The Ancient Egyptian Metaphysical Architecture

by Moustafa Gadalla

  The Neb (Golden) Proportion controls the proportions of innumerable living organisms which are manifested in the logarithmic/equal angle spirals.

  9.5 The Cosmic Proportion of the Human Figure

  Proportion is the commensuration of the various constituent parts with the whole. The human body is a prime example of such harmonic proportion, where the human frame has been formed with such propriety that the several members are commensurate with the whole.

  The Ancient Egyptian canon for the harmonic proportion of human figures differed only between children and adults. The differences were reflective of the actual physical differences at these two stages. At birth, it is the navel that divides the height of child into two halves. Upon maturation (reaching puberty), the junction of both legs (reproductive organs) is at mid-height of the adult figure. The position of the navel now divides the height into unequal parts that make the parts and the whole in compliance with the Neb (Golden) Proportion.

  The oldest discovered records from the 5th Dynasty show that the highest defined point along the vertical axis is the hairline of the person’s head, when presented in the earthly realm.

  Egyptian figurations carefully mark—with a headband, crown, diadem, or joint—a dividing line for the top of the skull of the earthly man, thus separating the crown of the skull. The height of the body was measured exclusive of the crown, as shown herein in this recovered Ancient Egyptian grid.

  The representation of the neteru (gods/goddesses) and/or human beings in the afterlife are shown on an 18-square grid, for the full height to the top of the head (i.e. including the crown of the head).

  The difference in the height between the two realms reflects the Ancient Egyptian’s deep understanding of the physiology and role of humans on Earth.

  The removal of this part of the human brain (the crown of the head) leaves man alive but without discernment, hence with no personal judgment. The person is in a vegetative state, living and acting only as the executant of an impulse that he receives, without actual choice. It is like a person in a coma.

  The navel is located about 11.1 grid squares from the bottom of the heel on the 18-square grid system (or the same equivalent ratio 0.618 for grid or non-grid systems). Such division follows the laws of harmony between the two parts themselves, and the parts to the whole, as per the following two relationships:

  1. The ratio between the Two (top and lower) Parts of the divine height (18 grid squares) are harmonic.

  Top : Lower is 0.618

  Lower : Top is 1.618

  2. Between the Two Parts to the whole Unity (divine height)—taking the full height (to the hairline of the earthly man’s head) as 1—the body from the feet to the navel, in the Egyptian canon, is equal to the reciprocal of the Neb (Golden) Proportion (1/N), i.e. 0.618. The portion from the navel to the hairline of the head equals the power 2 of the reciprocal of the Neb (Golden) Proportion (1/N2), or 0.382.

  1 / N + 1 / N2 = 1

  0.618 + 0.382 = 1

  where N = the Neb (Golden) Proportion (1.618)

  Because of the intimate relationship between the Summation Series and the Neb (Golden) proportion, we find that the different parts of the figure also follow the Summation Series [as shown in the depicted grid].

  More about mathematics in Ancient Egypt in the appendices section of this book.

  Chapter 10 : Combined—Arithmetic and Graphic Harmonic Design of Egyptian Buildings

  10.1 The Harmonic Design Parameters

  Harmonic design in Ancient Egyptian architecture was achieved through a unification of two systems:

  1. Arithmetic (significant numbers).

  2. Graphic (square, rectangles, and a few triangles).

  The union of the two systems reflects the relationship of the parts to the whole, which is the essence of harmonic design.

  This union of arithmetic and graphic design follows the elements described below.

  1. The Arithmetic System Consisted of:

  1-a. The Active Axes

  An axis is an imaginary and ideal line about which a moving body revolves. In geometry, an axis is equally imaginary—a line without thickness.

  The Egyptian temple was regarded as an organic, living unity. It is in constant motion; its intricate alignments and its multiple asymmetries make it oscillate about its axes. This movement takes place within a rhythm given by the “module” or the particular coefficient of the thing or idea to be defined.

  Ancient Egyptian architectural design is conspicuous for its strong apparent symmetry around a longitudinal axis. This is the result of the Ancient Egyptian knowledge of cosmic laws. The Egyptian designer reflected such slight cosmic asymmetry by ensuring that elements on either side of the axis are not exactly identical to one another. While most of them are balanced, elements are not symmetrical.

  The axis line can be found in a few recovered architectural drawings or sketches on papyri and tablets from various periods. They were, presumably, workmen’s notations, and in spite of their practical purpose, they still feature the axis line drawn in the same conventional way as in modern drawings.

  In the buildings themselves, the axis is marked by an engraved line on the stones of the upper course of a foundation slab, such as the case at Luxor Temple.

  1-b. Significant Points (Along the Axis)

  Significant points were determined along the design axis. These points mark the intersection with transverse axes, the alignment of a central doorway, the position of an altar, the center of the sanctuary, etc. These significant points follow a precise arithmetic progression. In many of the best plans, these significant points are at harmonic distances from one another, and their distances from one end to the other express the figures of the Summation (so-called Fibonacci) Series, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . . The harmonic analysis shows a series of significant points readable from both ends, i.e. if inverted, a system of significant points would also correspond to the Series with the reference point starting at the opposite end of the plan.

  High numbers of the Summation Series were crystallized in the Egyptian monuments ever since the Old Kingdom. The design of the pyramid temple of Khafra (Chephren) reaches the figure of 233 cubits in its total length, as measured from the pyramid, with a complete series of TEN significant points.

  The Karnak Temple follows the Summation Series’ figures up to 610 cubits, i.e. TWELVE significant points. [See diagrams of both temples in the next chapter].

  2. The Graphic System Consisted of:

  2-a. The Telescopic Triangles

  The typical Egyptian temple plan increases in width and height from the sanctuary towards the front. This over-all delimitation was based upon a “telescopic system” of design since the Old Kingdom. The increase in width was accomplished by the use of consecutive 1:2, 1:4, and 1:8 triangles from one or more significant point(s). [See diagram of Karnak Temple (partial) below.]

  The same telescopic configuration applied to the vertical plan, whereby the floor of the temple descended and the roofs ascended outwardly towards the temple’s pylons; as shown in several temples in an earlier chapter of this book.

  2-b. The Rectangular Perimeters

  The general horizontal and vertical outlines are basically rectangular in shape, for the overall plan as well as its constituent parts. The most common configurations that were used are:

  A simple square, such as that utilized in the Pyramid Temple of Khafra (Chephren) in Giza.

  A double square or 1:2 rectangle, such as the Zoser Complex at Saqqara, the inner enclosure at Karnak, and the festival hall of Twt Homosis III

  Root Rectangles—numerous examples [shown below].

  The Neb (Golden) Rectangle, where the “numerical value” of the ratio between the two sides equals 1.618—numerous examples such as in the Pyramid Temple of Khafra in Giza [shown earlier].

  10.2 The Vertical Plane

  The Ancient Egyptians were masters of the vertical principle as well as the horizontal line.
Vertical heights followed the same proportional increase as horizontal widths as additions were made to the front of monuments—an aspect characteristic of the Egyptian temples.

  Harmonic proportion was applied by the Ancient Egyptians in all three dimensions, such as:

  The pyramids (square bases and triangle volume).

  The striking case of the King’s Room in Khufu (Cheops) Pyramid, which affords exact relations for the great diagonal in space with respect to the dimension of the side. [See diagram in Chapter 11.]

  Pylons. [See diagram in Chapter 11.]

  Doorways/portals/gates. [See diagram in Chapter 11.]

  Vertical heights followed the same proportional increase as horizontal widths, as additions were made to the front of monuments—an aspect characteristic of the Egyptian temples.

  Various applications of harmonic design in Ancient Egyptian works throughout its recovered history—and throughout the land—are found in the next chapter of this book.

  Chapter 11 : Harmonic Analysis of Ancient Egyptian Works

  11.1 General

  The Ancient Egyptians manifested their knowledge in harmonic proportion long before its pre-dynastic era, continuing throughout its dynastic history. The selected diagrams shown in this chapter are just a few examples, spread along Egypt’s long-known history. Please note:

  1. The diagrams are based on measurements by various and independent sources [see Sources & Notes for each specific reference].

  2. In order not to overwhelm the reader with crowded drawings and columns, many details of the Ancient Egyptian buildings are not shown on the following diagrams. Simpler layout drawings will make it easier for the reader to see the consistent application of harmonic proportion in Egyptian works.

  3. In some cases, distances shown in these diagrams were converted into Egyptian cubits so that the Ancient Egyptian knowledge and consistent use of the Summation (so-called Fibonacci) Series becomes very clear.

  4. Of the example buildings used in this chapter (and the entire book), none of them have been touched during foreign rule, so there is not the slightest doubt that Egyptians had this knowledge long before any foreigners ever set a foot in Egypt.

  11.2 Pre-Dynastic Era (5000-2575 BCE)

  Because of the remote age of the pre-dynastic era, only mastaba-type tombs survived in the remote areas of Egypt. The superstructure of the mastaba tombs, even during the pre-dynastic era, followed harmonic proportions, as evident in tombs in the Abydos, Memphis, and Giza areas.

  A large number of Egyptian mastabas were documented by Aug. Mariette’s General catalogue of the monuments of Abydos, Paris 1880. Mariette found about 800 of them during his excavations at Abydos.

  Most of the simplest tombs conformed to the 5:8 Neb (Golden) rectangle.

  Several tombs were composed of a combination of a square and a 5:8 Neb (Golden) rectangle.

  To have hundreds of tombs throughout the country with such harmonic proportion shows that it was common knowledge, even at that early age.

  • • •

  Since temples require restoration every few decades/centuries, we find that every temple in Egypt includes references that they were built in pre-dynastic times. As such, temples from various dynastic eras are generally restorations of pre-dynastic works.

  11.3 Old Kingdom (2575-2150 BCE)

  Mastaba Tombs

  In order to show that harmonic proportion was common knowledge, here are a few examples of mastaba-type tombs in Giza. The rectangular superstructures are oriented north-south, and show harmonic designs, as shown below.

  – Mastaba Tomb 6 (Giza)

  The constructional diagram consists of a square and a 5:8 Neb (Golden) rectangle.

  Most other tombs are a simple 5:8 Neb (Golden) rectangle, as per the indicated measurements:

  – Mastaba Tomb 86 (Giza): 12.6′ (3.85 m) x 20.1′ (6.17 m)

  – Mastaba Tomb 87 (Giza): 19.1′ (5.82 m) x 31.2′ (9.52 m)

  – Mastaba Tomb 105 (Giza): 9.7′ (2.95 m) x 15.6′ (4.75 m)

  Khufu (Cheops) Pyramid’s Granite Room

  Khufu’s pyramid is located in Giza, and was built during his reign (2551-2528 BCE). The floor plan of the room is a double square (2 x 1 rectangle), 20 x 10 Egyptian cubits (34′-4″ x 17′-2″, 10.5 x 5.2 m).

  The double square, divided by a single diagonal CA, forms two right triangles, each having a base of 1 and a height of 2. The diagonal CA is equal to the square root of 5 (2.236), i.e. 22.36 cubits in actual length.

  The height of the room is designed to be one half the length of the floor diagonal CA, i.e. √5/2, which is 11.18 cubits (19′-2″ or 5.8 m) in actual length.

  This choice of CD, as the height of the room, will make the diagonal DB (in the triangle DCB) equal to 15 cubits. The result is that the three sides of the triangle ABD are in relation of 3:4:5.

  The harmonic proportion of this room shows the intimate relationship between 1:2:3:4:5 and demonstrates the relationship in the divine harmonic proportion (sacred geometry) between process and structure. It also shows that the right-angle triangle principle (so-called Pythagoras) was practiced in Egyptian design regularly, 2,000 years before Pythagoras walked this earth.

  • • •

  Complete analysis of the interior and exterior of masonry pyramids in Egypt is detailed in Egyptian Pyramids Revisited by Moustafa Gadalla.

  Pyramid Temple of Khafra

  This temple was built during Khafra’s (Chephren) reign (2520-2494 BCE), and is located in Giza next to his pyramid. Points of interesting harmonic proportion in this massive, yet very exact structure include:

  1. The almost symmetrical plan.

  2. It consists of two squares connected by a 5:8 Neb (Golden) rectangle.

  3. All significant points [explained in earlier chapter] are clearly connected, even to the corners of the massive piers.

  4. The significant points along the longitudinal axis correspond to the numbers of the Summation (so-called Fibonacci) Series [3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,…]. The total length to the pyramid is 233 cubits, while the width is89 cubits.

  Menkaura’s (Mycerinus) Pyramid

  This is the last masonry pyramid built during the Pyramid Age. This pyramid is the smallest and youngest of the three pyramids on the Giza Plateau. It was built by Menkaura (2494-2472 BCE), and has the following interesting harmonic design characteristics:

  1. The base is a perfect square with four triangular-shaped surfaces in space.

  2. Its cross section is very nearly a 5:8 triangle, representing the Neb (Golden) triangle.

  3. The ratio of the height to half the diagonal is 8:9 (the perfect musical tone).

  Menkaura’s pyramid represents the perfect harmony for sight and sound.

  This pyramid signified the end of the Pyramid Age.

  • • •

  A complete analysis of the interior and exterior of masonry pyramids in Egypt is detailed in Egyptian Pyramids Revisited by Moustafa Gadalla.

  11.4 Middle Kingdom (2040-1783 BCE)

  Peripteral Chapel Of Sen-usert (Sesostris) I

  The pavilion of Sen-usert I (1971-1926 BCE), at the Karnak Temple Complex, incorporates geodesic knowledge in its design, and it also provides a wealth of geodesic information on its walls. It has a list of all the provinces of Egypt with their respective land surface areas, proving that actual surveys were made. Major towns are listed, the total length of Egypt is given, and the normal height of the Nile flood at three principal points along the length of the river is noted. Much additional useful information is also provided on these walls.

  The constructional diagram is a square flanked by a 5:8 Neb (Golden) rectangle on each side, delimiting the length of both stairways.

  Tomb of Wahka

  Wakha’s rock-cut tomb was constructed ca. 1900 BCE at Qaw near Asyut, and is partly cut into the cliff. The layout is terraced, featuring a pillared portico, sloping causeway, stairway, courtyard, and superimposed columned porticos.

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bsp; Points of interesting harmonic proportions include:

  1. The upper part of the complex, stretching on many levels, consists of squares connected by a scissors-like lattice of 5:8 Neb (Golden) triangles/rectangles.

  2. The complex, excluding the front stairway, is proportioned based on five numbers of the Summation (Fibonacci) Series.

  This example shows that far away from the populated centers of Memphis and Thebes, harmonic proportion was common knowledge throughout the country.

  11.5 New Kingdom (1550-1070 BCE)

  Karnak Temple Complex

  The original sanctuary of the Karnak Complex at Luxor (Thebes) was built during the Middle Kingdom. This is the largest complex of temples in Egypt, where the temples, pylons, courts, columns and reliefs were continually added to from the Middle to the Late Kingdom for over 1,500 years.

  Although dating from various periods, the temples comply with the principles of harmonic design. This is important evidence to support the existence of archives where records of the projects were kept for reference.

  This is a good example of a temple constructed by accretion through successive additions.

  If we consider only the main axis (west-east) temple, we will find that more than one Summation (Fibonacci) Series [3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …] of significant points shows the application of comprehensive harmonic design along three different scales. The greatest distance is 610 cubits from the external rear to the axis of the triple shrine, in the front courtyard.