The Eye of Ra and the Golden Pyramid. The Egyptian Book of Changes

Transcription

1 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 1 The Eye of Ra and the Golden Pyramid are The Egyptian Book of Changes by Douglass A. White, PhD Delta Point Press

2 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 2 The Eye of Ra and the Golden Pyramid are The Egyptian Book of Changes In this article I explain how the Egyptian Book of Changes binary set of 64 archetypes relates to the pure mathematics of a pyramid. The discussion of pyramid mathematics is partly based on The Great Pyramid Tree of Life by William Eisen in his unusual book, The Universal Language of Cabalah: The Master Key to the God Consciousness. The Egyptians were adept at mathematics and symbology. They captured the essence of the entire Book of Changes in two symbols. n f Wejat Mer Simple Pyramid Mathematics First we will examine the abstract mathematics of an ideal pyramid. can be of any physical size. The pyramid What is important is the mathematical shape of the pyramid. The ideal pyramid has a square base. Each side of the base is 2 units. Thus the total perimeter of the base is 8 units. Ogdoad (8 Trigrams). This represents the Primordial In ancient Egypt the Ogdoad consisted of four divine couples. Each couple represented a cardinal direction and one of the classical elements. The Ogdoad and the Base of the Pyramid Deity Element Quality Direction Amen Air Invisible (unconsciousness) West (South) Amenet Air Awareness (consciousness) West (North) New Fire Desire South (East) Newet Fire Stars, Space South (West) Heh Water Time North (East) Hehet Water Evolution North (West) Kek Earth Inertia, Darkness East (North) Keket Earth Bliss East (South)

3 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 3 The classical Egyptian deities derived from these Primordial Ogdoad deities. I tentatively identify them as Amen-Ra, Mut-Hathor, Tem-Geb, Nut, Heh, Seshat, Khnem/Set, Nekhebet/Nephthys/Styt Since the pyramid has the value 2 for each side, half a side is then 1 unit. The profile of the pyramid consists of the base, the altitude, and the apothem. The apothem of a pyramid is the distance from the base at the perimeter to the apex. The altitude is the distance from the center of the base (inside the perimeter) to the apex. The profile of the pyramid forms two golden right triangles. Since half the base is 1 unit, the apothem is φ = 1.62 units and the altitude is φ = 1.27 units. (I round off the decimals.) The altitude is also very close to 4/π = 1.27 units. Thus we can express the values of the pyramid in terms of either pi or phi. Half the perimeter of the base divided by the altitude gives us the value of pi: π = 3.14 units. The apothem divided by half the base is phi: φ/1 = φ = 1.62 units. View from above. Half perimeter of base (4 units) is emphasized. Benben Pyramidion The Egyptians had a special name for the pyramidion or apex stone on a pyramid. They called it the Benben and considered it to be very sacred. In Heliopolis there was even a temple dedicated to the concept of the Benben. If we take our ideal pyramid of half-base = φ^0 = 1, apothem φ^1 = φ, and altitude φ as the Benben pyramidion of a larger pyramid, we can grow our larger pyramid as follows.

4 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 4 If we let the layer underneath the Benben expand so that the apothem extends by half the Benben base width (i.e. by 1 unit), then the new half-base becomes φ. Thus the larger pyramid s apothem becomes φ + 1 = 2.62 units. The altitude gains by (1 / φ) =.79 and becomes φ + (1 / φ) = 2.06 units. The ratios continue to form a golden triangle. The distance between the center of the base of the larger pyramid and the perimeter of the Benben is φ. This means that in the extension layer there is an identical golden triangle with the sides φ, 1, and φ. Moreover, the half base width equals the apothem of the Benben. With this understanding we can iterate the downward expansion of the pyramid by adding successive layers, each of which extends the apothem by an amount equal to the half-base width of the previous pyramid layer. The extensions of the apothem will therefore be by unit increments of the power of phi: φ^0 = 1, which is our first increment. Successive increments are φ^1 = φ, φ^2, φ^3, φ^4.... We can continue this incremental growth of the pyramid as far as we like and each of the dimensions of each successive layer increments by a multiple of φ in each of its dimensions. Here is a brief chart of this expanding pyramid. Notice that we build the pyramid from the Benben stone at the top downward just as the myth says the Egyptians built their pyramids. The top-down concept is from the mathematical design point of view, not from the engineering and construction point of view. In the chart below the Benben forms an upright triangle and the apothem is the hypotenuse. The extension layers flip the triangle on its side so the hypotenuse forms the new half-base. half-base altitude apothem Benben φ^0 = 1 φ φ half-base alt. ext. apothem ext. diagonal Layer 1 φ 1/ φ 1 φ Layer 2 φ^2 φ φ φ φ Layer 3 φ^3 φ φ φ^2 φ^2 φ Layer 4 φ^4 φ^2 φ φ^3 φ^3 φ The Benben is rotated to form the layers beneath it. It forms two similar triangles. A smaller one is rotated 90 degrees relative to the Benben. The long leg of the Benben triangle becomes the hypotenuse, and the Benben s short leg becomes the long leg of

5 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 5 the smaller triangle. The second triangle is exactly the same size as the Benben triangle, except that it is flipped mirror image and turned so that the Benben hypotenuse apothem becomes the new half-base of the pyramid expansion. Each expansion layer increments by a power of phi from Layer 1 on down. An Idealized Pyramid and the Golden Ratio The golden ratio is φ = (approx.) The Great Pyramid of Khufu at Giza with its pyramidion was originally m in altitude and the length of a side at the base was m. Half the base length is then m. The ratio / = The ideal Golden Pyramid Ratio for these dimensions would be φ = Khufu s pyramid is amazingly close to the ideal and may be off due to inaccurate estimates of the original shape or due to a slight distortion of the shape over time. We know that not all pyramids were built to this ideal. The Egyptians experimented and evolved the architectural concept and the practical engineering technology over a period of time. For example, the pyramid of Khafre (Chefren) is off the mark by a significant amount: The smaller pyramid of Menkaure (Mycerinus) comes in at a ratio of about 1.27, quite close to the ideal. If we go to the earlier pyramids we find that the step pyramid does not even have a square base. Sneferu s red pyramid is about.94545, which is way off. The bent pyramid of Sneferu is of course even worse. The planned height was m with a base of m, which gives , which is further off than Khafre. Userkaf: Sahure: The smaller 5 th and 6 th dynasty pyramids come in as follows:

6 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 6 Niuserre: Neferirkare: Neferefre: (incomplete) Unas: Teti: Pepy I: Pepy II: Merenre: From the data it seems that the only truly Golden Pyramid was the Great Pyramid of Giza, with Menkaure and Niuserre also quite close. The other pyramids were either early experiments or later rough imitations. Having built one grand ideal, the Egyptians apparently were then satisfied to build a pyramid that was reasonably close to the ideal. People used the Great Pyramid as the structural ideal. The golden ratio phi can be expressed exactly in the following way. φ = (1 + 5) / 2 There is a special series that relates to phi called the Fibonacci series. 0+1 = = = = = = = = = 55 Fn = Fn-1 + Fn-2, where the seed values are F0 and F1. The ratios of the consecutive members of this series approach phi as their limit. The above samples from the beginning of the series show that by the time we reach the ratio 55/34 = the ratio is already very close to phi. In the 19 th century the French mathematician Edouard Lucas realized that by using the

7 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 7 ratio of phi in the exact formula given above as the definition for the series, the ratio between successive members of the series would always be exactly equal to phi. Again, the series evolves to its next member by simply adding together the two previous members. However, because the ratio is always phi, this is also a geometric power series of the powers of phi. φ = (1 + 5) / Each member of this remarkable series is a compound of a Lucas integer (the first integer on the left) plus a corresponding Fibonacci integer multiplied by 5 to form the right hand component. Each member in the series also is the sum of the two previous members of the series and has an exact phi ratio with the member that precedes it and the member that follows it even though the Lucas and Fibonacci numbers are all integers. The Fibonacci number multiplied by 5 is close to the value of the corresponding Lucas number in the compound. As the series progresses the value of this product oscillates slightly above and below the corresponding Lucas value and approaches as its limit the same value. Thus the total value of the compound is close to and approaches as its limit twice the value of the Lucas number. If we move upward into the Benben we get the following values for the Lucas components by subtraction. 1-2 = -1 2 (-1) = = -4 3-(-4) = = -11 We discover that the Lucas numbers invert like a mirror image and have the same values except that they alternate positive and negative signs.

8 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 8 The compounds look like this (the sums are rounded off approximations): = = = = =.18 Since the two components of the compound number are close to each other in value and are opposite in sign, the total gets smaller and smaller and approaches 0, which is what we would expect as we go up the Benben toward its apex that is a single mathematical point and therefore has no size from a theoretical viewpoint. If you go back to the sketch of the phi pyramid, you can imagine that it is simply the Benben and that we are zigzagging upward to smaller and smaller iterations of the Benben within itself. Another interesting feature of the Fibonacci series is that you can take any two numbers (even two identical numbers) as your starting point and start the Fibonacci process and the series will approach the Golden Ratio of phi as its limit. However, the Fibonacci-Lucas series is special in that it always gives the Golden Ratio. The Eye of Ra Now that we have introduced the fundamental geometry of the Golden Pyramid, we will turn our attention to the Eye of Ra. n ] The first example above is a mathematical drawing of the Eye, and the second example is an artistic drawing of the Eye. This particular Eye is also called the Left

9 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V909 9 Eye of Horus. The name Horus (Heru) originally derives from the word and glyph for a face. The ancient Egyptians conceived of the sky as a face in which the sun and the moon were the two eyes. The mathematical aspect of the Eye of Ra derives from the Egyptian tradition that the Right Eye of Ra-Horus symbolizes the sun and the Left Eye of Ra-Horus symbolizes the moon. The moon passes through phases in which its shape appears to change as the shadow of the earth projects onto it from different angles. From this observation the Egyptians conceived the notion of using the Left Eye of Horus to represent fractions. They divided the glyph for the eye into six component parts. The whole eye n represented the full moon and thus unity. Sign Fraction Part of Eye b 1/2 Inner Corner a 1/4 Pupil o 1/8 Eyebrow c 1/16 Outer Corner d 1/32 Curl (crow s foot) V 1/64 Tear Duct The fractions were as follows: The sequence is a little strange. I would put the eyebrow first so that the sequence follows the natural flow of the calligraphy. Perhaps there was a reason that has been lost over time Another little problem arises that when you add up all the fractions associated with the six components of the eye, you get a complete eye graphically, but the total is still 1/64 short of a mathematical unity. This last tiny piece has the same size as the Tear duct glyph V, and this reminds us of the Benben situation in which the Benben triangle repeats itself in the first layer of expansion. The Egyptians later developed various notations to deal with this and other refinements to the system. If the complete Eye symbolized the full moon and wholeness, then the missing tiny 1/64 piece would correspond to the New Moon. The Egyptians sometimes used { (pronounced nehes) to represent this phase. Nehes means to awaken from sleep. The glyph shows the eyelid lifting. The Egyptians called the 30 th day of a lunar month Nehes indicating that this was when the lunar eye in the sky began to

10 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V lift its lid and reopen. With these symbols the Egyptians could represent any portion of a whole as long as it was divided into no more than 64 equal components. The formula for the series is like this, starting from the whole Eye: 1 / 2^n --> 1/2^(n+1) [n = 0, 1, 2, 3,.... ]. Thus, the series is 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64,.... This system became the basis for Egyptian weights and measures and is still used to some extent in England and with more accuracy in the United States even though most of the world has switched to the metric system. In the U.S. we still measure fluids by the gallon, half-gallon (1/2), quart (1/4), pint (1/8), cup (1/16), gill (1/32), nip cup (1/64), and ounce (1/128), although the gill (pronounced jill ) and the nip cup (archaic name nipperkin ) are now used mostly by the drinkers of alcoholic beverages. Of course, the tipplers are some of the most conservative people in the world in their own way and might be expected to retain the ancient Egyptian vocabulary in their liquor lingo. The nip cup, or nipperkin as it was known in the old days, probably goes back to ancient Egypt. The archaic form suggests that nip is short for nipper. Neper was an epithet of Osiris in his role as the Egyptian god of grain and ken was an ear of corn as well as a liquid measure, probably for the brew made with the corn. The transition to kin in the sense of a diminutive was natural as kernels of grain are small and the cup was also small. The words corn, grain, kernel, and granule all come from the Egyptian root ke[r]n. A nip also came to mean a small sip, usually of liquor, and a nipper is a tool with small pincers for picking up grains and other small items. The Eye Becomes the Pyramid The next step in our discussion is to show that the mathematical series of the Lunar Eye can be exactly equivalent to the pyramid. To show this we will first study the formula for the volume of a pyramid. This turns out to be ab/3 where a is the altitude and B is the area of the base. We will do a demonstration for the pyramid with a square base, since that is what the Egyptians used. We begin with a cube with side s. The volume of the cube is then s^3. We can

11 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V draw four lines passing through the center to connect all the opposite corners of the cube. This divides the cube into six pyramids, each with a base on one of the six sides of the cube and an apex at the center of the cube. They are all of equal size, so each has a volume of (s^3)/6. If we consider only one pyramid, then its base area is s^2 and its altitude is s/2. If we take the product of these two, we get (s^3)/2, which is 3 times too large for the volume, so we know at least for this case the volume must be ab / 3. We can use calculus to get the general case. However, the example of the cube shows us how to get a pyramid that is just like the eye. We start with one of the pyramids from a cube that we arbitrarily set at s = 2. The pyramid has a half-base of 1 and an altitude of 1. The apothem is then 2. The Binary Pyramid This pyramid is not as steep as the Golden Pyramid. Its squat silhouette is probably why the pharaohs preferred the Golden Pyramid ratio. However, the binary pyramid doubles at each level that expands downward from the Benben growing by powers of 2, whereas the Golden Pyramid expands by incrementing powers of phi. At each expansion layer of the Binary Pyramid the horizontal and vertical dimensions are powers of 2. The diagonal dimensions are the same powers of 2 multiplied by 2. The Egyptians thought of the pyramid as the projection of the sun s rays outward and downward from the apex to the earth. That is why the pyramid begins with its seed form in the Benben and then grows and expands. At each expansion the altitude doubles and so does the half-base. This means that if you start with a pyramid with a half-base and altitude of 64, the next smaller version has 1/2 those dimensions, the next smaller version has 1/4th those dimensions, the

12 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V next has 1/8th, the next 1/16th, the next smaller has 1/32th the dimensions, and the next smaller version has 1/64 th the original dimensions. The Benben has the dimension of 1 unit for both the half-base and the altitude. Thus we can assign a component of the eye to each of the expansion layers on the Binary Pyramid. The List of Fractions V 1/64 d 1/32 dv 3/64 c 1/16 c V 5/64 cd 3/32 cd V 7/64 o 1/8 o V 9/64 od 5/32 od V 11/64 oc 3/16 oc V 13/64 ocd 7/32 ocd V 15/64

13 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V a 1/4 a V 17/64 a d 9/32 a dv 19/64 a c 5/16 a c V 21/64 a cd 11/32 a cd V 23/64 a o 3/8 a o V 25/64 a od 13/32 a od V 27/64 a oc 7/16 a oc V 29/64 a ocd 15/32 a ocd V 31/64 b 1/2 b V 33/64 b d 17/32

14 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V b dv 35/64 b c 9/16 b c V 37/64 b cd 19/32 b cd V 39/64 b o 5/8 b o V 41/64 b od 21/32 b od V 43/64 b oc 11/16 b oc V 45/64 b ocd 23/32 b ocd V 47/64 b a 3/4 b a V 49/64 b a d 25/32 b a dv 51/64 b a c 13/16 b a c V 53/64

15 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V b a cd 27/32 b a cd V 55/64 b a o 7/8 b a o V 57/64 b a od 29/32 b a od V 59/64 b a oc 15/16 b a oc V 61/64 b a ocd 31/32 b a ocd V 63/64 n 64/64 z { Nehes: to awaken 1/64. This was the name of the 30 th day of the lunar month and represented the darkest phase of the moon and the initiation of the new moon. A lunar cycle is 29 solar days, 12 hours, 44 minutes, and seconds in duration ( minutes). One sixty-fourth of that is minutes or 11 hours 4 minutes and seconds. Thus, each division of a lunar month is a bit under half a solar day. However, there are only 59 such half days plus a few extra minutes. Also, the fractions had to wax and wane with the moon, so the process went from Nehes (new moon) to Wejat (full moon) and then ran in a backward sequence to return to Nehes. Thus, theoretically each phase lasted 5 hours 32 minutes and seconds, or a little less than a quarter of a 24-hour day.

16 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V How the Egyptians handled the extra phases in mapping their fraction system to the lunar-solar cycles is a problem. We know that they solved the problem of mapping the solar year to their system of 12 solar months of 30 days each (360 days or 36 decans) by adding a five-day half-decan dedicated to the birthdays of the Egyptian national divine family. Later they also added a leap day to keep the solar calendar from slipping by a day every four years. The theoretically neat system of binary fraction phases does not fit the lunar cycle with precision. We can divide the moon s waxing and waning into as many phases as we like, but the Egyptians did not use watches and lived by the solar cycles of day and night. The lunar phases are too slow for most people to notice on a day to day basis. It takes three or four days for a shift in the moon s shape to become obvious to a casual observer. More obvious from day to day is the shift in the time of moonrise and moonset. The moon rises and sets an average of about minutes later each day. This is not a precise hour. It also is not the same every day. Moonrise and moonset are about equal at the first and third quarters, but at new moon moonrise tends to be about 70 minutes later each day and moon set is about 30 minutes later. At full moon this relationship reverses. Thus there is a roughly sinusoidal oscillation between the two time lapses. Nevertheless, a shift of half an hour to a little over an hour from day to day is quite noticeable, whereas the shift of the sunlight of about 1 minute per day would not have been noticeable to anyone in Egypt except for a real specialist who could use the shadow of the Great Pyramid as his calculator. We know the ordinary Egyptians divided their day into morning (dawn until noon), afternoon (noon until dusk), dusk until midnight, and midnight until dawn. The evidence for this is the special boats they assigned to the sun for each of these periods. The morning boat is the Manjtet, the afternoon boat is the Sektet, in the first hours of evening the sun continues to sail in the Sektet, but enters the underworld of darkness. From the fourth hour of night the Sektet has to be towed through the deep realms of darkness that correspond to the six to eight hours of sleep a person has. The Egyptians no doubt calculated rough lunar months as 29 days or 30

17 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V days, and the priest astronomers would decide when to celebrate the new moon. The Chinese lunar calendar gives a pretty good glimpse of how it worked. The difference in New Year celebration is not important here. If we apportion about six hours for each quarter of a 24-hour day, we end up at the end of a lunar cycle of 29 days having traversed 58 phases, once for waxing and once for waning (116 total). However, there are 64 mathematical phases. In a long.lunar month of 30 days 60 phases would be accounted for. Thus, for each lunar month they were left with either 4 or 6 extra mathematical phases that did not fit the lunar cycle as it was partitioned by the circadian cycles. One solution might have been to make the first four or six phases double up. Another possible solution might have been to put an extra one at each of the quarterly celebrations (new, 1 st quarter, full, 3 rd quarter). In short months they could add an extra one for the new and full phases to stretch them to match the long months. I favor the quarterly insertion of intercalary phases as the best answer, but the whole question will have to await further research that may reveal exactly how the ancient Egyptians worked it out. At this point all we know is that the Left Eye of Horus represented the moon, and its component glyphs symbolized a set of binary fractions that stood for the phases or components of the moon or of any process, object, or collection of items. The Egyptians later developed symbols for finer gradations in their weights and measures. However, in the original system there is within the Nehes of the Eye and the Benben of a pyramid a mathematically infinite series that continues after 1/64: 1/128, 1/256, 1/512, 1/1024,.... It turns out that when you add up all the fractions in this infinite series, they come to exactly 1/64. Thus the infinitely fine gradations in the lifting of the eyelid as the Eye awakens are just like the infinitely tiny pyramids that hide inside the Benben. In fact this is more than just an analogy. Compare the definition of the Eye with that of phi, the key to the Golden Pyramid at Giza. 1 / 2^n --> 1/2^(n+1) [n = 0, 1, 2, 3,.... ]. φ = (1 + 5) / 2

18 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V The only difference between 1/2 and phi is the extra 5. However, both these ratios are constant throughout their respective series. Thus, we can design a Binary Pyramid that has the constant ratio 1/2 from layer to layer. This brings us back to our list of ratios achieved by the pyramid builders and throws light on one of the great mysteries of the pyramids. Sneferu, the founder of the Fourth Dynasty and immediate predecessor (and father) of Khufu, builder of the Great Pyramid at Giza, built TWO very large and special pyramids at Dahshur. Archaeologists generally believe that the first to be built was the strange bent pyramid. This was the first true pyramid built in Egypt and was preceded only by Djoser s step pyramid at Saqqara, which, as we noted was not a true pyramid. The bent pyramid is the fourth largest in Egypt. Its original design apparently was for a height of m with a base length of m. This would have given it a ratio of and an angle of When it was about two-thirds of the intended height, the builders changed plan and made the top portion have an angle of This gave it the top portion an estimated tan ratio of about.945. The tan ratio for a perfect Binary Pyramid is Sneferu then went on to build his famous Red Pyramid, according to this revised plan. The Red Pyramid is the third largest after Khufu s Golden Pyramid and that of Khafre at Giza. It had an original height of 104 m with a base length of 220 m. The angle is 43 22, which is the same as the top portion of the Bent Pyramid. The ratio for this pyramid is This is the closest pyramid to a tan ratio of UNITY (with an angle of 45 ) and thus is the closest approximation to a perfect Binary Pyramid in all of Egypt. Thus we discover that Sneferu built a Binary Pyramid originally clothed in white tura limestone that was a shining wonder in its day and inspired his son, Khufu, to build the Great Golden Pyramid at Giza. There may have been structural reasons why Sneferu went with instead of 45. With our new insights it may be possible for engineers to determine why. The agreement between the Red Pyramid and the top section of the Bent Pyramid regarding the angle suggests that the choice was deliberate. The Qabbalah and the Egyptian Book of Changes The Jews and Phoenicians who lived in Egypt or had extensive dealings with the Egyptians understood the Egyptian mathematical system and how it integrated with the Egyptian philosophy of life and cultural symbols. Thus, when they adopted an alphabet system based on the Egyptian model, these Semitic foreigners chose o for the Eye in the Sky and D for the Great Pyramid and combined these two symbols to

19 Eye of Ra and Golden Pyramid Douglass A. White, 2009 V form the word od (OD) which means eternity in their language. This strongly suggests that these neighbors of Egypt understood the symbolic meaning of the Egyptian symbols and their relationship with the calendar as well as their system of liquid measures. [ Suggested Reading: Guide to the Pyramids of Egypt by Alberto Siliotti with a preface by Zahi Hawass. New York: Barnes and Noble, 1997.