7.1 Sacred Geometry of Divine Architecture
Geometry to the Ancient Egyptians was much more than a study of points, lines, surfaces, and solids and their properties and measurement. The harmony inherent in geometry was recognized in Ancient Egypt as the most cogent expression of a divine plan that underlies the world—a metaphysical plan that determines the physical.
To the Ancient Egyptians, geometry was the means by which humanity could understand the mysteries of the divine order. Geometry exists everywhere in nature: its order underlies the structure of all things, from molecules to galaxies. The nature of the geometric form allows its functioning. A design using the principles of sacred geometry must achieve the same goal: using form to serve/represent a function.
Herodotus, the father of history and a native Greek, stated, in 500 BCE:
Now, let me talk more of Egypt for it has a lot of admirable things and what one sees there is superior to any other country.
The Ancient Egyptian’s works, large or small, are admired by all because they are proportionally harmonious and, as such, appeal to our inner as well as our outer feelings. This harmonic design concept is popularly known as sacred geometry, where all figures can be drawn or created using a straight line (not even necessarily a ruler) and compass – i.e. without measurement (dependent on proportion only).
7.2 The Egyptian Sacred Cord [tool]
Since sacred geometry is based on harmonic proportion, the unit distance (length) can theoretically be any unit. The only needed tool is a cord consisting of 12 equally spaced distances. The unit distance can be small or large, so as to fit the required design of artwork on a canvas, statues, or the layout of buildings.
Temples and other buildings in Ancient Egypt were laid out in a religious ceremony. This laying out was performed by very knowledgeable people who are known to the Greeks as harpedonaptae.
The harpedonaptae are the people who strictly adhered to the principles of sacred geometry (using only a straight line and a compass). Their cord was (and still is, in parts of present-day Egypt) a very special cord that consists of a 13-knot rope with 12 equally-spaced distances of one Egyptian cubit (1.72′ or 0.5236 m).
Any equally-spaced 13-knot cord is the basic tool used to establish various geometric shapes.
7.3 General Layout of Geometric Shapes
Triangles are the building blocks of any design.
The simplest formation is the equilateral triangle, which can be set out with the Egyptian rope knotted at twelve equal intervals and wound around three pegs so that it formed three sides, each measuring four units.
The line joining from any corner to the middle of the opposite side is its perpendicular.
However, the origin of the historic building layout was the setting out of the 3:4:5 triangle with the Egyptian rope, wound around three pegs so that it formed three sides measuring three, four, and five units, which provides a 90° angle between its 3 and 4 sides.
It was a relatively simple task to lay out rectangles and other more complex geometrical figures after establishing the 3:4:5 right-angle triangle.
A square EBCF, for example, can be established as shown herein:
(A) Construct two 3:4:5 triangles with a common diagonal AC.
(B) Connect FE where FC = EB = 3 units.
The Egyptian cord can be used as a compass to draw circular curves, as shown in the diagram below.
Other shapes, such as the 8:5 Neb (Golden) triangle or rectangle, as shown below, can also be established with the Egyptian cord.
[To see the formation of a wide variety of geometrical shapes, read Sacred Geometry and Numerology by this same author.]
The hieroglyphic symbol for the neter (god) Re, the cosmic creative force, is the circle. When the cord is looped as a full circle, the archetype of creation, we find that the radius of this sacred circle equals 1.91 cubits. In converting this measurement of 1.91 cubits of the radius into the metric system, we get 1 meter exactly (1.91 x 0.5236). 1 meter = 1/100,000th – part of the quarter of the Earth’s meridian. In other words, this particular 13-knotted Egyptian rope and the Egyptian unit of measurement known as a cubit are based on the measurement of the Earth’s circumference.
Throughout this book, you will find this cord to be the only tool needed to establish all sacred geometric shapes, from a straight line to a curve to other shapes.
7.4 The Sacred Circle of Re
The cosmic creative force Re is written as a circle with a dot or point in the center. It is a circle moving in a circle, one and solitary. The circle symbolically represents the Absolute, or undifferentiated, Unity.
The Circle Index is the functional representation of the circle. It is the ratio between the circumference of the circle to its diameter. It is popularized by Western academia by the Greek letter pi and given a value of 3.1415927.
The Circle Index and the Neb (Golden) Proportion were seen by the Ancient Egyptians not in numerical terms, but as emblematic of the creative or generative function. One cannot just reduce a process/function to a meaningless, unmeasurable “value” and then call it an “irrational number”.
The Ancient Egyptians were not interested in abstract “number gymnastics”.
That the Ancient Egyptians knew how to inscribe a polygon within a circle is proven beyond doubt by their invention of capitals and column shafts that are polygonal in cross-section.
The Egyptians built their capitals with nine elements (and occasionally with seven), in addition to 6, 8, 11, and 13-sided polygons, because they knew the properties of the circle and its relationship to perpendicular coordinates and other geometric figures. Their executed work is sufficient evidence of such knowledge.
The Egyptians manifested their knowledge of the circle properties and other curves as early as their surviving records. A 3rd Dynasty (~2630 BCE) record shows the definition of the curve of a roof in Saqqara by a system of coordinates.
This shows that their knowledge of the circle enabled them to calculate the coordinates along this vertical curve. Accordingly, the construction workers followed precise dimensions in their executed circular curves.
Such an application was evident in Egypt at least 2,000 years before Archimedes walked this earth.
7.5 Squaring the Circle—The Manifestation of Creation
“Squaring the circle” for the Ancient Egyptians represented the realization of creation—the transformation process of the concept of creation into its actual manifestation.
Such transformation is reflected and evident in all Ancient Egyptian “mathematical” papyri. In all these papyri, the area of a circle was obtained by squaring the circle. The diameter was always represented as 9 cubits. The Ancient Egyptian papyri equates the 9-cubit diameter circle to a square with the sides of 8 cubits.
The number 9, as the diameter, represents the Ennead, the group of 9 neteru (gods, goddesses) who produced the ingredients of creation. The 9 are all aspects of Re, the primeval cosmic creative force whose symbol is/was the circle.
8 corresponds to the physical world as we experience it. 8 is the number of Thoth, and at Khmunu (Hermopolis), Thoth is known as the Master of the City of Eight.
Musically, the ratio 8:9 is the Perfect Tone. The 8:9 ratio is present in Ancient Egyptian works, such as the proportion of the inner chamber of the top sanctuary at Luxor Temple.
The underlying metaphysical patterns of the manifested universe are represented in the relationship of squaring the circle (Re and Thoth—conceived and manifested).
Thoth transformed the creation concept (symbolized in a circle) into a physical and metaphysical reality. Such transformation is reflected in the Ancient Egyptian process of “squaring the circle”.
The area of a circle with 9 cubits as its diameter = 63.61725.
The area of the squared circle with 8 cubits as its side = 64.
The difference = 64 – 63.61725 = 0.38.
Such a difference = 0.6%, which reflects the Ancient Egyptian consideration of a slight dev
iation from perfection in the manifested world.
A good example of this slight imperfection is the orbit of the Earth around the sun, which follows an elliptical shape and not a perfect circle.
Musically, the ratio 8:9 is the Perfect Tone.
The ratio 8:9 = 2 to its 3rd power and 3 to its 2nd power. This is the perfect relationship between the reciprocals of 2 and 3 to their reciprocal powers of 3 and 2. The numbers 2 and 3 are the two primary cosmic numbers, as will be discussed in Chapter 9 of this book.
The walls of the Egyptian temple were covered with animated images—including hieroglyphs—to facilitate the communication between the above and the below.
The Ancient Egyptian framework was usually a square, representing the manifested world (squaring of the circle). Additionally, the square grid itself had symbolic meaning for the manifested world, which also made it easy to construct the root rectangles of 2, 3, and 5, on/by the square(s) background. The corners of squares and root rectangles were defined by notches along the perimeter or carefully defined by incised lines.
7.6 Triangles
The following is an overview of the geometric configuration of three Egyptian triangles.
The Thoth (Ibis) Triangle
Plutarch, in his Moralia Vol. V about Ancient Egypt, wrote:
By the spreading of Ibis’ feet, in their relation to each other and to her bill, she makes an equilateral triangle.
Ibis is the sacred bird of Thoth, whose words created the world.
An equilateral triangle could be set out with the Egyptian rope knotted at twelve equal intervals and wound about three pegs so that it formed three sides, each measuring four units.
The line joining from any corner to the middle of the opposite side is its perpendicular. With the Egyptian cord, all perpendiculars can be established without any measurements whatsoever.
The Osiris (Union) Triangle
The 3:4:5 triangle, where the height is to the base as 3 is to 4, was called the “Osiris” Triangle by Plutarch. This triangle was set out with the Egyptian rope, wound about three pegs so that it formed three sides measuring three, four, and five units, which provides a 90° angle between its 3 and 4 sides.
It is a historical lie to call it the Pythagorean Triangle. It was used in Ancient Egypt for thousands of years before Pythagoras walked this earth. It is very clear, from Plutarch’s testimony below, that the ancient Egyptians knew that 3:4:5 is a right-angle triangle, since 3 is called ‘upright’ and 4 is ‘the base’, forming a 90° angle. Plutarch wrote about the 3:4:5 right-angle triangle of ancient Egypt in Moralia, Vol. V:
The Egyptians hold in high honor the most beautiful of the triangles, since they liken the nature of the Universe most closely to it, as Plato in the Republic seems to have made use of it in formulating his figure of marriage. This triangle has its upright of three units, its base of four, and its hypotenuse of five, whose power is equal to that of the other two sides. The upright, therefore, may be likened to the male, the base to the female, and the hypotenuse to the child of both, and so Osiris may be regarded as the origin, Isis as the recipient, and Horus as perfected result. Three is the first perfect odd number: four is a square whose side is the even number two; but five is in some ways like to its father, and in some ways like to its mother, being made up of three and two. And panta (all) is a derivative of pente (five), and they speak of counting as “numbering by fives”. Five makes a square of itself.
The Neb (Golden) Triangle
The Neb (Golden) triangle, which is commonly known as 5:8 isosceles triangle, is by far the most widely used in constructional and harmonic diagram in Egyptian architecture and art, and it was no whim for Viollet-le-Duc to call it the Egyptian Triangle.
Numerous Egyptian amulets, representing the mason’s level, have been discovered and are now scattered throughout the museums of the world (Turin, Louvre, etc.). The Neb (Golden) triangle represents the greatest percentage of these shapes which included the 3:4:5 right-angle triangle and equilateral triangle.
7.7 The Combined Square-Triangles 3-D Pyramids
The pyramid shape consists of a square base and triangle volume.
For detailed information about the configurations and the associated sacred geometric designs of the Egyptian masonry pyramids, read Egyptian Pyramids Revisited by Moustafa Gadalla.
Chapter 8 : The Generative Square Root Rectangles
8.1 The Root Rectangles—From Circle to Square to Rectangles
The role of a root in a plant is the same exact role/function as that of the root in geometry. The root of a plant assimilates, generates, and transforms energies to the rest of the plant.
Likewise, the geometric root is an archetypal expression of the assimilative, generating, transformative function and process, whereas fixed whole numbers are the structures that emerge to build upon these principles of process.
As stated earlier, the concept of creation was manifested in the act of squaring the sacred circle of Re. The square is the basic geometric shape from which all root rectangles can be generated.
The diagonals serve as the generators of root rectangles. When we start with a square whose side is one, the diagonal is √2. From the square root of two, other root rectangles are produced directly by simply drawing with compasses, i.e. applying sacred geometry (producing without measurement) by using squares and rectangles and their diagonals.
The Ancient Egyptians were able to obtain root rectangles without measurements through various ways, such as:
Start with a square whose side is unity.
The √2 rectangle is produced from the square by setting the compass at the length of the diagonal and producing the base line to meet it. This makes the length of the long side equal to the square root of 2, taking the short side as unity.
The √3 rectangle is produced from the diagonal of the √2 rectangle.
The √4 rectangle (double square) is produced from the diagonal of the √3 rectangle.
The √5 rectangle is produced from the double square rectangle.
Design that is based on root rectangles is called generative dynamic design, which only the Egyptians practiced. Egyptian sacred objects and buildings have geometries based upon the division of space attained by the root rectangles and their derivatives, such as the Neb (Golden) Proportion, as will be shown throughout this book.
8.2 The Cosmic Solids
From the roots of Two, Three, and Five, all harmonic proportions and relationships can be derived. The interplay of these proportions and relations commands the forms of all matter—organic and inorganic—and all processes and sequences of growth.
The three sacred roots are all that are necessary for the formation of the five cosmic solids [shown above] which are the basis for all volumetric forms (where all edges and all interior angles are equal). The manifestation of these five volumes are generated from the Egyptian Ennead.
8.3 The Generative 1:2 Rectangle—The Double Squares
As stated earlier, the circle is the archetype of creation in Ancient Egypt. Dividing the circle by its diameter produces the 1:2 ratio, which is the musical octave. The manifested world through this division is symbolized by the inscribed two equal squares, representing the balance between our physical and metaphysical worlds [see diagram below].
The 1:2 geometric outline of the twin squares represents the diapason; the octave. The octave represents renewal or self-replication.
In Egyptian architectural design, the 1:2 double square rectangle assumed great importance in the elements or the general outline of the plan. Such outlines represented the octave and served as the renewal place for the physical and metaphysical well-being of the pharaoh.
• The earliest surviving of such 1:2 rectangular complexes is the Zoser Complex (2630–2611 BCE) at Saqqara. This vast sanctuary is in the form of a double square (1,000 x 500 cubits) whose walls are oriented exactly along the cardinal directions. It contains the Step Pyramid, several buildings, colonnades, and temples.
It was a very active site for all successive Pharaohs. The main function of the Zoser sanctuary was to serve as the Heb-Sed site.
Heb-Sed was the most important festival, from the point of view of the Kingship. Being a divine medium, the Egyptian King was not supposed (or even able) to reign unless he was in good health. The Heb-Sed festival was a rejuvenation of the King’s vital force.
This vast sanctuary set the pattern for later holy places in Egypt and elsewhere, such as:
• The Festival Hall (Akh-Menu) of Tuthomosis III at the Karnak Temple was also used for the Heb-Sed Festival. It also consists of double-square outlines.
• On the vertical plane, the doorways of the Ancient Egyptian temples were also proportioned 1:2.
The Neb (Golden) Proportion is obtained from the diagonal of by a rectangle with sides of 1:2—the root-five diagonal. [More details later in this book.]
8.4 Generating Root Rectangles from the Double Square
From a double square, all three sacred square roots can also be obtained, as shown herein.
√2 is the diagonal of a square.
√3 is obtained from the √2 square by setting on point O and drawing an arc with a radius = the side of the original square (OA), at point A, to meet the side of the √2 square at B. In the right angle triangle COB, the hypotenuse CB = the square root of [ (√2)2 + (1)2 ] = [√(2 + 1)] = √3.
√5 is the diagonal of a double square.
The Ancient Egyptian Metaphysical Architecture Page 8
