Philip Mistlberger
26th May 2005, 03:20 PM
The Right Eye of Horus
Sacred Geometry and the Design of Existence
© by Philip Teertha Mistlberger, All Rights Reserved.
The significance of "Sacred Geometry" and "Divine Proportion" ultimately lies in relationship -- the relationship between parts of a whole. As such, it reveals on open window to grasping via direct experience the significance of "right relationship". What this means is that in seeing and understanding the harmony of proportion via relationship, we have a glimpse into the importance of letting go of attachment -- by sensing the invisible hand behind all divine harmony that lies underneath existence, the pure archetypal templates of the universe that exist prior to the outer forms of the universe itself. Divine proportion and symmetries, and sacred geometry brings us face to face with the Design and the Designer of existence, and reveals to us the guiding intelligence that animates the outer universe, and, thus, shows us the route back to this Source of creation as well -- the Source of our own consciousness.
In sensing such divine harmony, we are afforded a deeper understanding of the illusoriness of the separate self -- i.e., we begin to see and understand, from the point of view of intellect, how nothing is truly Separate from anything else. Divine proportion speaks to the profound interrelationship and connection between all things. It hints at the Oneness inherent to existence, and finally to the One Spirit that is energizing all of life.
Many of the ancient cultures understood this integral model, and as such, many of the ancient monuments that bear their stamp have implicit within them readily apparent ratios and proportions that are examples of a sacred design. It was as if the architects and priests of these cultures were intentionally leaving behind signposts for future cultures to decode and learn from, unlocking the keys to the awakening of consciousness. Further, seeing and understanding these geometric ratios and the harmonizing essence of sacred geometry is an essential piece in the development of the capacity to experience awe and wonder, two inner states necessary for the opening into expanded consciousness.
What follow is a brief overview of some of the basics of sacred geometry.
1. Empty Space
http://www.plokta.com/plokta/issue13/space.gif
What is empty space? In a sense, the answer to this question is the most important of all. In terms of the separate universe of space and form, empty space defines the underlying matrix or Source of all things. For empty space is the unborn, and the undying, and that which is untouched by the universe of separation and its foundations of space and time.
The Buddhist tradition in particular exhalted the concept of empty space, designating it by the Sanskrit word shunyata and defining it as the essence and true nature of the "self". As empty space, this self -- pure consciousness -- is seen ultimately to be empty of form or content, which is why the Buddhist concept of "self" is anatta, or "no-self".
Empty space is the direct experience of Reality without conceptual thought. It is the ultimate symbol for seeing into the heart of Reality. For in empty space no separation is possible. The very emptiness is identical to pure Unity, or Oneness. Empty space is not "nothingness". It is pure Wholeness, without division or separation of any kind. It is pure awareness that is totally unidentified with thought.
The famous Heart Sutra of Mahayana Buddhism begins with,
Form is Emptiness, and Emptiness is Form.
"Form is Emptiness" speaks to the empty space or undifferentiated Unity that lies beneath all form. "Emptiness is Form" speaks to the pure potentiality of this empty space or undifferentiated Unity and of its mysterious, infinite capacity for Creation, and movement from Unity to separation -- a process that, as mentioned above, is going on all the time in our own minds.
Thus, empty space is symbolic of the ultimate womb of existence from which all things derive, as well as the purest possible representation of our True Identity, as empty space is no-thing, and is free of all projection, concept, or any possibility of distortion. It is untouched, immaculate, and unchangeable. For these reasons, empty space is the geometric symbol (or non-symbol) assigned par excellence for our ultimate identity, which is one and the same as the Source.
2. The Point
The point is unique in that it marks the first appearance of separation, for it is symbolic of the Primal Object, the first manifestation. As such, it is the perfect symbol of separate identity, and its ability to acquire individual status, even if the point is infinitesimally small.
However, the point has dual-symbolism. For while it is symbolic of our original separation from the Source (the appearance of the point of individual identity within the empty space of Source), it is also symbolic of the last stage prior to completely returning to Source. This last stage is characterized by the full awareness of our true nature that is beyond space, time, and dimension of any kind.
The point is also at the center and heart of all geometric forms. As these geometric forms are the building blocks of existence as manifested in the structure of the atom and molecules, DNA, and cells, and onward then we can conclude that symbolically the point is both the essence and the first manifestation of the universe.
3.The Line
The line is born when the point undertakes the action of extension. By moving and extending outwardly, a series of points occurs, which becomes the line.
When a series of lines all extending from one central point are connected from their outermost points, the circle is born. When the circle is born, so is the third dimension (volume), where the circle is known as the sphere.
4. The Circle, Sphere, and Pi
http://www.counton.org/pi/gallery4/images/art4_1.gif
The circle, being the two-dimensional shadow of the sphere, is a universal symbol of wholeness and completion. It is often symbolic of the purity of Spirit, and within its simple geometry is found a blueprint for all other geometries and forms. The circle in its purest form represents the return to Source via conscious relationship with the Whole. In three-dimensional form as the sphere, it is the perfect symbol for the totality of the manifest universe, and for this reason is also an apt symbol of both spiritual and physical projection, the original outward act of Creation (what science knows as the "Big Bang") that brought all manifest things into being, and is the standard shape of most cosmic bodies (stars and planets).
The simple well-known geometric ratio named Pi, symbolized by the Greek letter , is a value that represents the ratio between the diameter and circumference of a circle. (The diameter multiplied by Pi = the circumference). It is not a whole number, having the approximate value of 3.141592...etc., continuing endlessly, never ending at one precise value (which in the language of mathematics is called a "transcendental" number).
What does that mean in a symbolic, metaphysical sense? Simply, that the circle can never be precisely quantified by a so-called rational or linear value. It remains a mystery, just beyond the grasp of rational, numeric wholeness, and forever eluding exact definition and measurement. For this reason, the circle/sphere is also the consummate symbol of Spirit, that all-pervading life force or Spirit-energy that animates the universe and all life in it. Its function and nature remain mysterious and impossible to penetrate with the rational mind only.
Thus, we can view Creation symbolically as an Act resulting in the birth of the point of separate identity within the infinity and eternity of empty space (the Source). This point then undertakes the first Act of extension, resulting in the line. From the line is born the circle/sphere, from which follows all possible geometric templates representing the full Creation of the universe.
5. The Vesica Pisces
http://www.halexandria.org/images/scan0040.gif
When a second circle/sphere is created with its origin point occurring anywhere on the outer circle of the first circle/sphere, we have the birth of the symbol known in sacred geometry as the Vesica Pisces.
From within the central part of the Vesica Pisces (the overlapping arcs) is found the ratios that generate all significant root geometric templates (with the exception of the heptagram, explained below), such as the square, pentagram, hexagram, etc. Thus, it is seen that the circle is the symbol par excellence of the Mother of Existence (Spirit-energy, often called the "Goddess"), giving birth to the various archetypal forms from which all building blocks of matter and more complex life forms are based on.
The central form within the Vesica Pisces is suggestive of the eye, and indeed, the Vesica Pisces is the perfect symbol for conscious relationship, which is founded on the direct connecting of two separate minds truly seeing each other.
The ratios of the inner "eye" of the Vesica Pisces -- its height to its width -- is defined by the square root of three (1.732050808...). That is, the height multiplied by the square root of three = the length. Symbolically, this points to the fundamental significance of the value of three in all relationship, and how, in fact, there is no relationship between two things without a third element -- the all-important context -- being present. (For example, positive-negative-neutral; proton-electron-electromagnetic field; father-mother-child; etc.).
6. Phi and the Golden Mean Proportion
The Golden Mean Proportion, also known as the "Divine Proportion" or the "Golden Ratio" is a fundamental ratio that is both strikingly simple and deeply profound in its symbolic meaning. As with the value Pi, and the Vesica Pisces, it shows up in countless geometric forms throughout existence, being one of the chief archetypal templates of the manifest universe. Historically, this ratio has been recognized by philosophers, architects, and artists as one of primary importance. Many ancient temples and monuments (such as the Great Pyramid, Stonehenge, and the Parthenon) incorporated the Golden Mean Proportion in their design.
The Golden Mean Proportion is best explained with diagram, but essentially, it involves a straight line, divided into two segments, such that the shorter segment of the line = 1, and the longer segment = Phi, symbolized by the Greek letter , which in value, is 1.6180339887...*etc. (a value that never ends precisely, as with Pi).
The ratio then follows as thus: the length of the entire line is in proportion to the longer segment as this longer segment is in proportion to the shorter segment. Symbolically, this is representative of the act of Creation, or the primal separation that is outlined in the world's chief Creation myths. This is because it is the perfect symbol of the Unity of One being expressed in Two terms. Or, simply, the Creation from the One (Source), so that it now becomes Two (Separation) -- the birth of the manifest universe.
Imagine it this way. We begin with Unity. From itself, this Unity then divides, in such a way that the proportion of its division is harmonious, with the smaller part being in proportion to the larger, as the whole thing is in proportion to the larger part. Or, what is known symbolically as Divine Proportion, because the ratio itself is symbolic of perfection. The irony of this "perfect ratio" is that the actual numeric value, called Phi, is not a whole number, as given above. Thus, like Pi, its perfection is abstract and forever out of the reach of the linear quantification of the rational mind.
The most mystical of the four New Testament Gospels, attributed to John, begins with a line that perfectly expresses the Golden Ratio, or Divine Proportion, in words:
In the Beginning was the Word, and the Word was with God, and the Word was God.
This line, just as with the Divine Proportion of Phi, suggests the creative duality within Unity itself. The "Word" which is the same as Spirit-energy, is both within the Source, and part of the Source's creative duality as well. The relationship between the Word and God, or Spirit-energy and Source, is the same, symbolically, as that of the Golden Ratio or Divine Proportion represented in the division of the line at the point of the Phi ratio.
In other words, the entire line is symbolic of the Source, or the All, or empty space. The point of the Phi ratio, is symbolic of the first split, or the birth of ego-identity. The length of the larger segment can be seen as symbolic of Spirit-energy, and the shorter line can be understood to represent the World, or the manifest universe.
The same symbolism can be reversed as well. That is, the shorter segment can also be seen as the Source, with the longer length once again symbolizing Spirit-energy, and the whole line denoting the universe.
Either way, all three aspects are seen and understood to be harmonious, proportional aspects that are also an expression of one primal Unity. Likewise, Source, Spirit, and universe are ultimately an expression of the One, as only the One can exist. Miraculously, this One can divide itself, in accordance with the laws of Divine Proportion, but it always also remains only One. Or, as Buddhism phrases it,
Nirvana is Samsara
Which means, that the changeless, eternal Absolute Unity (Nirvana) is finally the same as the manifest universe of constant change (Samsara).
The Fibonacci spiral
The 13th century mathematician Fibonacci discovered a simple pattern that emerged from a sequence of numbers that involved the last two in the sequence being added together to produce the next one in the sequence, as follows:
0+1=1. 1+1=2. 1+2=3. 2+3=5. 3+5=8. 5+8=13. 8+13=21. 13+21=34. And so on. This results in the sequence:
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377, and so on.
As with the Golden Mean Proportion, this sequence of numbers is another superb symbol of the process of Creation. We begin with 0, which is Empty Space, or the Source, and then add 1, which is symbolic of the Point in Empty Space, the Primal Division from Unity, the first Separation. This then subdivides endlessly, but the sequence of numbers grows only by accessing numbers from within its own sequence. Just so, the Primal Unity of Existence can never be altered, although this Unity has the miraculous capacity to endlessly replicate itself outwardly (as in the Fibonacci sequence of numbers getting bigger), or it can also be seen as the infinite dividing itself inwardly (the sequence of numbers reducing to zero). Either way, as with the Golden Proportion, in the end there is only Unity.
Put numerically and symbolically, 0 = Infinity. The Beginning equals the End. The Nothing equals the All.
The Fibonacci sequence of numbers, when put in fractional ratios, result in quotients which gradually get closer and closer to the Golden Ratio (Phi) of approximately 1.618, as follows:
1/0=infinity. 1/1=1. 2/1=2 3/2=1.5 5/3=1.6667 8/5=1.6000 13/8=1.6250 21/13=1.6154 34/21=1.6190 55/34=1.6176
and so forth. Each subsequent quotient in the series of numbers gets closer and closer to the Phi ratio, but only perfectly duplicates it at infinity.
Thus, there is a direct relationship between Fibonacci's number sequence, and the Golden Mean Proportion. Further, when the Fibonacci sequence is placed out in a series of Golden Mean Rectangles, a Spiral is born from within these rectangles, as demonstrated below.
This Fibonacci Spiral is one of the key archetypal forms of manifest existence. It is the outward expression of the Original Act of Separation, in particular, in the formation of space and time, i.e., dimension.
The universality of the spiral is found within many key structures in the manifest universe, especially spiral galaxies and the spiral form of the DNA helix. Whirlpools and the spiral movement of the kundalini energy up the spine, as well as nautilus shells and numerous plant and animal proportions reveal the archetypal significance of this form ¨C itself the result of the Golden Mean Ratio, or the Divine Proportion.
The Pentagram
The pentagram, or five-pointed star, is an excellent symbol of the psychospiritual process that we can call "embracing the Shadow", that is, fully integrating our unconscious dark side. The pentagram is a symbol of power, and when the Shadow is integrated, we become empowered in the true, spiritual sense of the word.
It is a curious fact that around sixty nations on Earth use the five-pointed star on their flag, many from completely different cultures and religions. The symbol itself traces back into ancient times, mostly to Egypt, where it was depicted as the heiroglyph connected to the duat, or underworld, and the red god Set, a rebellious and destructive deity who was associated with storms and chaos, but who originally was just a symbol for the red desert and death via famine or flood.
The inverted pentagram at some time in the past picked up the German name drundenfuss, translating as "Witch's foot". In its right side up form it was connected also to Pan, the old Greek goat-foot deity. It was only in later times that the symbol became associated with a Christianized version of Satan, which by that time was a somewhat comedic metaphor for the repressed sexuality so rampant in those times, a cultural pathology that expressed eventually in part through the Inquisition burnings.
The Hexagram
The hexagram or six-pointed star, is an ancient symbol that was originally called the "Seal of Solomon". To Jews it is a sacred symbol termed the "Shield of David", and to Hindus it has been known as "Vishnu's Mark". Spiritually, the hexagram symbol has often been associated with the rising of aspiration (the base triangle) meeting the descent of grace (the upper triangle).
The Heptagram and the Properties of Seven
The number seven has some curious particularities. For example 1 divided by 7 gives a simple recurring decimal of six digits:
0.142857142857142857...
and this one multiplied successively by 2, 3, 4, 5, 6, gives products where we find the same numbers in the same order:
142857 x 2 = 285714 142857 x 3 = 428571 142857 x 4 = 571428 142857 x 5 = 714285 142857 x 6 = 857142
For the ancients, seven was known as "the virgin", as no number less than seven divides into it. For example, two divides four, six, eight, and ten; three divides six and nine; four divides eight; five divides ten; six divides six. But no number divides seven evenly. Thus seven is a "virgin", as no other number enters into it, symbolic of its numeric purity.
Among all prime numbers, seven is the first number that does not enter in the Euclidean circle. Dividing all other root numbers into 360 results in a whole number, except for seven, as follows:
360 / 1 = 360 360 / 2 = 180 360 / 3 = 120 360 / 4 = 90 360 / 5 = 72 360 / 6 = 60 360 / 7 = 51.42857143 etc., repeating irrationally. 360 / 8 = 45 360 / 9 = 40
Symbolically, this means that the heptad (seven) defies precise quantification, much as with the values of Pi (circle) and Phi (Golden Mean Ratio). Put another way, the heptad cannot be measured, or defined or captured, by linear rationalism.
The heptagon is the only basic geometric form that cannot be created via the Vesica Pisces, which is often regarded by sacred geometry as the "womb" of existence. It is because of this unique form that the value of seven is regarded as virginal, because it is not "born" through the geometric and arithmetic precision that the other root numbers are.
The steeper angle of the heptagram is 51.428* degrees. This is very close to the angle of slope of the sole surviving Wonder of the ancient world, the Great Pyramid of Giza, Egypt, which is about 51.8 degrees. Further, the lesser slope of the heptagram is essentially the same as the slope of the internal passageways in the Great Pyramid, about 26.5 degrees. If a double heptagram (fourteen point star) is constructed, then the form of the Great Pyramid is found in the center of interfacing lines, with a 51.428 degree angle of slope. Many ancient Greek temples (such as those dedicated to Athena, Minerva, and Neith) were based on seven-fold geometry. All this is suggestive of the importance ancient architects attached to this number seven.
The Design of Existence and Spiritual Awakening
The ultimate value and usefulness of sacred geometry and the mystical study of numbers, is very similar to the Zen concept of the koan. A koan is a paradox that has no rational answer, such as "Who am I?" "Where does the universe end?", etc. It is a device to engage the mind with, until the rational process of deduction wears itself out, at which point something called an apperception occurs -- the turning back of consciousness toward its own source.
At this point, the mind is quiet, and intuitive, direct, tacit understanding leaps forward. The question (koan) is solved be dissolving the question itself.
This is direct understanding, termed satori or kensho in Zen, is the experience of non-duality, even if only for a moment in time. It is a taste of the unborn consciousness within us, which sees and understands free of concept. Inherent within it this is limitless freedom and joy, and boundless love and gratitude, simply for what is.
To study the design of existence is useful only inasmuch as it leads to the experience of awe and wonder, and the ultimate relaxing of our intellect in its ceaseless struggle to understand. In that sense, true understanding happens only when we realize, fully and completely, that intellectual understanding alone is not enough to take us to the other shore. But paradoxically we do need a sharpening of intelligence, and this something sacred geometry can help to bring about, via the practice of contemplating its forms.
Intellectual understanding that leads to awe and wonder and gratitude is one path to inner, intuitive revelation, the Awakening to Ultimate Reality, our true nature.
Sacred Geometry and Divine Proportion
The Sattva System with its 14 stages of development ¨C seven stages of Ego growth, and seven stages of Returning to Source ¨C in terms of geometry is based on the heptagram, the seven-pointed star. The significance of this symbol is found within two of the most well known ancient sites on Earth, those of Giza, Egypt, and Stonehenge, England. Additionally, there are significant geometric ratios between the Great Pyramid, Stonehenge, the Earth, Moon, and Mars, some of which will be demonstrated shortly.
The significance of ˇ°Divine Proportionˇ± ultimately lies in relationship ¨C the relationship between parts of a whole. As such, it reveals on open window to grasping via direct experience the significance of ˇ°right relationshipˇ±. What this means is that in seeing and understanding the harmony of proportion via relationship, we have a glimpse into the importance of letting go of attachment ¨C exactly by sensing the invisible hand behind all divine harmony that lies underneath existence.
In sensing such divine harmony, we are afforded a deeper understanding of the illusoriness of the separate self ¨C i.e., we begin to see and understand, from the point of view of intellect, how nothing is truly separate from anything else. Divine proportion speaks to the profound interrelationship and connection between all things. It hints at the Oneness inherent to existence, and finally to the One Spirit animating all life.
Many of the ancient cultures understood this wholistic model (although certainly some did not), and as such, many of the monuments that bear their stamp have encoded within them readily apparent ratios and proportions that are examples of a sacred design. It was as if the architects and priests of these cultures were intentionally leaving behind signposts for future cultures to decode and learn from, unlocking the keys to the awakening of consciousness contained within them. Further, seeing and understanding these geometric ratios and the harmonizing essence of sacred geometry is an essential piece in the development of the capacity to experience awe and wonder, two inner states necessary for the development of higher consciousness.
For this discussion, we will limit our analysis to two of these sacred sites on Earth, the Giza plateau of Egypt, and Stonehenge of Britain.
A)The Great Pyramid
When a circle representing the dimensions of the Earth, and one representing the dimensions of the Moon, are placed adjacent to each other, with the Moon on top, and a triangle is drawn with its apex in the center of the Moon and its bottom base through the middle of the Earth, the resulting angle of 51 degrees 51 minutes of arc is exactly equal to the apothem slope of the Great Pyramid.
http://www.geocities.com/annubis33/TheDesignofExistence/GPEarthMoon.jpg
This striking geometric proportion is highlighted due to the unique properties of the Giza Great Pyramid, alone amongst about 100 extant pyramids in Egypt in possessing the complex inner chamber system it does, in addition to its encoded mathematical values of "pi" and "phi".
Such ratios speak of harmony, and flow, giving the suggestion of effortlessness and "fit" in a way that is strongly evocative of the surrender that is the essence of all ways of describing the return of consciousness to its Source, or what the Sattva System defines as the last seven stages on the Inner Arc to Unity.
In the case of the Great Pyramid, they speak to an obvious awareness attained by the pyramid builders -- not just of the proportions between the Earth and Moon, but to the significance of unifying disciplines, such as science, geometry, esthetics, and sacred proportions, all ending in a harmonious consciousness that is naturally at home wherever it is. As such, it refers to the profound sense of spirituality attained by many of the ancients, and notably the early Egyptians.
The Pyramid-Moon-Earth ratio is well known and has been discovered by researchers as far back as the 1970s (and possibly earlier).
I myself have discovered an equally interesting series of proportionate ratios between the inner chambers and overall dimensions of the Great Pyramid, and Earth and Mars. For now, I present a few of these:
In the diagram below, the Great Pyramid is superimposed over the Earth (top) and Mars (bottom), both shown to scale in their proportional diameters. On the right side of the diagram, the section lengths "A" and "B" are shown, representing the distances from the truncated Pyramid apex to the roof of the Queen's Chamber, and from the roof of the Q.C. to a key point in the subterranean chamber directly beneath the Great Pyramid (also known as the "Pit").
http://www.geocities.com/annubis33/TheDesignofExistence/GPEM.jpg
Point "C" is shown as the addition of points "A" and "B". This diagram demonstrates a simple proportion of very close ratios between both the radii of Earth and Mars and the orbital distances of Earth and Mars from the Sun, with some inner and outer dimensions of the Great Pyramid.
On the right of the diagram, note the section lengths "A" and "B". The bottom horizontal line connecting with "B" lines up with the equator of Mars, as well as the key point in the subterranean chamber that lies 3.1 feet below the floor of the underground horizontal passageway.
The top horizontal line connecting with "B" ends on the roof of the Queen's Chamber, aligned centrally in the pyramid. This also lies on the "north pole" of Mars in our diagram (and the "south pole" of Earth). The top horizontal line connecting with "A" ends on the truncated apex of the Great Pyramid, which is also the equator of the Earth.
The arcs "D" and "E" -- both of which align with the roof of the Queen's Chamber and the key point in the "Pit" below the pyramid respectively -- corresponds almost exactly (99.42% accuracy) to the orbits of Earth and Mars, and their mean proportional distances from the Sun.
The numeric analysis is as follows:
In the Great Pyramid, the distance from the truncated Apex (450 feet above the ground) of the Pyramid to the roof of the Queen's Chamber is 360 feet. The distance from the roof of the Queen's Chamber to the floor of the horizontal passageway entering the subterranean chamber is 188.42 feet. The ratio of 360 to 188.42 is 1 to 0.523.
The radius of the Earth is 3,959 miles. The radius of Mars is 2,106 miles. The ratio of 3,959 to 2,106 is 1 to 0.532.
.523 is 98.3% of .532. Superimposing such a margin of difference onto Mars would result in the floor of the horizontal passageway falling less than 36 miles from the exact equator of Mars.
However, the subterranean chamber (also known as the "Pit") extends deeper, and at a certain point hits the value of 191.5 feet below the roof of the Queen's Chamber (3.1 feet below the floor of the descending passageway leading into the Pit). This particular point corresponds exactly to the equator of Mars, when overlayed over the circle representing Mars in our diagram. In other words, the ratio of the Earth-Mars radii (3959/2106=1.8798) is exactly equal to the ratio of the distance from the top of the Great Pyramid to the roof of Queen's Chamber, and from the QC roof to this point in the Pit (360/191.5=1.8798). Thus, there is a direct geometric and numeric relationship between the inner design of the chambers in the Great Pyramid to its outer dimensions, and the relative radii of Mars and Earth.
Additionally, the ratio of "A" to "C" in the diagram, where "C" = "A" + "B", is 1 to 1.5319. This very closely corresponds to the semi-major axis ratios of the Earth and Mars. The Earth is 1 AU (Astronomical Unit = 93 million miles) from the Sun. Mars, on average, is 141,635,349 miles from the Sun, or 1.5230 AUs. The value of 1.5230 is 99.42% of 1.5319, or these ratios are within 0.58% of each other. Thus, the arcs touching on the roof of the Queen's Chamber and on the point in the subterranean chamber just below the floor of the horizontal passageway are proportionate not just to the radii of Earth and Mars, but to their respective orbital distances from the Sun as well (shown as "D" and "E" in the diagram).
Further, as a note of sidelong interest, the point at which the roof of the descending passageway, leading to the Pit, bisects the Mars arc, corresponds approximately to 41 degrees N latitude (line "F" on the diagram), which is the latitude of Cydonia, a region of mysterious structures on the Martian surface which many researchers have suspected to be artificial.
In this whole analysis, a key point should be noted. The top of the Great Pyramid is believed by some to have once held a capstone, which would have brought the elevation of the pyramid to 481 feet above the ground. However, there is no conclusive evidence that such a capstone ever existed. When we use a model of the Great Pyramid utilizing its current dimensions, with its apparently "truncated" apex of 450 feet above the ground, then the proportions just outlined all fall into place.
What would this suggest? From an intellectual standpoint it could be argued that this may indicate evidence that there never was any capstone, as the flat-topped pyramid at 450 feet elevation results in these harmonious ratios with Mars and Earth dimensions. However, from the intuitive point of view, what it more simply suggests is that the current proportions of the Great Pyramid allow for these ratios. Thus, with what is now, a harmony exists, suggesting a symbolic teaching that harmony can always be unearthed in present conditions, regardless of past circumstances -- provided that, in the Taoist language, we surrender to what is rather then struggling to get things to conform to what we think they should be.
An additional close ratio of significant geometric structures and the Earth/Mars radii ratio is as follows: Distance from *actual* GP apex to Queen's Chamber roof = 364 ft. Distance from QC roof to Subterranean chamber = 188 ft. 188/364 = approx. 0.52. This ratio is significant, as not only is it also essentially the ratio in radius between Earth and Mars, it is also the ratio in volume between a sphere and a cube, when the sphere is contained perfectly within the cube. (Cube=1, Sphere=0.524, in volume). This ratio is 98.5% the actual Earth Mars ratio of 1:0.532.
Further geometric interrelationships between the Great Pyramid (GP) and Mars are shown in the below diagram.
The 6 circles are all Mars-proportional in diameter, with Earth shown as the broken line circle. The GP, at 51.8 degree slope, is shown centrally. As in the above diagram, "south pole" Earth and "north pole" Mars meet at the roof of the Queen's Chamber (Q). Using simple geometry, the following is found:
http://www.geocities.com/annubis33/TheDesignofExistence/GP6Mars.jpg
1) Mars circles "B" and "F" are centered on the diagonal edges of the GP. Directly above them, are Mars circles "C" and "E", their centers aligned vertically with the centers of circles "B" and "F". When the centers of circles "B" and "E" are connected, the resulting angle of the line in relation to the base of the GP is 26.9 degrees. This is very close to the angle of the Grand Gallery and the ascending and descending passageways in the GP, which are about 26.5 degrees, thus being within 1% accuracy of the Mars circle ratios.
2) When the center of Mars circle "C" is connected to the Queen's Chamber, it corresponds very closely (40.5 degree angle) to the north face air shaft of the Queen's Chamber. (The most recent estimates of the QC airshafts angles, based on Rudolph Gantenbrink's update of Flinders Petrie's work, are about 39.5 degrees, but this is considered to have a wide possible margin of error, +/- over 1 degree).
3) When the center of Mars circle "E" is connected to the south base of the Queen's Chamber, it corresponds very closely to the south face air shaft (also 40.5 degrees) of the QC.
Trigonometric analysis at follows:
Base length of GP = 755ft = "Z".
Diameter of Mars circle = 383ft = "X".
Arctangent of 383/755 = 26 degrees 53 min/arc.
Length of "zero point" of QC air shaft to outer far point of GP = 363ft = "Y".
Length of distance from center of Mars circle "C" to point where line "Y" ends directly above center of circle "E" = 312ft = "W".
Arctan of 312/363 = 40 deg. 40 min/arc.
4) A perfect Fibonacci Spiral (based on the Fibonacci sequence of numbers) extends from the coffer in the King's Chamber (K) through the floor of the Queen's Chamber (Q), bisects the arc of the Earth-circle and the outer face of the GP, and ends up at the theoretical apex of the GP (top of missing capstone).
The above analysis suggests that the angles of the QC airshafts and Grand Gallery may be related to geometric proportions rather than astronomical positionings. As such, it suggests an alternate explanation to the theory of Robert Bauval and Adrian Gilbert, who had linked the placement of the QC airshafts to the positions of the stars Sirius and Kochab circa 2450 BC. This could suggest that the south face QC air shaft is not connected to Sirius at all, but is rather associated with geometric ratios, possibly involving encoded links to Mars. As Bauval and Gilbert's estimated construction date for the GP of 2450 BC was based on the positioning of Sirius at that time, any alternate hypothesis for the angles of the QC airshafts potentially throws the whole 2450 BC construction time in doubt. (This analysis is still in progress).
The diagram below shows in detail the movement of the Fibonacci spiral (from the floor of the King's Chamber to the apex of the pyramid), based on Golden Mean rectangles, in relation to the inner design of the Great Pyramid.
http://www.geocities.com/annubis33/TheDesignofExistence/fibonaccigp.jpg
As the Queen's Chamber as been involved in several of these geometric ratios involving Mars, it is interesting to note the following ratio relationships between the QC and the King's Chamber, and their close correspondence with the radii ratios between Mars and Earth.
The floor plan of the King's Chamber is rectangular, of 2 X 1 proportion. The Queen's Chamber is basically square. Thus, any apparent links with Earth/Mars radii ratios are not obvious at first glance. However, when we calculate comparative area of the chamber floors, and comparative volumes, we see something interesting, as follows (units in feet).
1) King's Chamber -- 34.3 by 17.17 = 588.82 sq ft.
2) Queen's Chamber -- 18.875 by 17.15 = 323.71 sq ft.
323.71 / 588.82 = 0.5497.
The actual ratio between Mars and Earth radii is 2106 / 3959 = 0.5319.
0.5319 is 97% of 0.5497. Thus, the square footage area of the KC floor in ratio to the QC floor is within 3% of the radii ratios of Earth and Mars.
When this is shown diagrammatically, with the squares representing the relative areas of the two chambers, and the circles the relative diameters of Mars and Earth, we have this:
http://www.geocities.com/annubis33/TheDesignofExistence/qckchambers.jpg
Now let us compute the volumes of the KC and QC.
KC -- 34.3 X 17.17 X 19.1 = 11,248.6 cubic ft.
QC -- 18.875 X 17.15 X 17.9 (average height of gabled roof) = 5,793 cubic ft.
5793 / 11248.6 = 0.515. This figure is also 97% of the Earth-Mars radii ratio of 0.5319.
Thus, there is a close resemblance in relative sizes of the two main chambers of the Great Pyramid and the relative sizes of Earth and Mars.
There are further suggestive Mars-related geometric ratios to the Giza plateau. The diagram below displays the alignment between the eccentric Martian orbit about the Sun, and the off-center alignment of the 3 pyramids on the Giza plateau.
http://www.geocities.com/annubis33/TheDesignofExistence/GizaMarsratio.jpg
The greater square ("C") shows the Giza plateau within a 10 by 10 grid. The Great Pyramid is "1", the 2nd Pyramid is "2", and the 3rd Pyramid is "3". The Sphinx is "S", shown at the bottom of the grid.
The circle (broken line) represents the orbit of Mars about the Sun. Length "D" is the Martian perehelion (closest point to Sun, 1.381 AU) and length "E" is the Martian aphelion (furthest point from Sun, 1.665 AU).
Around the Mars orbit, a square has been constructed ("C" by "C"). The center of the square is shown where the lower left hand corner (southeast corner on the Giza plateau) of the 2nd Pyramid is aligned. However, owing to the eccentric Martian orbit, the position of the "Sun" is actually at the point where lines "D" and "E" meet.
Straight lines drawn from the lower right hand corner of the outer square, aligned with both the true center of the square and the actual position of the Sun in the Martian orbital path, results in an angle of difference of approximately 2 degrees ("A"), as measured from the lower right corner of the square, connecting to the lower left corner of the 3rd Pyramid (SE corner). The alignment fits almost exactly with the placement of the 3 Giza pyramids. Thus, there is a proportionate ratio between the peculiar alignment of the Giza pyramids, and the eccentric Martian orbit.
Further, a square constructed based on length "D" (square "B") is in proportion to the greater square "C" by 1:2.15. Put another way, square "B" is about 46% the size of square "C". This is essentially the same size difference between the Great Pyramid, and the 3rd Pyramid, which some researchers have speculated to be associated with Mars.
Finally, it can be noted that the Sphinx is found with its paws overlapping the Mars orbit circle, and the bottom line of the grid. The possible significance of this will be explored later.
A simple equation can be derived that expresses a direct relationship between the peculiar shape of the orbit of Mars around the sun, the sacred goemetry "Golden Mean" value of "Phi", the perimeter of the Great Pyramid, and the length of the Grand Gallery. It follows as thus:
1) Mars aphelion (furthest point in its orbit from the Sun) = 1.665 AU (where 1 AU = 93 million miles, distance from Earth to Sun). Mars perehelion = 1.381 AU. The diagram below is a rough approximation of the eccentricity of Mars' orbit about the Sun. "A" = 1.381 AUs. "B" = 1.665 AUs. "C" = 1.52 AUs (average Martian semi-major axis). "D" = approx. 60 degrees. "E" = approx. 50 degrees.
http://www.geocities.com/annubis33/TheDesignofExistence/Marsorbit.jpg
2) "B"/"A" = 1.665/1.381 = 1.205. Or, very close to 6/5.
3) Phi, the Golden Mean ratio, = 1.618. (approx. value).
4) Pi, the ratio between circumference and diameter, = 3.14159 (rounded off).
There is a direct relationship between 6/5, Phi, and Pi. It is,
6/5 = Pi/Phi^2. (meaning Pi divided by Phi squared). As follows...
3.14159 (Pi) / 2.6179 (Phi squared) = 1.200042. Or, essentially 1.2 (6/5).
5) Thus, 6/5=Pi/Phi^2.
6) An Oregonian Crop Circle formation brought to my attention the relationship between the Grand Gallery of the Great Pyramid and the GP theoretical height of 481 feet. (481 feet was the circumference of the Oregon crop circle). The appromiximate diameter of the crop circle was thus 481/Pi=153 feet. This value of 153 feet is also the length of the Grand Gallery. Thus, the Grand Gallery ("GG") X Pi = 481 ft. = Height of GP, or "H".
7) It has long been known that the Great Pyramid encodes the value of Pi, as twice the height ("2H") X Pi = the Perimeter of the Pyramid ("P"). As follows....
2 X 481ft (GP Height = "H") X 3.14159 (Pi) = 3,022 ft., which = "P" (perimeter of the GP).
Thus, 2HPi=P Or,
Pi=P/2H
We also know that the Grand Gallery ("GG") X Pi = "H" (height of GP). Or,
GGPi=H. So,
GG X 2 X Pi X Pi = P. Condensed, this becomes
GG(2Pi^2)=P. (The length of the Grand Gallery times Two times Pi squared = the Perimeter of the Great Pyramid). Or,
GG = P/2Pi^2 Or,
Pi^2 = P/2GG Or,
Pi = /\P/2GG. (Square root of P/2GG)
Verification:
P = 3022 ft.
2GG = 153 ft X 2 = 306 ft.
3022/306 = 9.876.
Square root of 9.876 = 3.1426. (This is within 99.97% accuracy for Pi).
Bringing the Mars orbit, Phi, Pi and GP, Pi, H and P equations together is now simple as we have a common factor on both sides, being "Pi". We have...
Mars orbital shape (1.665AU/1.381AU) = "Alpha", where Alpha = 6/5.
Alpha = Pi/Phi^2. We also know from above that
Pi = /\P/2GG. Thus, combining the two, we have
Alpha = /\P/2GG / Phi^2.
Translating from math to English, this says that the variation in Mars' orbit about the Sun, expressed in fractional ratio, is equal to the Square Root of the Perimeter of the Great Pyramid over twice the length of the Grand Gallery divided by Phi Squared. Verification:
3022 (Perimeter of GP) / 306 (2 X Grand Gallery) = 9.876.
Square root of 9.876 = 3.1426.
Phi squared = 2.6179.
3.1426 / 2.6179 = 1.200.
Variation between Mars perehelion and aphelion is 1.205.
This value is within 99.6% of 1.200. Given the approximations of rounded off figures and pyramid and astronomic measurements, such an approximation may be considered as good as exact. So, what we have with the equation
Alpha = /\P/2GG / Phi^2
is the mathematical expression of the relationship in proportion between the slightly "warped" orbit of Mars around the Sun, the lengths of the Perimeter and Grand Gallery of the Great Pyramid, and the sacred geometry Golden Mean value of "Phi", which is basic to 5-sided symmetry, and the human form.
Finally, because the Great Pyramid also encodes the value of Phi (1.618...), the entire equation can be reduced to elements of the Great Pyramid.
The GP apothem slope = 611.5 ft. Half the base of the GP (or 1/8th its Perimeter) = 377.75 ft. 611.5/377.75 = 1.618, or Phi.
377.75/611.5 = 0.618. Arccosine (.618) = 51.8 degrees, which is the angle of the Great Pyramid.
Thus, Cosine (51.8 d.) = 0.618, or 1/Phi. So, Phi = 1/Cosine (51.8 d.). Inserting this into the above equation yields,
Alpha = /\P/2GG / (1/cos 51.8)^2.
Meaning, the fractional ratio of Mars' orbital eccentricity (1.2) is equal to the Square Root of the Great Pyramid Perimeter divided by twice the length of the GP Grand Gallery, all divided by the inverse of the cosine of 51.8 degrees (GP angle) Squared.
The other symbolic associations with such ratios between Earth-Mars, and the so-called "Queen's Chamber" (so named by the later Arabs for their tradition of burying women in gabled roof styles like that of the Queen's Chamber, though there is no evidence a queen was ever actually buried there) and the eerie and poorly understood subterrannean chamber are rich and will be explored in the future at this site.
As can be easily shown, the Heptagram (7-pointed star) is directly related to the outer slope angle of the Great Pyramid, as well as the angle of the ascending and descending passageways, and the Grand Gallery that leads to the King's Chamber.
http://www.geocities.com/annubis33/TheDesignofExistence/heptagramgp.jpg
From the diagram it can be noted the angles "A" of the pyramid slope and of the heptagram are essentially the same. (In fact, they are within 99.2% of each other. The pyramid angle is about 51 degrees 50 minutes of arc, or about 51.8 degrees. The steep angle of the heptagram is approximately 51.4 degrees). Angle "B", the shallow angle of the Heptagram, closely corresponds to the inner passageways of the GP (approximately 26 degrees).
The mystic properties of the heptagram, or 7-pointed star, are well known, owing mostly due to the fact that its slope angles cannot be precisely defined by numbers, unlike all other forms based on the other numbers. This is because 7 does not divide exactly into 360 (degrees), resulting in 51.42857143...going on forever, always an approximation. As a result of this many ancient temples and works of art were designed with the value of seven built into their geometry. The Great Pyramid is also based on 7-star proportions.
One of the most well recognized works of art is Leonardo da Vinci's "Vitruvian Man", which clearly demonstrates the geometric proportions of the human form.
http://www.geocities.com/annubis33/TheDesignofExistence/leonardogp.jpg
In superimposing the Great Pyramid angles onto Leonardo's Man, I noticed some fascinating things. First, when a GP is drawn at the base of the image, its angle of 51.8 degrees results in the apex ending at the navel of the man. This is very significant, as the navel region is universally recognized in mystical and spiritual traditions as being the true "center of being". It is also an area of profound biological import, being the link with the mother (umbilical cord). Psychologically and bioenergetically, it has been seen as the zone associated with so-called "will-power", especially as interpreted in the some Eastern traditions which refer to it as the source of life.
Of futher note is that angle "B" as shown, where the man's upraised arm joins the outer circle at his middle finger, as connected to the vertical axis of the drawing (center of the man's chest), and the horizontal arm beneath, results in an angle of just over 26 degrees, identical to the angle of the inner passageways and Grand Gallery of the Great Pyramid.
Additionally, the navel-pyramid apex is not just a zone associated with a "mystical" center, it is also the exact geometric center of the whole image, as can be seen. And, another GP drawn from the top down, results in both pyramids meeting in the center of the outer circle, with the capstones overlapping at the navel region.
Thus, we can see from these observations that the seven-pointed star, the Great Pyramid (and its absent "capstone"), and the human form all bear close geometric relationships to each other.
In the above analysis a geometric relationship between the GP, Earth, and Mars, and the human canon of proportions is suggested, and the diagram below offers more insights in this direction.
http://www.geocities.com/annubis33/TheDesignofExistence/dmgpvm.jpg
As can be seen, superimposed over Leonardo's Man are two forms. At the base is once again the 2-d version of the Great Pyramid, with its 51.8 degree slope, and its apex resting on the man's navel.
The second 5-sided form is the 2-d form of the structure at the Cydonia region of Mars that has come to be known as the D&M Pyramid. Its alignment with Leonardo's Man is shown in Richard Hoagland's "Monuments of Mars". What has not been demonstrated before, however, is how this 5-pointed structure, when superimposed over the canon of human proportions, shares a central point (the navel) with a triangle aligned with the base of the square circumscribing the man that yields a 51.8 degree apothem slope -- that of the Great Pyramid of Giza.
Conclusion: This demonstrates without question that there is a definite and simple geometric relationship between the 5-sided Mars D&M Pyramid as defined by Erol Torun and Hoagland, the 4-sided Great Pyramid, and the canon of human proportions.
All this is made possible by the relationship between the circle and the square, which Leonardo's canon is based on. These two simple geometric forms have long held powerful archetypal symbolism, with the circle being most often connected to "spirit" (or consciousness) and the square to "matter" (or the world). As suggested at the top of this webpage, it is my contention that the key element of the ancient mystery schools and their surviving physical legacies such as the Giza monuments and Stonehenge, is the relationship between consciousness and matter, or more simply, the subject and the object. The simple geometric relationship demonstrated in the above diagram is yet another signpost suggesting that the purpose and function of human consciousness is to bridge these two domains. I would further suggest that the objects at Cydonia and Giza are markers pointing to this "bridge" -- a bridge known and recognized in most wisdom traditions as the universal "life force" and most often associated with the navel or center region of the human form. This "life force" (known variously as "Ruach", "Prana", "Chi", "Holy Spirit", etc) is the final connecting principal in existence and perhaps is the key piece in any Unified Field Theory that theoretical physics has been reaching for in recent times, such as in Superstring theory, for example.
The Geometry of the Calstone Wellington, Wiltshire, Crop Circle, in relation the the Giza plateau
In the domain of Sacred Geometry and encoded numeric ratios, perhaps no better modern phenomena expresses these than "crop circles", mysterious formations seemingly appearing spontaneously in wheatfields and often bearing spectacular geometric formations that contain sophisticated patterns and precise angles.
The top diagram below shows a two dimensional pictogram of a crop circle formation that appeared in Wiltshire, England, on July 30th of 2000. As can be shown, series of simple geometric extrapolations from this formation yields the relative proportions of the 3 Giza pyramids.
<p align=center>http://www.geocities.com/annubis33/TheDesignofExistence/calcrop.jpg</p>
<p align=center>http://www.geocities.com/annubis33/TheDesignofExistence/calstonecrop.jpg</p>
Analysis:
1) The triangular shape of the formation is anchored by the two flanking circles. If two circles of equivalent diameter are placed on the central axis, one on top of the other, a line drawn from the top of the top circle, connecting with the centers of the flanking circles, yields a triangle with an angle of 50.5 degrees, +/- 1/2 degree (angle "E" in diagram). The angle of the 3rd Giza pyramid is 50.7 degrees.
2)A square drawn between the two flanking circles, and the lower central point and the top of the top circle, results in the square defined by line "C" in the diagram. This square, in proportion to a square drawn around one of the flanking circles, (line "A" in the diagram), results in a ratio of approximately 1:2.25. At Giza, the 3rd Pyramid is in ratio to the Great Pyramid in base size at about 1:2.22. These two ratios are within 1.5% of each other.
3)A square constructed from the center of the left flanking circle to the outer rim of the two central circles (line "B" in diagram) yields a square with a perimeter 94% of square "C". The ratio is 1:1.06. The 2nd Pyramid of Giza is in ratio to the Great Pyramid, in perimeter, at 1:1.07.
4)A line drawn connecting the center of the left flanking circle with the top point of square "B" results in a triangle with an angle of 53 degrees degrees, +/- 1/2 degree (angle "F" in diagram). The angle of the 2nd Pyramid of Giza is 53.1 degrees.
5)A line connecting from the top of the original triangle just above the small central square in the crop formation, to the base of the flanking circles, results in a triangle with an angle of 52 degrees, +/- 1/2 degree (angle "D" in diagram). The angle of the Great Pyramid is 51.8 degrees.
Thus, in summary, squares "C", "B", and "A" are in size ratio to Giza's Great Pyramid, 2nd Pyramid, and 3rd Pyramid respectively. Further, angles "D", "F", and "E" correspond in degrees to the Great Pyramid, 2nd Pyramid, and 3rd Pyramid respectively.
Ratio summary:
Square "C" = 1. Great Pyramid = 1.
Square "B" = .94. Second Pyramid = .94.
Square "A" = .44. Third Pyramid = .46.
Angle "D" = 52 degrees. Great Pyramid angle = 51.8 degrees.
Angle "F" = 53 degrees. Second Pyramid angle = 53.1 degrees.
Angle "E" = 50.5 degrees. Third Pyramid angle = 50.7 degrees.
What can such encoded information ultimately mean, and what good can it do us to "decipher" it?
Well, from the point of view of sacred geometry, certain angles and values have great redundancy and often come to aquire profound psychological or spiritual meaning. Their presence in both natural and myserious formations does not necessary imply a manipulating intelligence behind them, or even any separate, localized consciousness at all, but may merely be part of a sort of cosmic background signature of information that is constantly being transmitted. The angles and shapes themselves may be perfectly natural, and frequently pointing to certain pivotal values, such as those found at Giza.
What good it can do knowing them is a more subtle point, but there is no question that the higher sensitivities that are part of what Gurdjieff termed the "Higher Emotional Body" are stimulated by certain mathematical harmonies that express in art, architecture, and music. This is all part of the "Divine Proportion" that when attuned to, fine-tunes one's sensitivity and awareness of the innate wonder and perfection of existence, and all its interrelating parts -- seemingly independent, yet only part of one functioning Whole.
The Eye in the Triangle: Mona Lisa, the American dollar, and the missing Great Pyramid capstone
The proportions of sacred geometry can be found in numerous works of ancient and classic architecture -- as well as famous art. We have already touched on Leonardo da Vinci's "Vitruvian Man" as a case in point. There is no question that it can be seen from Leonardo's records and sketchs that he was conversant in sacred geometry, and used much of it not just in his art but in his scientific designs as well.
A brief further example of his encoded designs is shown below, in this geometric analysis of the Mona Lisa, probably the most beloved and well known portrait painting in the world since its creation by Leonardo's hand 500 years ago.
All-powerful in this painting are the eyes, of which more has been written about by art historians, and commented on by lay folk visiting the Louvre in France, perhaps than any other feature in the history of Western art. But what is less realized are the geometric mechanics behind the mesmerizing power of these eyes. Below are some observations.
http://www.geocities.com/annubis33/TheDesignofExistence/mona2.jpg
1) It is the left eye that is perfectly centered -- not both eyes. Two lines drawn from the upper corners of the painting, inclined at 33 degrees (the Masonic number of final initiation and completion), meet exactly at the left eye.
2) The painting is flanked by two pillars framing the background to Mona Lisa. A triangle constructed from the inner points of the two pillars, with a base across the horizontal pillar behind Mona Lisa, with a Great Pyramid angle of 51.8 degrees, ends in an apex just above the central left eye. Comparatively speaking, Mona Lisa's left eye rests in the position of the missing Great Pyramid capstone. Also, lengths "A", "B", and "C" are in near phi ratio to each other (the Golden mean proportion that is basic to the human form and the 5-point symmetry).
3) A triangle constructed from the base of the painting, joining with the apex of the 51.8 degree triangle, results in an angle of approximately 70 degrees. This is basically the same angle of the narrow pyramid drawn on the American dollar bill. Further, the left Mona Lisa eye corresponds exactly to the positioning of the eye on the raised capstone of the dollar bill -- when seen in the small triangle formed by a horizontal base aligned with her chin (see inset).
http://www.geocities.com/annubis33/TheDesignofExistence/amerdollar.jpg
None of this is to suggest that Leonardo was working with deliberation here, or that his Mona Lisa is somehow the source of the Eye in the Triangle occult symbolism and its eventual inclusion on the American currency. But what it does strongly suggest is that the sacred geometry he was working with is directly connected to certain high powered proportions and forms, many of which lie at the basis of the teachings transmitted by the ancient Mystery Schools. What is being suggested here is that the "Left Eye of Horus", the left eye of Mona Lisa, the Great Pyramid apothem angle of 51.8 degrees, its missing capstone, and the Eye in the Triangle of the American currency all have a common origin.
Sacred Geometry and the Design of Existence
© by Philip Teertha Mistlberger, All Rights Reserved.
The significance of "Sacred Geometry" and "Divine Proportion" ultimately lies in relationship -- the relationship between parts of a whole. As such, it reveals on open window to grasping via direct experience the significance of "right relationship". What this means is that in seeing and understanding the harmony of proportion via relationship, we have a glimpse into the importance of letting go of attachment -- by sensing the invisible hand behind all divine harmony that lies underneath existence, the pure archetypal templates of the universe that exist prior to the outer forms of the universe itself. Divine proportion and symmetries, and sacred geometry brings us face to face with the Design and the Designer of existence, and reveals to us the guiding intelligence that animates the outer universe, and, thus, shows us the route back to this Source of creation as well -- the Source of our own consciousness.
In sensing such divine harmony, we are afforded a deeper understanding of the illusoriness of the separate self -- i.e., we begin to see and understand, from the point of view of intellect, how nothing is truly Separate from anything else. Divine proportion speaks to the profound interrelationship and connection between all things. It hints at the Oneness inherent to existence, and finally to the One Spirit that is energizing all of life.
Many of the ancient cultures understood this integral model, and as such, many of the ancient monuments that bear their stamp have implicit within them readily apparent ratios and proportions that are examples of a sacred design. It was as if the architects and priests of these cultures were intentionally leaving behind signposts for future cultures to decode and learn from, unlocking the keys to the awakening of consciousness. Further, seeing and understanding these geometric ratios and the harmonizing essence of sacred geometry is an essential piece in the development of the capacity to experience awe and wonder, two inner states necessary for the opening into expanded consciousness.
What follow is a brief overview of some of the basics of sacred geometry.
1. Empty Space
http://www.plokta.com/plokta/issue13/space.gif
What is empty space? In a sense, the answer to this question is the most important of all. In terms of the separate universe of space and form, empty space defines the underlying matrix or Source of all things. For empty space is the unborn, and the undying, and that which is untouched by the universe of separation and its foundations of space and time.
The Buddhist tradition in particular exhalted the concept of empty space, designating it by the Sanskrit word shunyata and defining it as the essence and true nature of the "self". As empty space, this self -- pure consciousness -- is seen ultimately to be empty of form or content, which is why the Buddhist concept of "self" is anatta, or "no-self".
Empty space is the direct experience of Reality without conceptual thought. It is the ultimate symbol for seeing into the heart of Reality. For in empty space no separation is possible. The very emptiness is identical to pure Unity, or Oneness. Empty space is not "nothingness". It is pure Wholeness, without division or separation of any kind. It is pure awareness that is totally unidentified with thought.
The famous Heart Sutra of Mahayana Buddhism begins with,
Form is Emptiness, and Emptiness is Form.
"Form is Emptiness" speaks to the empty space or undifferentiated Unity that lies beneath all form. "Emptiness is Form" speaks to the pure potentiality of this empty space or undifferentiated Unity and of its mysterious, infinite capacity for Creation, and movement from Unity to separation -- a process that, as mentioned above, is going on all the time in our own minds.
Thus, empty space is symbolic of the ultimate womb of existence from which all things derive, as well as the purest possible representation of our True Identity, as empty space is no-thing, and is free of all projection, concept, or any possibility of distortion. It is untouched, immaculate, and unchangeable. For these reasons, empty space is the geometric symbol (or non-symbol) assigned par excellence for our ultimate identity, which is one and the same as the Source.
2. The Point
The point is unique in that it marks the first appearance of separation, for it is symbolic of the Primal Object, the first manifestation. As such, it is the perfect symbol of separate identity, and its ability to acquire individual status, even if the point is infinitesimally small.
However, the point has dual-symbolism. For while it is symbolic of our original separation from the Source (the appearance of the point of individual identity within the empty space of Source), it is also symbolic of the last stage prior to completely returning to Source. This last stage is characterized by the full awareness of our true nature that is beyond space, time, and dimension of any kind.
The point is also at the center and heart of all geometric forms. As these geometric forms are the building blocks of existence as manifested in the structure of the atom and molecules, DNA, and cells, and onward then we can conclude that symbolically the point is both the essence and the first manifestation of the universe.
3.The Line
The line is born when the point undertakes the action of extension. By moving and extending outwardly, a series of points occurs, which becomes the line.
When a series of lines all extending from one central point are connected from their outermost points, the circle is born. When the circle is born, so is the third dimension (volume), where the circle is known as the sphere.
4. The Circle, Sphere, and Pi
http://www.counton.org/pi/gallery4/images/art4_1.gif
The circle, being the two-dimensional shadow of the sphere, is a universal symbol of wholeness and completion. It is often symbolic of the purity of Spirit, and within its simple geometry is found a blueprint for all other geometries and forms. The circle in its purest form represents the return to Source via conscious relationship with the Whole. In three-dimensional form as the sphere, it is the perfect symbol for the totality of the manifest universe, and for this reason is also an apt symbol of both spiritual and physical projection, the original outward act of Creation (what science knows as the "Big Bang") that brought all manifest things into being, and is the standard shape of most cosmic bodies (stars and planets).
The simple well-known geometric ratio named Pi, symbolized by the Greek letter , is a value that represents the ratio between the diameter and circumference of a circle. (The diameter multiplied by Pi = the circumference). It is not a whole number, having the approximate value of 3.141592...etc., continuing endlessly, never ending at one precise value (which in the language of mathematics is called a "transcendental" number).
What does that mean in a symbolic, metaphysical sense? Simply, that the circle can never be precisely quantified by a so-called rational or linear value. It remains a mystery, just beyond the grasp of rational, numeric wholeness, and forever eluding exact definition and measurement. For this reason, the circle/sphere is also the consummate symbol of Spirit, that all-pervading life force or Spirit-energy that animates the universe and all life in it. Its function and nature remain mysterious and impossible to penetrate with the rational mind only.
Thus, we can view Creation symbolically as an Act resulting in the birth of the point of separate identity within the infinity and eternity of empty space (the Source). This point then undertakes the first Act of extension, resulting in the line. From the line is born the circle/sphere, from which follows all possible geometric templates representing the full Creation of the universe.
5. The Vesica Pisces
http://www.halexandria.org/images/scan0040.gif
When a second circle/sphere is created with its origin point occurring anywhere on the outer circle of the first circle/sphere, we have the birth of the symbol known in sacred geometry as the Vesica Pisces.
From within the central part of the Vesica Pisces (the overlapping arcs) is found the ratios that generate all significant root geometric templates (with the exception of the heptagram, explained below), such as the square, pentagram, hexagram, etc. Thus, it is seen that the circle is the symbol par excellence of the Mother of Existence (Spirit-energy, often called the "Goddess"), giving birth to the various archetypal forms from which all building blocks of matter and more complex life forms are based on.
The central form within the Vesica Pisces is suggestive of the eye, and indeed, the Vesica Pisces is the perfect symbol for conscious relationship, which is founded on the direct connecting of two separate minds truly seeing each other.
The ratios of the inner "eye" of the Vesica Pisces -- its height to its width -- is defined by the square root of three (1.732050808...). That is, the height multiplied by the square root of three = the length. Symbolically, this points to the fundamental significance of the value of three in all relationship, and how, in fact, there is no relationship between two things without a third element -- the all-important context -- being present. (For example, positive-negative-neutral; proton-electron-electromagnetic field; father-mother-child; etc.).
6. Phi and the Golden Mean Proportion
The Golden Mean Proportion, also known as the "Divine Proportion" or the "Golden Ratio" is a fundamental ratio that is both strikingly simple and deeply profound in its symbolic meaning. As with the value Pi, and the Vesica Pisces, it shows up in countless geometric forms throughout existence, being one of the chief archetypal templates of the manifest universe. Historically, this ratio has been recognized by philosophers, architects, and artists as one of primary importance. Many ancient temples and monuments (such as the Great Pyramid, Stonehenge, and the Parthenon) incorporated the Golden Mean Proportion in their design.
The Golden Mean Proportion is best explained with diagram, but essentially, it involves a straight line, divided into two segments, such that the shorter segment of the line = 1, and the longer segment = Phi, symbolized by the Greek letter , which in value, is 1.6180339887...*etc. (a value that never ends precisely, as with Pi).
The ratio then follows as thus: the length of the entire line is in proportion to the longer segment as this longer segment is in proportion to the shorter segment. Symbolically, this is representative of the act of Creation, or the primal separation that is outlined in the world's chief Creation myths. This is because it is the perfect symbol of the Unity of One being expressed in Two terms. Or, simply, the Creation from the One (Source), so that it now becomes Two (Separation) -- the birth of the manifest universe.
Imagine it this way. We begin with Unity. From itself, this Unity then divides, in such a way that the proportion of its division is harmonious, with the smaller part being in proportion to the larger, as the whole thing is in proportion to the larger part. Or, what is known symbolically as Divine Proportion, because the ratio itself is symbolic of perfection. The irony of this "perfect ratio" is that the actual numeric value, called Phi, is not a whole number, as given above. Thus, like Pi, its perfection is abstract and forever out of the reach of the linear quantification of the rational mind.
The most mystical of the four New Testament Gospels, attributed to John, begins with a line that perfectly expresses the Golden Ratio, or Divine Proportion, in words:
In the Beginning was the Word, and the Word was with God, and the Word was God.
This line, just as with the Divine Proportion of Phi, suggests the creative duality within Unity itself. The "Word" which is the same as Spirit-energy, is both within the Source, and part of the Source's creative duality as well. The relationship between the Word and God, or Spirit-energy and Source, is the same, symbolically, as that of the Golden Ratio or Divine Proportion represented in the division of the line at the point of the Phi ratio.
In other words, the entire line is symbolic of the Source, or the All, or empty space. The point of the Phi ratio, is symbolic of the first split, or the birth of ego-identity. The length of the larger segment can be seen as symbolic of Spirit-energy, and the shorter line can be understood to represent the World, or the manifest universe.
The same symbolism can be reversed as well. That is, the shorter segment can also be seen as the Source, with the longer length once again symbolizing Spirit-energy, and the whole line denoting the universe.
Either way, all three aspects are seen and understood to be harmonious, proportional aspects that are also an expression of one primal Unity. Likewise, Source, Spirit, and universe are ultimately an expression of the One, as only the One can exist. Miraculously, this One can divide itself, in accordance with the laws of Divine Proportion, but it always also remains only One. Or, as Buddhism phrases it,
Nirvana is Samsara
Which means, that the changeless, eternal Absolute Unity (Nirvana) is finally the same as the manifest universe of constant change (Samsara).
The Fibonacci spiral
The 13th century mathematician Fibonacci discovered a simple pattern that emerged from a sequence of numbers that involved the last two in the sequence being added together to produce the next one in the sequence, as follows:
0+1=1. 1+1=2. 1+2=3. 2+3=5. 3+5=8. 5+8=13. 8+13=21. 13+21=34. And so on. This results in the sequence:
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377, and so on.
As with the Golden Mean Proportion, this sequence of numbers is another superb symbol of the process of Creation. We begin with 0, which is Empty Space, or the Source, and then add 1, which is symbolic of the Point in Empty Space, the Primal Division from Unity, the first Separation. This then subdivides endlessly, but the sequence of numbers grows only by accessing numbers from within its own sequence. Just so, the Primal Unity of Existence can never be altered, although this Unity has the miraculous capacity to endlessly replicate itself outwardly (as in the Fibonacci sequence of numbers getting bigger), or it can also be seen as the infinite dividing itself inwardly (the sequence of numbers reducing to zero). Either way, as with the Golden Proportion, in the end there is only Unity.
Put numerically and symbolically, 0 = Infinity. The Beginning equals the End. The Nothing equals the All.
The Fibonacci sequence of numbers, when put in fractional ratios, result in quotients which gradually get closer and closer to the Golden Ratio (Phi) of approximately 1.618, as follows:
1/0=infinity. 1/1=1. 2/1=2 3/2=1.5 5/3=1.6667 8/5=1.6000 13/8=1.6250 21/13=1.6154 34/21=1.6190 55/34=1.6176
and so forth. Each subsequent quotient in the series of numbers gets closer and closer to the Phi ratio, but only perfectly duplicates it at infinity.
Thus, there is a direct relationship between Fibonacci's number sequence, and the Golden Mean Proportion. Further, when the Fibonacci sequence is placed out in a series of Golden Mean Rectangles, a Spiral is born from within these rectangles, as demonstrated below.
This Fibonacci Spiral is one of the key archetypal forms of manifest existence. It is the outward expression of the Original Act of Separation, in particular, in the formation of space and time, i.e., dimension.
The universality of the spiral is found within many key structures in the manifest universe, especially spiral galaxies and the spiral form of the DNA helix. Whirlpools and the spiral movement of the kundalini energy up the spine, as well as nautilus shells and numerous plant and animal proportions reveal the archetypal significance of this form ¨C itself the result of the Golden Mean Ratio, or the Divine Proportion.
The Pentagram
The pentagram, or five-pointed star, is an excellent symbol of the psychospiritual process that we can call "embracing the Shadow", that is, fully integrating our unconscious dark side. The pentagram is a symbol of power, and when the Shadow is integrated, we become empowered in the true, spiritual sense of the word.
It is a curious fact that around sixty nations on Earth use the five-pointed star on their flag, many from completely different cultures and religions. The symbol itself traces back into ancient times, mostly to Egypt, where it was depicted as the heiroglyph connected to the duat, or underworld, and the red god Set, a rebellious and destructive deity who was associated with storms and chaos, but who originally was just a symbol for the red desert and death via famine or flood.
The inverted pentagram at some time in the past picked up the German name drundenfuss, translating as "Witch's foot". In its right side up form it was connected also to Pan, the old Greek goat-foot deity. It was only in later times that the symbol became associated with a Christianized version of Satan, which by that time was a somewhat comedic metaphor for the repressed sexuality so rampant in those times, a cultural pathology that expressed eventually in part through the Inquisition burnings.
The Hexagram
The hexagram or six-pointed star, is an ancient symbol that was originally called the "Seal of Solomon". To Jews it is a sacred symbol termed the "Shield of David", and to Hindus it has been known as "Vishnu's Mark". Spiritually, the hexagram symbol has often been associated with the rising of aspiration (the base triangle) meeting the descent of grace (the upper triangle).
The Heptagram and the Properties of Seven
The number seven has some curious particularities. For example 1 divided by 7 gives a simple recurring decimal of six digits:
0.142857142857142857...
and this one multiplied successively by 2, 3, 4, 5, 6, gives products where we find the same numbers in the same order:
142857 x 2 = 285714 142857 x 3 = 428571 142857 x 4 = 571428 142857 x 5 = 714285 142857 x 6 = 857142
For the ancients, seven was known as "the virgin", as no number less than seven divides into it. For example, two divides four, six, eight, and ten; three divides six and nine; four divides eight; five divides ten; six divides six. But no number divides seven evenly. Thus seven is a "virgin", as no other number enters into it, symbolic of its numeric purity.
Among all prime numbers, seven is the first number that does not enter in the Euclidean circle. Dividing all other root numbers into 360 results in a whole number, except for seven, as follows:
360 / 1 = 360 360 / 2 = 180 360 / 3 = 120 360 / 4 = 90 360 / 5 = 72 360 / 6 = 60 360 / 7 = 51.42857143 etc., repeating irrationally. 360 / 8 = 45 360 / 9 = 40
Symbolically, this means that the heptad (seven) defies precise quantification, much as with the values of Pi (circle) and Phi (Golden Mean Ratio). Put another way, the heptad cannot be measured, or defined or captured, by linear rationalism.
The heptagon is the only basic geometric form that cannot be created via the Vesica Pisces, which is often regarded by sacred geometry as the "womb" of existence. It is because of this unique form that the value of seven is regarded as virginal, because it is not "born" through the geometric and arithmetic precision that the other root numbers are.
The steeper angle of the heptagram is 51.428* degrees. This is very close to the angle of slope of the sole surviving Wonder of the ancient world, the Great Pyramid of Giza, Egypt, which is about 51.8 degrees. Further, the lesser slope of the heptagram is essentially the same as the slope of the internal passageways in the Great Pyramid, about 26.5 degrees. If a double heptagram (fourteen point star) is constructed, then the form of the Great Pyramid is found in the center of interfacing lines, with a 51.428 degree angle of slope. Many ancient Greek temples (such as those dedicated to Athena, Minerva, and Neith) were based on seven-fold geometry. All this is suggestive of the importance ancient architects attached to this number seven.
The Design of Existence and Spiritual Awakening
The ultimate value and usefulness of sacred geometry and the mystical study of numbers, is very similar to the Zen concept of the koan. A koan is a paradox that has no rational answer, such as "Who am I?" "Where does the universe end?", etc. It is a device to engage the mind with, until the rational process of deduction wears itself out, at which point something called an apperception occurs -- the turning back of consciousness toward its own source.
At this point, the mind is quiet, and intuitive, direct, tacit understanding leaps forward. The question (koan) is solved be dissolving the question itself.
This is direct understanding, termed satori or kensho in Zen, is the experience of non-duality, even if only for a moment in time. It is a taste of the unborn consciousness within us, which sees and understands free of concept. Inherent within it this is limitless freedom and joy, and boundless love and gratitude, simply for what is.
To study the design of existence is useful only inasmuch as it leads to the experience of awe and wonder, and the ultimate relaxing of our intellect in its ceaseless struggle to understand. In that sense, true understanding happens only when we realize, fully and completely, that intellectual understanding alone is not enough to take us to the other shore. But paradoxically we do need a sharpening of intelligence, and this something sacred geometry can help to bring about, via the practice of contemplating its forms.
Intellectual understanding that leads to awe and wonder and gratitude is one path to inner, intuitive revelation, the Awakening to Ultimate Reality, our true nature.
Sacred Geometry and Divine Proportion
The Sattva System with its 14 stages of development ¨C seven stages of Ego growth, and seven stages of Returning to Source ¨C in terms of geometry is based on the heptagram, the seven-pointed star. The significance of this symbol is found within two of the most well known ancient sites on Earth, those of Giza, Egypt, and Stonehenge, England. Additionally, there are significant geometric ratios between the Great Pyramid, Stonehenge, the Earth, Moon, and Mars, some of which will be demonstrated shortly.
The significance of ˇ°Divine Proportionˇ± ultimately lies in relationship ¨C the relationship between parts of a whole. As such, it reveals on open window to grasping via direct experience the significance of ˇ°right relationshipˇ±. What this means is that in seeing and understanding the harmony of proportion via relationship, we have a glimpse into the importance of letting go of attachment ¨C exactly by sensing the invisible hand behind all divine harmony that lies underneath existence.
In sensing such divine harmony, we are afforded a deeper understanding of the illusoriness of the separate self ¨C i.e., we begin to see and understand, from the point of view of intellect, how nothing is truly separate from anything else. Divine proportion speaks to the profound interrelationship and connection between all things. It hints at the Oneness inherent to existence, and finally to the One Spirit animating all life.
Many of the ancient cultures understood this wholistic model (although certainly some did not), and as such, many of the monuments that bear their stamp have encoded within them readily apparent ratios and proportions that are examples of a sacred design. It was as if the architects and priests of these cultures were intentionally leaving behind signposts for future cultures to decode and learn from, unlocking the keys to the awakening of consciousness contained within them. Further, seeing and understanding these geometric ratios and the harmonizing essence of sacred geometry is an essential piece in the development of the capacity to experience awe and wonder, two inner states necessary for the development of higher consciousness.
For this discussion, we will limit our analysis to two of these sacred sites on Earth, the Giza plateau of Egypt, and Stonehenge of Britain.
A)The Great Pyramid
When a circle representing the dimensions of the Earth, and one representing the dimensions of the Moon, are placed adjacent to each other, with the Moon on top, and a triangle is drawn with its apex in the center of the Moon and its bottom base through the middle of the Earth, the resulting angle of 51 degrees 51 minutes of arc is exactly equal to the apothem slope of the Great Pyramid.
http://www.geocities.com/annubis33/TheDesignofExistence/GPEarthMoon.jpg
This striking geometric proportion is highlighted due to the unique properties of the Giza Great Pyramid, alone amongst about 100 extant pyramids in Egypt in possessing the complex inner chamber system it does, in addition to its encoded mathematical values of "pi" and "phi".
Such ratios speak of harmony, and flow, giving the suggestion of effortlessness and "fit" in a way that is strongly evocative of the surrender that is the essence of all ways of describing the return of consciousness to its Source, or what the Sattva System defines as the last seven stages on the Inner Arc to Unity.
In the case of the Great Pyramid, they speak to an obvious awareness attained by the pyramid builders -- not just of the proportions between the Earth and Moon, but to the significance of unifying disciplines, such as science, geometry, esthetics, and sacred proportions, all ending in a harmonious consciousness that is naturally at home wherever it is. As such, it refers to the profound sense of spirituality attained by many of the ancients, and notably the early Egyptians.
The Pyramid-Moon-Earth ratio is well known and has been discovered by researchers as far back as the 1970s (and possibly earlier).
I myself have discovered an equally interesting series of proportionate ratios between the inner chambers and overall dimensions of the Great Pyramid, and Earth and Mars. For now, I present a few of these:
In the diagram below, the Great Pyramid is superimposed over the Earth (top) and Mars (bottom), both shown to scale in their proportional diameters. On the right side of the diagram, the section lengths "A" and "B" are shown, representing the distances from the truncated Pyramid apex to the roof of the Queen's Chamber, and from the roof of the Q.C. to a key point in the subterranean chamber directly beneath the Great Pyramid (also known as the "Pit").
http://www.geocities.com/annubis33/TheDesignofExistence/GPEM.jpg
Point "C" is shown as the addition of points "A" and "B". This diagram demonstrates a simple proportion of very close ratios between both the radii of Earth and Mars and the orbital distances of Earth and Mars from the Sun, with some inner and outer dimensions of the Great Pyramid.
On the right of the diagram, note the section lengths "A" and "B". The bottom horizontal line connecting with "B" lines up with the equator of Mars, as well as the key point in the subterranean chamber that lies 3.1 feet below the floor of the underground horizontal passageway.
The top horizontal line connecting with "B" ends on the roof of the Queen's Chamber, aligned centrally in the pyramid. This also lies on the "north pole" of Mars in our diagram (and the "south pole" of Earth). The top horizontal line connecting with "A" ends on the truncated apex of the Great Pyramid, which is also the equator of the Earth.
The arcs "D" and "E" -- both of which align with the roof of the Queen's Chamber and the key point in the "Pit" below the pyramid respectively -- corresponds almost exactly (99.42% accuracy) to the orbits of Earth and Mars, and their mean proportional distances from the Sun.
The numeric analysis is as follows:
In the Great Pyramid, the distance from the truncated Apex (450 feet above the ground) of the Pyramid to the roof of the Queen's Chamber is 360 feet. The distance from the roof of the Queen's Chamber to the floor of the horizontal passageway entering the subterranean chamber is 188.42 feet. The ratio of 360 to 188.42 is 1 to 0.523.
The radius of the Earth is 3,959 miles. The radius of Mars is 2,106 miles. The ratio of 3,959 to 2,106 is 1 to 0.532.
.523 is 98.3% of .532. Superimposing such a margin of difference onto Mars would result in the floor of the horizontal passageway falling less than 36 miles from the exact equator of Mars.
However, the subterranean chamber (also known as the "Pit") extends deeper, and at a certain point hits the value of 191.5 feet below the roof of the Queen's Chamber (3.1 feet below the floor of the descending passageway leading into the Pit). This particular point corresponds exactly to the equator of Mars, when overlayed over the circle representing Mars in our diagram. In other words, the ratio of the Earth-Mars radii (3959/2106=1.8798) is exactly equal to the ratio of the distance from the top of the Great Pyramid to the roof of Queen's Chamber, and from the QC roof to this point in the Pit (360/191.5=1.8798). Thus, there is a direct geometric and numeric relationship between the inner design of the chambers in the Great Pyramid to its outer dimensions, and the relative radii of Mars and Earth.
Additionally, the ratio of "A" to "C" in the diagram, where "C" = "A" + "B", is 1 to 1.5319. This very closely corresponds to the semi-major axis ratios of the Earth and Mars. The Earth is 1 AU (Astronomical Unit = 93 million miles) from the Sun. Mars, on average, is 141,635,349 miles from the Sun, or 1.5230 AUs. The value of 1.5230 is 99.42% of 1.5319, or these ratios are within 0.58% of each other. Thus, the arcs touching on the roof of the Queen's Chamber and on the point in the subterranean chamber just below the floor of the horizontal passageway are proportionate not just to the radii of Earth and Mars, but to their respective orbital distances from the Sun as well (shown as "D" and "E" in the diagram).
Further, as a note of sidelong interest, the point at which the roof of the descending passageway, leading to the Pit, bisects the Mars arc, corresponds approximately to 41 degrees N latitude (line "F" on the diagram), which is the latitude of Cydonia, a region of mysterious structures on the Martian surface which many researchers have suspected to be artificial.
In this whole analysis, a key point should be noted. The top of the Great Pyramid is believed by some to have once held a capstone, which would have brought the elevation of the pyramid to 481 feet above the ground. However, there is no conclusive evidence that such a capstone ever existed. When we use a model of the Great Pyramid utilizing its current dimensions, with its apparently "truncated" apex of 450 feet above the ground, then the proportions just outlined all fall into place.
What would this suggest? From an intellectual standpoint it could be argued that this may indicate evidence that there never was any capstone, as the flat-topped pyramid at 450 feet elevation results in these harmonious ratios with Mars and Earth dimensions. However, from the intuitive point of view, what it more simply suggests is that the current proportions of the Great Pyramid allow for these ratios. Thus, with what is now, a harmony exists, suggesting a symbolic teaching that harmony can always be unearthed in present conditions, regardless of past circumstances -- provided that, in the Taoist language, we surrender to what is rather then struggling to get things to conform to what we think they should be.
An additional close ratio of significant geometric structures and the Earth/Mars radii ratio is as follows: Distance from *actual* GP apex to Queen's Chamber roof = 364 ft. Distance from QC roof to Subterranean chamber = 188 ft. 188/364 = approx. 0.52. This ratio is significant, as not only is it also essentially the ratio in radius between Earth and Mars, it is also the ratio in volume between a sphere and a cube, when the sphere is contained perfectly within the cube. (Cube=1, Sphere=0.524, in volume). This ratio is 98.5% the actual Earth Mars ratio of 1:0.532.
Further geometric interrelationships between the Great Pyramid (GP) and Mars are shown in the below diagram.
The 6 circles are all Mars-proportional in diameter, with Earth shown as the broken line circle. The GP, at 51.8 degree slope, is shown centrally. As in the above diagram, "south pole" Earth and "north pole" Mars meet at the roof of the Queen's Chamber (Q). Using simple geometry, the following is found:
http://www.geocities.com/annubis33/TheDesignofExistence/GP6Mars.jpg
1) Mars circles "B" and "F" are centered on the diagonal edges of the GP. Directly above them, are Mars circles "C" and "E", their centers aligned vertically with the centers of circles "B" and "F". When the centers of circles "B" and "E" are connected, the resulting angle of the line in relation to the base of the GP is 26.9 degrees. This is very close to the angle of the Grand Gallery and the ascending and descending passageways in the GP, which are about 26.5 degrees, thus being within 1% accuracy of the Mars circle ratios.
2) When the center of Mars circle "C" is connected to the Queen's Chamber, it corresponds very closely (40.5 degree angle) to the north face air shaft of the Queen's Chamber. (The most recent estimates of the QC airshafts angles, based on Rudolph Gantenbrink's update of Flinders Petrie's work, are about 39.5 degrees, but this is considered to have a wide possible margin of error, +/- over 1 degree).
3) When the center of Mars circle "E" is connected to the south base of the Queen's Chamber, it corresponds very closely to the south face air shaft (also 40.5 degrees) of the QC.
Trigonometric analysis at follows:
Base length of GP = 755ft = "Z".
Diameter of Mars circle = 383ft = "X".
Arctangent of 383/755 = 26 degrees 53 min/arc.
Length of "zero point" of QC air shaft to outer far point of GP = 363ft = "Y".
Length of distance from center of Mars circle "C" to point where line "Y" ends directly above center of circle "E" = 312ft = "W".
Arctan of 312/363 = 40 deg. 40 min/arc.
4) A perfect Fibonacci Spiral (based on the Fibonacci sequence of numbers) extends from the coffer in the King's Chamber (K) through the floor of the Queen's Chamber (Q), bisects the arc of the Earth-circle and the outer face of the GP, and ends up at the theoretical apex of the GP (top of missing capstone).
The above analysis suggests that the angles of the QC airshafts and Grand Gallery may be related to geometric proportions rather than astronomical positionings. As such, it suggests an alternate explanation to the theory of Robert Bauval and Adrian Gilbert, who had linked the placement of the QC airshafts to the positions of the stars Sirius and Kochab circa 2450 BC. This could suggest that the south face QC air shaft is not connected to Sirius at all, but is rather associated with geometric ratios, possibly involving encoded links to Mars. As Bauval and Gilbert's estimated construction date for the GP of 2450 BC was based on the positioning of Sirius at that time, any alternate hypothesis for the angles of the QC airshafts potentially throws the whole 2450 BC construction time in doubt. (This analysis is still in progress).
The diagram below shows in detail the movement of the Fibonacci spiral (from the floor of the King's Chamber to the apex of the pyramid), based on Golden Mean rectangles, in relation to the inner design of the Great Pyramid.
http://www.geocities.com/annubis33/TheDesignofExistence/fibonaccigp.jpg
As the Queen's Chamber as been involved in several of these geometric ratios involving Mars, it is interesting to note the following ratio relationships between the QC and the King's Chamber, and their close correspondence with the radii ratios between Mars and Earth.
The floor plan of the King's Chamber is rectangular, of 2 X 1 proportion. The Queen's Chamber is basically square. Thus, any apparent links with Earth/Mars radii ratios are not obvious at first glance. However, when we calculate comparative area of the chamber floors, and comparative volumes, we see something interesting, as follows (units in feet).
1) King's Chamber -- 34.3 by 17.17 = 588.82 sq ft.
2) Queen's Chamber -- 18.875 by 17.15 = 323.71 sq ft.
323.71 / 588.82 = 0.5497.
The actual ratio between Mars and Earth radii is 2106 / 3959 = 0.5319.
0.5319 is 97% of 0.5497. Thus, the square footage area of the KC floor in ratio to the QC floor is within 3% of the radii ratios of Earth and Mars.
When this is shown diagrammatically, with the squares representing the relative areas of the two chambers, and the circles the relative diameters of Mars and Earth, we have this:
http://www.geocities.com/annubis33/TheDesignofExistence/qckchambers.jpg
Now let us compute the volumes of the KC and QC.
KC -- 34.3 X 17.17 X 19.1 = 11,248.6 cubic ft.
QC -- 18.875 X 17.15 X 17.9 (average height of gabled roof) = 5,793 cubic ft.
5793 / 11248.6 = 0.515. This figure is also 97% of the Earth-Mars radii ratio of 0.5319.
Thus, there is a close resemblance in relative sizes of the two main chambers of the Great Pyramid and the relative sizes of Earth and Mars.
There are further suggestive Mars-related geometric ratios to the Giza plateau. The diagram below displays the alignment between the eccentric Martian orbit about the Sun, and the off-center alignment of the 3 pyramids on the Giza plateau.
http://www.geocities.com/annubis33/TheDesignofExistence/GizaMarsratio.jpg
The greater square ("C") shows the Giza plateau within a 10 by 10 grid. The Great Pyramid is "1", the 2nd Pyramid is "2", and the 3rd Pyramid is "3". The Sphinx is "S", shown at the bottom of the grid.
The circle (broken line) represents the orbit of Mars about the Sun. Length "D" is the Martian perehelion (closest point to Sun, 1.381 AU) and length "E" is the Martian aphelion (furthest point from Sun, 1.665 AU).
Around the Mars orbit, a square has been constructed ("C" by "C"). The center of the square is shown where the lower left hand corner (southeast corner on the Giza plateau) of the 2nd Pyramid is aligned. However, owing to the eccentric Martian orbit, the position of the "Sun" is actually at the point where lines "D" and "E" meet.
Straight lines drawn from the lower right hand corner of the outer square, aligned with both the true center of the square and the actual position of the Sun in the Martian orbital path, results in an angle of difference of approximately 2 degrees ("A"), as measured from the lower right corner of the square, connecting to the lower left corner of the 3rd Pyramid (SE corner). The alignment fits almost exactly with the placement of the 3 Giza pyramids. Thus, there is a proportionate ratio between the peculiar alignment of the Giza pyramids, and the eccentric Martian orbit.
Further, a square constructed based on length "D" (square "B") is in proportion to the greater square "C" by 1:2.15. Put another way, square "B" is about 46% the size of square "C". This is essentially the same size difference between the Great Pyramid, and the 3rd Pyramid, which some researchers have speculated to be associated with Mars.
Finally, it can be noted that the Sphinx is found with its paws overlapping the Mars orbit circle, and the bottom line of the grid. The possible significance of this will be explored later.
A simple equation can be derived that expresses a direct relationship between the peculiar shape of the orbit of Mars around the sun, the sacred goemetry "Golden Mean" value of "Phi", the perimeter of the Great Pyramid, and the length of the Grand Gallery. It follows as thus:
1) Mars aphelion (furthest point in its orbit from the Sun) = 1.665 AU (where 1 AU = 93 million miles, distance from Earth to Sun). Mars perehelion = 1.381 AU. The diagram below is a rough approximation of the eccentricity of Mars' orbit about the Sun. "A" = 1.381 AUs. "B" = 1.665 AUs. "C" = 1.52 AUs (average Martian semi-major axis). "D" = approx. 60 degrees. "E" = approx. 50 degrees.
http://www.geocities.com/annubis33/TheDesignofExistence/Marsorbit.jpg
2) "B"/"A" = 1.665/1.381 = 1.205. Or, very close to 6/5.
3) Phi, the Golden Mean ratio, = 1.618. (approx. value).
4) Pi, the ratio between circumference and diameter, = 3.14159 (rounded off).
There is a direct relationship between 6/5, Phi, and Pi. It is,
6/5 = Pi/Phi^2. (meaning Pi divided by Phi squared). As follows...
3.14159 (Pi) / 2.6179 (Phi squared) = 1.200042. Or, essentially 1.2 (6/5).
5) Thus, 6/5=Pi/Phi^2.
6) An Oregonian Crop Circle formation brought to my attention the relationship between the Grand Gallery of the Great Pyramid and the GP theoretical height of 481 feet. (481 feet was the circumference of the Oregon crop circle). The appromiximate diameter of the crop circle was thus 481/Pi=153 feet. This value of 153 feet is also the length of the Grand Gallery. Thus, the Grand Gallery ("GG") X Pi = 481 ft. = Height of GP, or "H".
7) It has long been known that the Great Pyramid encodes the value of Pi, as twice the height ("2H") X Pi = the Perimeter of the Pyramid ("P"). As follows....
2 X 481ft (GP Height = "H") X 3.14159 (Pi) = 3,022 ft., which = "P" (perimeter of the GP).
Thus, 2HPi=P Or,
Pi=P/2H
We also know that the Grand Gallery ("GG") X Pi = "H" (height of GP). Or,
GGPi=H. So,
GG X 2 X Pi X Pi = P. Condensed, this becomes
GG(2Pi^2)=P. (The length of the Grand Gallery times Two times Pi squared = the Perimeter of the Great Pyramid). Or,
GG = P/2Pi^2 Or,
Pi^2 = P/2GG Or,
Pi = /\P/2GG. (Square root of P/2GG)
Verification:
P = 3022 ft.
2GG = 153 ft X 2 = 306 ft.
3022/306 = 9.876.
Square root of 9.876 = 3.1426. (This is within 99.97% accuracy for Pi).
Bringing the Mars orbit, Phi, Pi and GP, Pi, H and P equations together is now simple as we have a common factor on both sides, being "Pi". We have...
Mars orbital shape (1.665AU/1.381AU) = "Alpha", where Alpha = 6/5.
Alpha = Pi/Phi^2. We also know from above that
Pi = /\P/2GG. Thus, combining the two, we have
Alpha = /\P/2GG / Phi^2.
Translating from math to English, this says that the variation in Mars' orbit about the Sun, expressed in fractional ratio, is equal to the Square Root of the Perimeter of the Great Pyramid over twice the length of the Grand Gallery divided by Phi Squared. Verification:
3022 (Perimeter of GP) / 306 (2 X Grand Gallery) = 9.876.
Square root of 9.876 = 3.1426.
Phi squared = 2.6179.
3.1426 / 2.6179 = 1.200.
Variation between Mars perehelion and aphelion is 1.205.
This value is within 99.6% of 1.200. Given the approximations of rounded off figures and pyramid and astronomic measurements, such an approximation may be considered as good as exact. So, what we have with the equation
Alpha = /\P/2GG / Phi^2
is the mathematical expression of the relationship in proportion between the slightly "warped" orbit of Mars around the Sun, the lengths of the Perimeter and Grand Gallery of the Great Pyramid, and the sacred geometry Golden Mean value of "Phi", which is basic to 5-sided symmetry, and the human form.
Finally, because the Great Pyramid also encodes the value of Phi (1.618...), the entire equation can be reduced to elements of the Great Pyramid.
The GP apothem slope = 611.5 ft. Half the base of the GP (or 1/8th its Perimeter) = 377.75 ft. 611.5/377.75 = 1.618, or Phi.
377.75/611.5 = 0.618. Arccosine (.618) = 51.8 degrees, which is the angle of the Great Pyramid.
Thus, Cosine (51.8 d.) = 0.618, or 1/Phi. So, Phi = 1/Cosine (51.8 d.). Inserting this into the above equation yields,
Alpha = /\P/2GG / (1/cos 51.8)^2.
Meaning, the fractional ratio of Mars' orbital eccentricity (1.2) is equal to the Square Root of the Great Pyramid Perimeter divided by twice the length of the GP Grand Gallery, all divided by the inverse of the cosine of 51.8 degrees (GP angle) Squared.
The other symbolic associations with such ratios between Earth-Mars, and the so-called "Queen's Chamber" (so named by the later Arabs for their tradition of burying women in gabled roof styles like that of the Queen's Chamber, though there is no evidence a queen was ever actually buried there) and the eerie and poorly understood subterrannean chamber are rich and will be explored in the future at this site.
As can be easily shown, the Heptagram (7-pointed star) is directly related to the outer slope angle of the Great Pyramid, as well as the angle of the ascending and descending passageways, and the Grand Gallery that leads to the King's Chamber.
http://www.geocities.com/annubis33/TheDesignofExistence/heptagramgp.jpg
From the diagram it can be noted the angles "A" of the pyramid slope and of the heptagram are essentially the same. (In fact, they are within 99.2% of each other. The pyramid angle is about 51 degrees 50 minutes of arc, or about 51.8 degrees. The steep angle of the heptagram is approximately 51.4 degrees). Angle "B", the shallow angle of the Heptagram, closely corresponds to the inner passageways of the GP (approximately 26 degrees).
The mystic properties of the heptagram, or 7-pointed star, are well known, owing mostly due to the fact that its slope angles cannot be precisely defined by numbers, unlike all other forms based on the other numbers. This is because 7 does not divide exactly into 360 (degrees), resulting in 51.42857143...going on forever, always an approximation. As a result of this many ancient temples and works of art were designed with the value of seven built into their geometry. The Great Pyramid is also based on 7-star proportions.
One of the most well recognized works of art is Leonardo da Vinci's "Vitruvian Man", which clearly demonstrates the geometric proportions of the human form.
http://www.geocities.com/annubis33/TheDesignofExistence/leonardogp.jpg
In superimposing the Great Pyramid angles onto Leonardo's Man, I noticed some fascinating things. First, when a GP is drawn at the base of the image, its angle of 51.8 degrees results in the apex ending at the navel of the man. This is very significant, as the navel region is universally recognized in mystical and spiritual traditions as being the true "center of being". It is also an area of profound biological import, being the link with the mother (umbilical cord). Psychologically and bioenergetically, it has been seen as the zone associated with so-called "will-power", especially as interpreted in the some Eastern traditions which refer to it as the source of life.
Of futher note is that angle "B" as shown, where the man's upraised arm joins the outer circle at his middle finger, as connected to the vertical axis of the drawing (center of the man's chest), and the horizontal arm beneath, results in an angle of just over 26 degrees, identical to the angle of the inner passageways and Grand Gallery of the Great Pyramid.
Additionally, the navel-pyramid apex is not just a zone associated with a "mystical" center, it is also the exact geometric center of the whole image, as can be seen. And, another GP drawn from the top down, results in both pyramids meeting in the center of the outer circle, with the capstones overlapping at the navel region.
Thus, we can see from these observations that the seven-pointed star, the Great Pyramid (and its absent "capstone"), and the human form all bear close geometric relationships to each other.
In the above analysis a geometric relationship between the GP, Earth, and Mars, and the human canon of proportions is suggested, and the diagram below offers more insights in this direction.
http://www.geocities.com/annubis33/TheDesignofExistence/dmgpvm.jpg
As can be seen, superimposed over Leonardo's Man are two forms. At the base is once again the 2-d version of the Great Pyramid, with its 51.8 degree slope, and its apex resting on the man's navel.
The second 5-sided form is the 2-d form of the structure at the Cydonia region of Mars that has come to be known as the D&M Pyramid. Its alignment with Leonardo's Man is shown in Richard Hoagland's "Monuments of Mars". What has not been demonstrated before, however, is how this 5-pointed structure, when superimposed over the canon of human proportions, shares a central point (the navel) with a triangle aligned with the base of the square circumscribing the man that yields a 51.8 degree apothem slope -- that of the Great Pyramid of Giza.
Conclusion: This demonstrates without question that there is a definite and simple geometric relationship between the 5-sided Mars D&M Pyramid as defined by Erol Torun and Hoagland, the 4-sided Great Pyramid, and the canon of human proportions.
All this is made possible by the relationship between the circle and the square, which Leonardo's canon is based on. These two simple geometric forms have long held powerful archetypal symbolism, with the circle being most often connected to "spirit" (or consciousness) and the square to "matter" (or the world). As suggested at the top of this webpage, it is my contention that the key element of the ancient mystery schools and their surviving physical legacies such as the Giza monuments and Stonehenge, is the relationship between consciousness and matter, or more simply, the subject and the object. The simple geometric relationship demonstrated in the above diagram is yet another signpost suggesting that the purpose and function of human consciousness is to bridge these two domains. I would further suggest that the objects at Cydonia and Giza are markers pointing to this "bridge" -- a bridge known and recognized in most wisdom traditions as the universal "life force" and most often associated with the navel or center region of the human form. This "life force" (known variously as "Ruach", "Prana", "Chi", "Holy Spirit", etc) is the final connecting principal in existence and perhaps is the key piece in any Unified Field Theory that theoretical physics has been reaching for in recent times, such as in Superstring theory, for example.
The Geometry of the Calstone Wellington, Wiltshire, Crop Circle, in relation the the Giza plateau
In the domain of Sacred Geometry and encoded numeric ratios, perhaps no better modern phenomena expresses these than "crop circles", mysterious formations seemingly appearing spontaneously in wheatfields and often bearing spectacular geometric formations that contain sophisticated patterns and precise angles.
The top diagram below shows a two dimensional pictogram of a crop circle formation that appeared in Wiltshire, England, on July 30th of 2000. As can be shown, series of simple geometric extrapolations from this formation yields the relative proportions of the 3 Giza pyramids.
<p align=center>http://www.geocities.com/annubis33/TheDesignofExistence/calcrop.jpg</p>
<p align=center>http://www.geocities.com/annubis33/TheDesignofExistence/calstonecrop.jpg</p>
Analysis:
1) The triangular shape of the formation is anchored by the two flanking circles. If two circles of equivalent diameter are placed on the central axis, one on top of the other, a line drawn from the top of the top circle, connecting with the centers of the flanking circles, yields a triangle with an angle of 50.5 degrees, +/- 1/2 degree (angle "E" in diagram). The angle of the 3rd Giza pyramid is 50.7 degrees.
2)A square drawn between the two flanking circles, and the lower central point and the top of the top circle, results in the square defined by line "C" in the diagram. This square, in proportion to a square drawn around one of the flanking circles, (line "A" in the diagram), results in a ratio of approximately 1:2.25. At Giza, the 3rd Pyramid is in ratio to the Great Pyramid in base size at about 1:2.22. These two ratios are within 1.5% of each other.
3)A square constructed from the center of the left flanking circle to the outer rim of the two central circles (line "B" in diagram) yields a square with a perimeter 94% of square "C". The ratio is 1:1.06. The 2nd Pyramid of Giza is in ratio to the Great Pyramid, in perimeter, at 1:1.07.
4)A line drawn connecting the center of the left flanking circle with the top point of square "B" results in a triangle with an angle of 53 degrees degrees, +/- 1/2 degree (angle "F" in diagram). The angle of the 2nd Pyramid of Giza is 53.1 degrees.
5)A line connecting from the top of the original triangle just above the small central square in the crop formation, to the base of the flanking circles, results in a triangle with an angle of 52 degrees, +/- 1/2 degree (angle "D" in diagram). The angle of the Great Pyramid is 51.8 degrees.
Thus, in summary, squares "C", "B", and "A" are in size ratio to Giza's Great Pyramid, 2nd Pyramid, and 3rd Pyramid respectively. Further, angles "D", "F", and "E" correspond in degrees to the Great Pyramid, 2nd Pyramid, and 3rd Pyramid respectively.
Ratio summary:
Square "C" = 1. Great Pyramid = 1.
Square "B" = .94. Second Pyramid = .94.
Square "A" = .44. Third Pyramid = .46.
Angle "D" = 52 degrees. Great Pyramid angle = 51.8 degrees.
Angle "F" = 53 degrees. Second Pyramid angle = 53.1 degrees.
Angle "E" = 50.5 degrees. Third Pyramid angle = 50.7 degrees.
What can such encoded information ultimately mean, and what good can it do us to "decipher" it?
Well, from the point of view of sacred geometry, certain angles and values have great redundancy and often come to aquire profound psychological or spiritual meaning. Their presence in both natural and myserious formations does not necessary imply a manipulating intelligence behind them, or even any separate, localized consciousness at all, but may merely be part of a sort of cosmic background signature of information that is constantly being transmitted. The angles and shapes themselves may be perfectly natural, and frequently pointing to certain pivotal values, such as those found at Giza.
What good it can do knowing them is a more subtle point, but there is no question that the higher sensitivities that are part of what Gurdjieff termed the "Higher Emotional Body" are stimulated by certain mathematical harmonies that express in art, architecture, and music. This is all part of the "Divine Proportion" that when attuned to, fine-tunes one's sensitivity and awareness of the innate wonder and perfection of existence, and all its interrelating parts -- seemingly independent, yet only part of one functioning Whole.
The Eye in the Triangle: Mona Lisa, the American dollar, and the missing Great Pyramid capstone
The proportions of sacred geometry can be found in numerous works of ancient and classic architecture -- as well as famous art. We have already touched on Leonardo da Vinci's "Vitruvian Man" as a case in point. There is no question that it can be seen from Leonardo's records and sketchs that he was conversant in sacred geometry, and used much of it not just in his art but in his scientific designs as well.
A brief further example of his encoded designs is shown below, in this geometric analysis of the Mona Lisa, probably the most beloved and well known portrait painting in the world since its creation by Leonardo's hand 500 years ago.
All-powerful in this painting are the eyes, of which more has been written about by art historians, and commented on by lay folk visiting the Louvre in France, perhaps than any other feature in the history of Western art. But what is less realized are the geometric mechanics behind the mesmerizing power of these eyes. Below are some observations.
http://www.geocities.com/annubis33/TheDesignofExistence/mona2.jpg
1) It is the left eye that is perfectly centered -- not both eyes. Two lines drawn from the upper corners of the painting, inclined at 33 degrees (the Masonic number of final initiation and completion), meet exactly at the left eye.
2) The painting is flanked by two pillars framing the background to Mona Lisa. A triangle constructed from the inner points of the two pillars, with a base across the horizontal pillar behind Mona Lisa, with a Great Pyramid angle of 51.8 degrees, ends in an apex just above the central left eye. Comparatively speaking, Mona Lisa's left eye rests in the position of the missing Great Pyramid capstone. Also, lengths "A", "B", and "C" are in near phi ratio to each other (the Golden mean proportion that is basic to the human form and the 5-point symmetry).
3) A triangle constructed from the base of the painting, joining with the apex of the 51.8 degree triangle, results in an angle of approximately 70 degrees. This is basically the same angle of the narrow pyramid drawn on the American dollar bill. Further, the left Mona Lisa eye corresponds exactly to the positioning of the eye on the raised capstone of the dollar bill -- when seen in the small triangle formed by a horizontal base aligned with her chin (see inset).
http://www.geocities.com/annubis33/TheDesignofExistence/amerdollar.jpg
None of this is to suggest that Leonardo was working with deliberation here, or that his Mona Lisa is somehow the source of the Eye in the Triangle occult symbolism and its eventual inclusion on the American currency. But what it does strongly suggest is that the sacred geometry he was working with is directly connected to certain high powered proportions and forms, many of which lie at the basis of the teachings transmitted by the ancient Mystery Schools. What is being suggested here is that the "Left Eye of Horus", the left eye of Mona Lisa, the Great Pyramid apothem angle of 51.8 degrees, its missing capstone, and the Eye in the Triangle of the American currency all have a common origin.