• • •
• A double square [1:2 rectangle] could be obtained from two intersecting circles as each’s circumference passes through the center of the other circle.
• The three sacred square roots are shown herein:
In the red equilateral triangle ABC, the perpendicular line AE = √3, since the base = 1 and the hypotenuse = 2. As such, a hexagon could be drawn by utilizing AD and DE as two sides of the hexagon that can be drawn on the right circle on the right side. Two of the remaining four sides can be drawn from A and E with an arc = AD = ED, to points C and F.
From point F as a center, draw an arc with the same length of the hexagon side, to intersect the circle at point G—the sixth point on the hexagon.
8.5 The Root Five Rectangle and the Golden Proportion
The √5 rectangle is obtained from the double square. The diagonal of the 1:2 rectangle is √5.
To find the relationship between the root five rectangle and the Golden Proportion (N), begin from a basic square (the manifested universe), such as DAIJ.
Find the midpoint (O) between edges A and D.
Draw a semi-circle from center O with a radius OA.
From the intersecting point G, establish the square GBCF.
Extend GF to H and E. The rectangle ADEH is a root five rectangle that contains two combinations:
1. Two reciprocal Neb (Golden) rectangles: ACFH (1 x 1.618) and CDEF (1 x 0.618).
2. A square (BCFG) plus two lateral Neb (Golden) rectangles. ABGH and CDEF—each is proportioned 1:0.618 (which is equal to 1.618—the Golden Proportion.
• • •
The inscribed square in the upper half of the circle represents the physical manifestation of the world. The resultant proportions were essential parts of the Ancient Egyptian design, as will be shown throughout this text.
• • •
Regarding the relationship between the square root of 5 and the five-sided pentagon, see Sacred Geometry and Numerology by Moustafa Gadalla.
8.6 Proportioning a Line According to the Golden Proportion
To proportion a line (say AB) according to the Neb (Golden) Proportion, establish BC = 1/2 of AB and perpendicular to it.
Draw diagonal AC, which is equal to the square root of the sum of the square of the distance BC and AB:
√[(BC)2 + (AB)2] = √(1+4) = √5
Set on corner C with an arc = CB, find the point x
The line Cx = 1
Therefore Ax = (2.236 – 1) = 1.236
Set on A as a center, with a radius Ax, draw an arc to y.
Ay = Ax = 1.236
yB = (AB – Ay) = (2 – 1.236) = 0.764
The ratio 1.236 / 0.764 = 1.618 = The Neb (Golden) Section/Proportion (N).
This proportioning explains the uniquely reciprocal relationship between two unequal parts of a whole in which the small part stands in the same proportion to the large part as the large part stands to the whole. Such is the formula for the Neb (Golden) Section/Proportion.
The two parts of the Golden Section are often referred to as a minor and a major.
8.7 Neb: The Golden Segment
Neb is an Ancient Egyptian term meaning gold (traditionally, the finished perfected end product, the goal of the alchemist), Lord, master.
The hieroglyph denoting Neb is a segment of a circle whose central angle is 140o. The ratio of this angle to the whole circle (length of arc to whole circumference) = 0.3889, which constitutes the second power of 0.625. The second power spiritually constitutes reaching to a higher level. Neb means exactly that.
8.8 The [Whirling Squares] Spirals
The spiral in nature is the result of continued proportional growth. This type of spiral is known mathematically as the constant angle or logarithmic spiral. Logarithmic expansion is the basis for the geometry of spirals. The fetus of man and animals, which are the manifestation of the generation laws, are shaped like the logarithmic spiral. Manifestations of spirals are evident in vegetable and shell growth, spider webs, the horn of the dall sheep, the trajectory of many subatomic particles, the nuclear force of atoms, the double helix of DNA, and most of all, in many of the galaxies. Patterns in the mental realm, as well, are also generated in spiraling motions.
The logarithmic spiral is the product of the combined effect of addition and multiplication, which is a progressive addition just like the Summation (Fibonacci) Series (2, 3, 5, 8, 13, 21, 34…). As will be shown below, the progression of the spiral curve maintains the same ratio/proportion rhythm of the Neb (Golden) Proportion. The sides of each golden rectangle maintain the ratio between the sides of each added rectangle to the constant ratio of the Neb (Golden) Proportion (the more things change, the more they stay the same).
Logarithmic spirals are characterized by the golden section properties. A logarithmic spiral is formed by progressive addition by means of “whirling squares” consisting of squares and Neb (Golden) rectangles growing in harmonic progression from center Α outward. Each consecutive stage of growth is encompassed by a Neb (Golden) rectangle that is by a square larger than the previous one. In other words, the progression of the Neb (Golden) Proportion yields the whirling squares.
You start with a square ABCD. Then you add the Neb (Golden) rectangle EFBA, as shown above.
Find the mid-point in DA (i.e. point x).
Set on point x and draw an arc with a radius of xB, to point E.
The Neb (Golden) rectangle EFBA is formed.
It should be noted how the two diagonals (DF and BE) are always perpendicular between the smaller and larger Neb (Golden) rectangles. It is therefore that the spiral is called a right angle spiral. It should be noted that the ratio between the two sides (here EF and DE) corresponds with the Golden Proportion (1.618).
Continue the same process, as indicated below.
squares + Neb (Golden) rectangles = Neb (Golden) rectangles
A B C D + E F B A = E F C D
H E D G + E F C D = H F C G
I J F H + H F C G = I J C G
J K L C + I J C G = I K L G etc.
Since logarithmic spirals follow the same process as the Summation Series, they are subsequently characterized by the Neb (Golden) Proportion. The two dashed diagonals (like all diagonals of the compounded Neb rectangle) are in Neb (Golden) ratio to each other (1.618).
• • •
Logarithmic spirals can also be built by whirling triangles that make use of an isosceles triangle that has a top angle of 36°, i.e. by dividing the circle into 10 divisions.
8.9 Dynamic Design Applications
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 the symbolic meaning of 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.
Following are a few examples of the generative dynamic design layout:
i. A simple theme in the square root of two (√2) is exhibited in the figure below of the netert (goddess) Nut, the personification of the sky as matrix of all.
The spaces between the bars on either side of the figure were filled with hieroglyphic writing [removed here in order to show the geometric outlines].
– ABCD is a square.
– The diagonal BD = √2
– Point E was determined so that BE = BD = √2
– Lines GG and FF were located based on the principle of inscribing a square into a half circle. [See similar diagram in the next section, ii.]
– The center of action is the hip joint of Nut.
• • •
ii. Here we have a square that is defined by bars cut into the stone at the top and bottom of the composition. The are
a is dynamically divided for a pictorial composition. The plan of this arrangement is depicted below.
– ABCD is a square.
– A root-five rectangle was used in the center of a square to determine the vertical lines GG and HH.
– The horizontal line EF forms a 5:8 rectangle ABEF.
• • •
iii. The Egyptian bas-relief composition [below] shows that its designer proportioned the picture, as well as the groups of hieroglyphs, by the application of whirling square rectangles to a square. The outlines of the major square are carefully incised into the stone by four bars, two of which have slight pointed projections on either end.
The following are just a few highlights of the design layout:
– ABCD is a square.
– A root-five rectangle was used in the center of a square to determine the vertical lines at points G and H.
– The horizontal line EF forms a 5:8 rectangle ABEF.
– The general construction plan was that of figure (a) above.
– Spacing for the grouping of the hieroglyphic writing is in figure (b) above.
– Spacing for additional elements of the design is shown in figure (c) above.
• • •
iv. Generative rectangles in Karnak’ Pylon
An interesting observation regarding the significance of the differently proportioned rectangles is found on the pylon at the Temple of Khonsu, at the Karnak Temple Complex.
This pylon shows the falcon, vulture, and ibis, each on a differently proportioned rectangle.
The falcon of Horus stands on a 1:2 rectangle, which represents the octave—a self-replication.
The vulture represents Mut, the assimilative power. Therefore, the ratio between the sides of the rectangle is the square root of the Neb (Golden) Proportion. The roots are symbols of pure archetypal, assimilative, generating, and transformative processes.
The ibis symbol of Thoth is atop a Neb (Golden) rectangle 5:8.
• • •
v. Typical Egyptian Temple Gate:
The typical Ancient Egyptian doorway layout incorporated both sacred ratios (pi and phi), as shown and explained herein.
1. The overall outline in the vertical plane is the double-square, 1:2 ratio. [H = 2B]
2. The opening width is based on a square inscribed within a semicircle, the typical Ancient Egyptian way of proportioning a rootfive rectangle. Thus, the thickness of the doorjamb is 0.618 the width of the opening.
3. The height of the aperture (h) = 3.1415 = pi
The incorporation of both sacred ratios [Pi and Phi] in a single unit is found in other Ancient Egyptian works such as the Great Pyramid of Giza. [For more details, read Egyptian Pyramids Revisited by Moustafa Gadalla].
• • •
vi. Examples of root rect. in Luxor Temple’s Triple sanctuary
The triple sanctuary at the southern extremity of the temple represents the Triple Word, the three-in-one. The separate sanctuaries are symbolic descriptions of the three aspects of the single creative power.
The central chamber is proportioned exactly 8:9; that is to say the ratio of the first musical note of the octave. The sanctuary then “grows” by alternate whole numbers and roots, geometrically expressing the underlying cosmic principle of generation. The root generates the square whose diagonal is in turn the irrational root generating the next square.
This is a geometric expression of the manner in which creation manifests itself and grows. The temple is therefore a man-made recreation in stone of the metaphysical and cosmic laws of genesis. The temple grows as does the universe.
• • •
More example applications of dynamic design in Ancient Egyptian works are found in Chapter 11 of this book.
Chapter 9 : The Arithmetic Generative Progression
9.1 Number Mysticism
In the animated world of Ancient Egypt, numbers did not simply designate quantities but instead were considered to be concrete definitions of energetic formative principles of nature.
For Egyptians, numbers were not just odd and even. These animated numbers in Ancient Egypt were referred to by Plutarch in Moralia, Vol. V, when he described the Egyptian 3-4-5 triangle:
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.
The vitality and the interactions between these numbers shows how they are male and female, active and passive, vertical and horizontal, etc.
All the design elements in Egyptian art and buildings (dimensions, proportions, numbers, etc.) were based on the Egyptian number symbolism.
[For more information about number mysticism read Egyptian Cosmology: the Animated Universe by Moustafa Gadalla.]
9.2 The Generative Numbers
For the Ancient Egyptians, the two primary numbers in the universe are 2 and 3. All phenomena, without exception, are polar in nature and treble in principle. As such, the numbers 2 and 3 are the only primary numbers from which other numbers are derived.
Two symbolizes the power of multiplicity—the female, mutable receptacle – while Three symbolizes the male. This was the music of the spheres—the universal harmonies played out between these two primal male and female universal symbols of Osiris and Isis, whose heavenly marriage produced the child Horus. Plutarch confirmed this Egyptian knowledge in Moralia, Vol. V:
Three (Osiris) is the first perfect odd number: four is a square whose side is the even number two (Isis); but five (Horus) is in some ways like to its father, and in some ways like to its mother, being made up of three and two…
The significance of the two primary numbers 2 and 3 (as represented by Isis and Osiris) was made very clear byDiodorus of Sicily [Book I, 11. 5]:
These two neteru (gods), they hold, regulate the entire universe, giving both nourishment and increase to all things…
9.3 Progression of Growth and Proportion
The sequence of numerical creation of Isis followed by Osiris followed by Horus is 2, 3, 5, . . .
It is a progressive series where you start with the two primary numbers in the Ancient Egyptian system, i.e. 2 and 3. Then you add their total to the preceding number, and on and on—any figure is the sum of the two preceding ones. The series would therefore be:
2
3
5 (3 + 2)
8 (5 + 3)
13 (8 + 5)
21 (13 + 8)
34 (21 + 13)
55 (34 + 21)
89, 144, 233, 377, 610, . . .
The Summation Series is reflected throughout nature. The number of seeds in a sunflower, the petals of any flower, the arrangement of pine cones, the growth of a nautilus shell, etc.—all follow the same pattern of these series.
Since this Series was in existence before Fibonacci (born in 1179 CE), it should not bear his name. Fibonacci himself and his Western commentators did not even claim that it was his “creation”. Let us call it as it is—a Summation Series.
The Summation Series conforms perfectly with (and can be regarded as an expression of) Egyptian mathematics, which has been defined by everyone as an essentially additive procedure. This additive process is obvious in their reduction of multiplication and division into the same process: by breaking up higher multiples into a sum of consecutive duplications. It involves a process of doubling and adding. This progressive doubling lends itself to speedy calculation. It is significant that the methods used in modern calculators and computers are closely related to the Egyptian method.
The overwhelming evidence indicates that the Summation Series was known to the Ancient Egyptians. Many Ancient Egyptian plans of temples and tombs throughout the history of Ancient Egypt show along their longitudinal axis and transversely dimensions in cubits (one cubit =1.72′ (0.523 m), giving in clear and consecutive terms the Summation Series 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 . . .
There has b
een evidence about the knowledge of the Summation Series ever since the Pyramid (erroneously known as mortuary) Temple of Khafra (Chephren) at Giza, built in 2500 BCE about 3700 years before Fibonacci.
The essential points of the temple [shown below] comply with the Summation Series, which reaches the figure of 233 cubits in its total length, as measured from the pyramid, with TEN consecutive numbers of the series.
9.4 The Summation Series and the Golden Proportion
This series was the origin of Ancient Egyptian harmonic design. It offers the true pulsation of natural growth. The ratio between each group of two consecutive numbers follows the pulsation:
3:2 = 1.5
5:3 = 1.667
8:5 = 1.60
13:8 = 1.625
21:13 = 1.615
34:21 = 1.619
55:34 = 1.618
89:55 = 1.618
144:89 = 1.618, . . .
So, as the series progresses, the ratio between successive numbers tends towards the Neb (Golden) Proportion (which numerically = 1.618), to which Western academia has recently assigned an arbitrary symbol—the Greek alphabet letter φ (phi) – even though it was known and used long before the Greeks. This proportion is also known in Western texts as ‘Golden’ and ‘Divine’.
Western academia has even misrepresented the Neb (Golden) Proportion by calling it the Golden Number. A proportion is not a number, it is a relationship. ‘Number’ implies the capacity to enumerate.
The term ‘Golden Section’ did not come into use in Western texts until the 19th century. In most Western mathematical books and journals, the common symbol for the Neb (Golden) Proportion is tau (τ) instead of phi (φ)—presumably because tau is the initial letter of the Greek word for section.
The Ancient Egyptian Metaphysical Architecture Page 9