pyramid measuring approximately fifty-seven meters by fifty-seven meters and eight meters in height, quite typical at the time. This structure clearly conformed to the (early, at least) symbolic and ritual requirements probably laid down in great detail. Besides, the entire building process followed a very strict ritual, as everything we know about the foundation ceremonies of similar, later complexes suggests: at night, under a starlit sky, the high priests defined the four corners of the complex; they stretched a cord from one corner to the next and then let it drop onto the ground; using a wooden shovel they dug out the foundations forming an offering pit, etc. Rituals also determined many key features of these structures. In the burial complex of Djoser, the walls featured fourteen carved “gates,” which served as portals through which the ka, the pharaoh’s spirit, could come and go. The, presumably few, mere mortals who were allowed access to the interior of the complex had to use the one and only true entrance to it. Chambers and rooms without functional, but with only symbolic, entrances were also intended for the dead king’s use. As per custom, the burial chamber was in the ground under the truncated pyramid, but a shaft allowed the pharaoh’s ka to communicate with the outside world.

However, the truncated pyramid built for Djoser’s tomb most likely failed to meet the special requirements of as an activity that converts a conception of the human mind into tangible object. Successive extensions gave it increasingly greater volume, increasingly greater mass. The initial extension on all sides was followed by an addition to one side only, and then again by a substantial addition, which transformed it into a stepped pyramid of four horizontal tiers. This construction was extended in width and height to produce in its final form a six-tiered stepped pyramid measuring 62 meters in height and approximately 109 meters long and 125 meters wide, visible now from the Nile.

Some Egyptologists argue that the changes in the design of the central building, which transformed it from a mastaba into a stepped pyramid, were due to changes in the intended symbolism or the burial rituals—yet it is strange that this occurred five times in a few years. In matters of doctrine, however, what we may perceive as a slight differentiation, the parties involved may perceive as huge. In any case, if the symbolic and ritual requirements had indeed changed, it is likely that this also had to do with whether the structure that gradually came into being could accommodate them. In contrast to several art forms, particularly to contemporary art, in the end result—the final form of the building—is infinitely more significant than the steps followed to achieve it. A stairway to the heavens (assuming that this was the new intended symbolism) had to be as tall as a twenty-story building; clearly, the initial eight-meter structure fell short of enabling even a pharaoh to reach the North Star. An abstract idea gained the desired status, and became convincing and esteemed when it was transubstantiated into an accumulation of matter in thousands upon thousands of tons.

The person who had the foresight to realize the building’s inability to meet its purpose four times in a row and to insist on a new extension is arguably justly regarded as the forefather of all architects. His official capacity was High Priest; his name was Imhotep and he was bestowed an exceptional honor, albeit two millennia after his death: he was deified, an honor enviable even to holders of the Pritzker Prize. The fact that he was deified as a doctor and healer raises no doubts about his accomplishments as architect: the name of Imhotep became associated with that of Ma’at, the goddess who personified justice and cosmic order out of chaos. Order was fundamental to both and medicine, considering that in the ancient world, disease was often attributed to a disturbance of the order sustaining the body and disturbing the balance of its parts. Besides, was not Claude Perrault, the trusted architect of Colbert and of Louis XIV, who designed the famous east wing of the Louvre and translated Vitruvius into French, a doctor, and in that capacity one of the first members of the French Academy of Sciences?

In the great pyramids of Giza, revealed with utmost clarity its dual nature: the ethereal, completely detached from anything that belongs to the material world, conception of an edifice, and its earthly materialization fully dependent on physical things. Without doubt, in the last 5,000 years, when people have talked about the “form” of a building, they have meant not only its geometric features, but also its size and mass. “The ship on which Theseus sailed with the youths and returned in safety, the thirty-oared galley, was preserved by the Athenians down to the time of Demetrius Phalereus. They took away the old timbers from time to time, and put new and sound ones in their places, so that the vessel became standing illustration for the philosophers … some declaring that it remained the same, others that it was not the same vessel,” mentions Plutarch.1 Since the time of Imhotep, architects have known that both concept and matter are equally important components of buildings.