THE CELESTIAL TETRAHEDRON
For the Quabbalists, the AIN-SOPH is the absolute, Supreme Deity, the one eternal state of being, and symbolized by the circle, within this circle all worlds are manifested, thus creating all this is inside and all that is outside.
In the process of creation, the manifest life of the AIN-SOPH proceeds from the circumference to the center, and establishes a point. The two dimensional symbol is now shown as plat no. 1 .
With the circle as the first limitation defining the limits of space and the second limitation as the point, a universal germ, AIN-SOPH no longer shines through space, but rather upon space from an established first point.
The ZOHAR further establishes 40 rates of vibrations or subdivisions of the manifested AIN-SOPH, represented symbolically as concentric circles within the primary circle of the limits of space.
For informational purposes, the 40 circles of creation from the AIN-SOPH, are divided into 4 great circles of world chains, consisting of ten subdivisions each. The four world chains (A, B ,C, D) are called:
A l- l0, Atziluth or Boundless world of Divine Names;
B l- l0, Briah, or Archangelic world of Creations;
C l- l0, Yetzirah or Hierarchal world of Formations;
D l- l0, Assiah or Elemental world of Substances.
Now, these four world chains between the great circle and the central point do not concern us for the purpose of the celestial tetrahedron. What is important is that the ZOHAR pre-establishes two additional circles between the great circle and the first subdivision called Atziluth. Thus, the internal, unmanifested nature of AIN-SOPH is divided into three parts represented by the concentric circle spaces respectively between the outer circle (x) as follows:
X to X-l, AIN, the vacuum of pure spirit;
X-l to X-2, AIN-SOPH, the limitless and Boundless;
X-2 to X-3, AIN-SOPH-AUR, the limitless Light.
Thus, from the outer circle (x) to the central point, are three circles representing the three in one unity of the unmanifested AIN-SOPH. From this unmanifested trinity comes the manifested worlds represented by 40 great subdivisions. The ZOHAR symbol for the celestial tetrahedron is now as shown with each successive circle being a limit on the preceding.
As stated by Manley P. Hall,
"It should be borne in mind that in the beginning the Supreme Substance, AIN, alone permeated the area of the circle; the inner rings had not yet come into manifestation. As the Divine Essence concentrated Itself, the rings X 2 and X 3 became apprehensible, for AIN SOPH is a limitation of AIN, and AIN SOPH AUR, or Light, is a still greater limitation. Thus the nature of the Supreme One is considered to be threefold, and from this threefold nature the powers and elements of creation were reflected into the Abyss left by motion of AIN SOPH towards the center of Itself. The continual motion of AIN SOPH towards the center of Itself resulted in the establishment of the dot in the circle. The dot was called God, as being the supreme individualization of the Universal Essence. The name of this point is I AM, called by the Hebrews Eheieh and it controls and contains all other emanations."
Further the central point represents the crown or Kether of the Sephirothic Tree. There are four trees, one in each of the four world chains. The central point controls and contains all the 39 circles within the four world chains which it encloses. Thus, the ZOHAR establishes the concept of the three forces of the trinity outside the central dot as unmanifested universe and the central dot as a separate world unto itself consisting of 40 subdivisions divided over 4 world chains of manifested universe.
This cosmic seal set forth in the ZOHAR is the esoteric wisdom of one of the three books of the Quabbala. Despite the ravings of H.P. Blavatsky against the Hebrew alphabet as essentially phallic and the medieval books of the Quabbala as altered to suit Christianity, she unequivocally accepts the ZOHAR as representing the Hindu cosmology described in The Secret Doctrine as follows:
"The entire scheme is in the Chaldean Book of Numbers, and even in the ZOHAR, if one only understood the meaning of the apocalyptic hints. First comes Ain-Soph, the "Concealed of the Concealed," then the Point, Sephirah and the later Sephiroth; then the Atzilatic World, a World of Emanations that gives birth to three other worlds: (the first, the Briatic World or World of Creation), called the Throne, the abode of pure Spirits; the second, the World of Formation, or Yetziratic, the habitat of the Angels who sent forth the Third, or World of Action, the Asiatic World, which is the Earth or our World; and yet it is said of it that this world, also called Olam ha-Kliphoth, containing the (six other) Spheres,Galgallim), and matter, is the residence of the "Prince of Darkness." This is as clearly stated as can be; for Metatron, the Angel of the second or Briatic World, means Messenger, Angel, called the great Teacher; and under him are the Angels of the Third World, or Yetziratic, whose ten and seven classes are the Sephiroth,of whom it is said that "they inhabit and vivify this world as Essential Entities and Intelligences, whose correlatives and contraries inhabit the third or Asiatic World." These "Contraries" are called the "Shells," (Kliphoth), or demons, who inhabit the seven habitations called Sheva Hekhaloth, which are simply the seven zones of our globe. Their prince is called in the Kabala Sammael, the Angel of Death, who is also the seducing serpent Satan; but that Satan is also Lucifer, the bright angel of Light, the Light-and Life-bringer, the "Soul" alienated for the Holy Ones, the other angels, and for a period, anticipating the time when they would have descended on Earth to incarnate in their turn."
In, conclusion, the l4th century ZOHAR repeating the wisdom of the ancient Chaldean Book of Numbers, clearly defines the unmanifested universe of the celestial tetrahedron as a trinity of three concentric circles localizing a central dot, which dot in turn, represents the manifested universe and its subdivisions.
2. CIRCLE-TRIANGLE
A geometrical evolution has occurred from the early representation of the unmanifested world as a symbol of three circles and a center point. For example, G.I. Gurdjieff in explaining his law of reciprocal maintenance sets forth a series of twelve essence groups called the Trogoantoegocrat. At the top of the ladder he represents the absolute or unmanifested deity as a circle, with an inscribed triangle and a central dot. Likewise, the Quabbalists represent the first three Sephiroth, Kether, Binah and Hochma as a triangle, represented by the letter 'shin', and circumscribed by a circle. Also, all the major books of theosophy represents the triple aspects of Deity with a triangle, symbolic of the higher unity of Manas-Buddha-Atman.
In fact, every exoteric religion has evolved the symbol of the three aspects of Deity into a circle, inscribed with a triangle and central dot, replacing the three circles of the Zohar with a central triangle as follows:
3. GOD AS TRINITY.
The reason for the evolution of the three circles into the triangle is that upon recognition of the three aspects of God - Absolute- Ain-Soph etc., the triangle is more geometrically representative. Thus, the outer circle still symbolizes the limits of space in which the deity acts but the deity itself becomes represented by the triangle. The triad or triangle has become a universal exoteric symbol of deity in its three aspects.
Again this took historical time to achieve because the evolution of exoteric religions, also progressed in stages.
The first cultural epoch or first religion represented by the Hindu-Brahman can be thought of as a plurality of
gods in unity as a circle. The second cultural epoch represented by the Persian-Zoroastrian, can be thought of
as the emergence of Duality, as Ahura-Madza and Ahrihman. Not until the third cultural epoch as represented by
the Egyptian-Chaldean, do we see the first emergence of the three aspects of god as a trinity or triangle. Thenceforth,
god can be represented as a trinity in all religions as follows:
| RELIGION | TRINITY | ||
| Hindu | Brahma | Shiva | Vishnu |
| Christian | Father | Son | Holy Spirit |
| Hebrew | Kether | Hochma | Binah |
| Egyptian | Osiris | Isis | Horus |
| Persian | Ahura-Mazda | Vohuman | Asha Vahishta |
| Scandinavian | Thor | Freya | Odin |
| Druid | Taulec | Fan | Mollec |
| Babylonian | Anu | Ea | Bel |
"The first period, the primeval Indian, developed a religion that seems like an inner lighting-up, an inner repetition, in ideas and feelings, of the very first period, when sun and moon were still bound up with the earth. The spirit who, in the first condition of the earth, in the primeval mist, connected himself with all angels, archangels, high gods, and spiritual being, was for Indian consciousness summed up as a single high individuality under the name of Brahm or Brahma. This first post-Atlantean culture recapitulated in the spirit what had happened earlier. It is a repetition of the first epoch of the earth, in its inner aspect.
"Now let us look at the second cultural period. In the principles of light and darkness we have the religious consciousness of the primeval Persian period. The great initiate saw an opposition between two beings, one of the which was personified in the sun and the other in the moon. Ahura Mazdao or Ormuzd, the Light-aura, is the being whom the Persians venerated as the highest god. Ahriman is the evil spirit, the representative of all the beings who belonged to earth-plus-moon. The religion of the Persians is a remembrance of the second epoch of the earth.
"In the third cultural epoch, man had to say to himself, "In me are the forces of the sun and of the moon; I am a son of the sun and a son of the moon. All the forces of the sun and of the moon appear as my father and my mother." Thus we have UNITY in the primeval past as the attitude of the Indian; while the DUALITY that appeared with the separation of the sun is reflected in the religion of the Persians; and in the religious views of the Egyptians, Chaldeans, Assyrians, and Babylonians we find the trinity that appeared in the third cultural epoch. Trinity appears in all religions of the third period, and in Egypt is exemplified in Osiris, Isis and Horus."
Of course, the Hindu and Persian had a trinity in their esoteric cosmology, but these were not revealed by the Vedas and Manes, respectively, until after the establishment of the Egyptian trinity.
4. LAW OF THREE
Now, the triangle represents everything with a threefold nature and as such, its possibilities of representation are infinite. However, we do not want to confuse the triangle as unmanifested deity with the triad as manifested in the created universe. The celestial tetrahedron is the three aspects of deity unmanifested and represented, for now, as a triangle. The manifestations of deity in the created world are also threefold such as the triads of the primary colors, the equilateral triangle in Masonry, triad of sun, moon and earth, etc. However, the latter triads are manifestations of the law of three in physical, created universe of nature and do not concern us in discussion of the celestial tetrahedron. However, an explanation of this fundamental occult law will help later in explaining how deity contains within itself the three forces, universally recognized as the occult trinity.
The medieval Christians were the first to reveal as an exoteric symbol the principal of the law of three. In an effort to explain in an appropriate symbol the Christian doctrine of the triune nature of God as Father, Son and Holy Ghost as separate, yet one unity, many European churches (built by anti-Catholic masons of course) display the following symbol of God as trinity.
Here we see the Father, Son and Holy Ghost separated as three circles but connected by two aspects of two forces called EST and Non-EST. These represent the active and passive forces. In other similar symbols, we see reference to a third force called the neutralizing force. Together, these three forces, active, passive and neutralizing, represent the three aspects of Deity unmanifested. As manifested, we need a fourth resultant combining the three into the matter of physics.
Although the trinity of three forces appear in numerous esoteric occult teachings such as the Salt, Sulphur and Mercury of Alchemy, no clear exoteric explanation is given until the publication of The Secret Doctrine in l888 by H.P. Blavatsky and later In Search of the Miraculous by P.D. Ouspensky in l949. In the latter, G.I. Gurdjieff explains the law as follows:
"Every phenomenon, on whatever scale and in whatever world it may take place, from molecules to cosmic, is the result of the combination or the meeting of three different and opposing forces. One force, or two forces recognized by modern science as positive and negative, alone can never produce a phenomenon. The presence of a third force is always necessary. The first may be called active or positive; the second, passive or negative and the third, neutralizing. The first two forces are obvious to any person as positive or negative, male-female, good-evil, etc. However, the third neutralizing force is difficult of observation.
"Also, all these forces are just names for what in reality appears as all three forces are equally active, passive and neutralizing only in their meeting points and only in relation to one another at a given moment in time. For example, the element carbon is much more active in the organic world than it is in the mineral world as is silicon more active in the formation of the earth's crust than it is in the organic world.
"The relationship of the three forces to the trinity is that the three forces acting together form a trinity which produces a new phenomenon. Thus, the three in one, when manifested form a fourth complete whole as matter and subdivisions, thereof: This concept is the essence of the celestial tetrahedron, the three forces as trinity produce a fourth, manifested as matter, the first subdivision of material, created and physical universe which can be counted, weighed, built upon, lifted and observed. The tetrahedron represents the three forces as trinity of god (spirit) and the fourth as matter.
Further, Gurdjieff explains:
"The four aspects or states of every substance have definite names.
When a substance is the conductor of the first or the active force it is called 'carbon,' and, like the carbon of chemistry, it is designated by the letter C.
When a substance is the conductor of the second or the passive force, it is called 'oxygen,' and, like the oxygen of chemistry, it is designated by the letter O.
When a substance is the conductor of the third or neutralizing force, it is called 'nitrogen,' and like the nitrogen of chemistry, it is designated by the letter N.
When a substance is taken without relation to the force manifesting itself through it, it is called 'hydrogen,' and, like the hydrogen of chemistry, it is designated by the letter H.
The active, the passive, and the neutralizing forces are designated by the figures l, 2, 3. and the substances by the letters C, O, N, and H."
Although the basic concept of the law of three is ancient, the modern notation of C-O-N-H used by Gurdjieff was most likely borrowed from Madame H.P.B. in The Secret Doctrine. She says,
"The numbers 3 and 4 are respectively male and female, spirit and matter and their union is the emblem of life eternal in spirit on its ascending arc and in matter as the ever resurrecting element by procreation and reproduction ... The former (the 3) is invisible (as spirit) and the latter (the 4) is on the plane of objective perception (as matter). This is why all the matter of the universe, when analyzed by science to its ultimates, can be reduced to four elements only C-O-N-H; and why the three primaries, the noumena of the four, or graduated spirit or force have remained a terra incognita."
In conclusion, God as Trinity, represented as an equilateral triangle, for exoteric purposes revealed as the triple aspect of deity, while for esoteric purposes is concealed as the ancient law of three.
5. SEPHIROTH TREE
The principal of three as Spirit and unmanifested cosmos passing into four as matter and manifested cosmos has its prima-facie origin in Western occultism in the SEPHER YETZIRAH book of the quabbalah. The esoteric symbol of the Quabbalah is the Sephiroth Tree of Life (from Kircher). From the AIN SOPH represented, as before, by the O circle beyond and above all number, emanate a succession of ten Sephiroth.
Do the first three Sephiroth, shown as Kether, Binah and Holhmah, represent the trinity as Law of Three? Consulting some Quabbalahistic sources we find: IBN Geberol saying, "The triad is the root of everything"; the Idrah Zootah says; "The Ancient, whose name be blessed! has three heads which are yet one head"; and Issac Myer saying, "The triad of the Ancient Quabbalah is Kether, the Father; Binah, the Holy Spirit or Mother; and Hokhomah, the Word or Son; which three are distinguished from the other Sephiroth and are called the 'upper holy triad' and together form the Intellectual World (A'Tzeel atic world) and govern the head."
This upper holy triad has always been distinguished from the lower triads emanating from the lower seven Sephiroth, because in fact, the Quabbalahists intend this first triad to represent the trinity. From Kether, emanates the opposition, the female and male, the negative and positive principals. Together, the three are one, in unity.
How then does the three as trinity pass into the four as matter in the Quabbala? From the third Sephiroth Hokhomah, emanates the fourth sephiroth known as HESED. The Zohar says that from the breast of Hokhomah, the word, the divine son emanated six principals or Sephiroth called the "Sephiroth of Construction". Issac Myer says these symbolize the dimensions of matter, the six sides of the entire universe, the six days of building (from Genesis), be it an atom or universe, as represented in the three dimensions of Length, Breadth, Depth, and the positive and negative poles of each of these. These six dimensions of the material or manifested cosmos are represented as the six sided cube with a numerical Hebraic script value of 72.
Thus, the trinity as the upper holy triad of the first three sephiroth pass into the fourth sephiroth HESED and its emanations as the constructed or manifested world represented by the dimensions of matter.
Another interesting way we see the three pass into the four as represented in the Zohar is by the great name of the Tetragrammaton represented in the name of God written as Jehovah (YHVH and pronounced as ADONI). The Zohar says, "There are four lights! Three of them are concealed and one is revealed.
The letter of Kether if Y-Yod, of Binah H. Heh and of Hokhmah which is a complete symbol in itself of the name of God as representative of the trinity. However, a fourth letter is always added to the name of God making the complete Tetragrammaton of four letters. The letter H is added at the end, thus YHVH. The reason for this is explained by Issac Myer as the last letter H of this ineffable name being always applied to the six lower and last Sephiroth represented, as above state, in the Fourth Sephiroth of Construction, HESED. The first three sephiroth represented the upper holy triad in the first three letters Y-H-V and the lower Sephiroth of Construction in the fourth letter repeated as H as HESED. Taken together, we have the sacred holy name of Jehovah, the Tetragrammaton, as YHVH expressed as three letters plus one making the four letter name of God.
This mystery of the holy name as representative of the principal of three passing into the four has been aptly commented upon. "If thou art attentive to that most holy name of the Deity (that is the Tetragrammaton or four lettered name YHVH) as found explained in the mysteries of mysteries, thou beholdest, a name of three letters, thou of four symbols, one of them a twin, being written twice, that is, the H-HED. If anybody examines them, he can see that the very name represents both one and three, truly each pointing to the oneness of the substance, so that it shows the trinity of persons."
The first and second letters form YH, the second and third letters form HV and the third and fourth letters VH, joined together they form the one great name YHVH. Thus, the three letters (of the trinity) form a unity of four (the constructed world of matter).
6. TETRAHEDRON AS MATTER
The tetrahedron like, the first circle of the ZOHAR, is the first subdivision to divide the universe into two parts: all of the universe inside the system and all the universe outside its system. Buckminister Fuller has examined the tetrahedron as the sum of the forces of the law of three in several ways.
In solving the relationship of plane geometry and Galileo's parallelogram of forces, he thought of the basic mathematical problem of ships on a collision course on the ocean. One ship was drawn on a plane running at a certain velocity in a given direction. Multiply its weight times its velocity and make a length of a line called a vector. Then likewise, vector the other ship. Then, make two other lines parallel with the first set of two vectorial lines and make a parallelogram with the first two vectors.
Next, make a line diagonal in the parallelogram from the point of collision to its diagonally opposite corner. The external equidistant extension of the diagonal of the parallelogram was called the resultant of forces. From this diagram the two ships after collision are supposed to go in the same somewhere direction for some miles. Of course, this does not really happen after the collision because one ship sinks to the bottom of the ocean.
Now what Fuller recognized was that modern science did not recognize what we said earlier about the neutralizing force. The third force is very difficult for the average person to see. Although, science grasped the idea of the fourth force or resultant, (our hydrogen as the combination of the three forces), it did not grasp or recognize the third force or nitrogen principal. Fuller found that in reality four forces are operating on the ships at sea. Two forces accelerate conically together, rising from the earth, plus gravity, plus the resultant. So when the two ships collide, they really want to rise up against gravity, and they could if they had enough speed, but instead they sink or go a few yards in the direction of the resultant of forces. The pattern of real forces looks very much like a music stand, three vectorial legs spread out with a fourth vertical vector.
Fuller shows the creation of the first subdivision of universe as a tetrahedron in plat no. 1 . The ball on the string is anything. The string however long is the force of gravity as the combined mass attraction exerted upon one by all the stars in universe. It can move in any direction but it cannot get out of or absolutely away from the universe, so the single string is always attached to it.
Turning the mass attractions of the stars into two hemispheres, those to the left and those to the right. We find itself like a ping pong ball suspended in the middle of a one string fiddle with two restraints. It can still move but only on a plane perpendicular to the string, in circles, or cloverleaves, etc.
With three restraints, It moves only as if in the middle of drum head and in a line perpendicular to the plane of the drum head or web triangle. With one restraint, the ball was moving omnidirectionally and multidimensionally. With two restraints, it was moving in a plane and with three restraints, only in a line. Finally, with four restraints, the ball is immobilized and represents a minimum system.
Although restrained from directional motion, the center ball can still create or move as the center dot in the ZOHAR represented 39 subdivisions of influences. The ball can still turbine, rotate, and precess locally in space without altering the geometric position at volume center of the celestial tetrahedron from whose four corners the four vectors of restraint were imposed. It can rotate in place around an axis and that axis can incline at many angles like a gyroscope top without altering its volumetric center.
The tetrahedron is a triangularly faceted polyhedron of four faces. It is unique as the minimum system. All matter of physical universe, as manifested, is in the primary form of a tetrahedron. Modern science recognizes this in organic chemistry, mineralogy and crystallography.
The dot was the monad or number l; The line, the power of 2; the triangle, the power of 3 and the solid tetrahedron, the power of 4. Note, when he established the solid or primary matter as the fourth power, he did not use the cube (the traditional ancient symbol of Earth) as the form, he used the tetrahedron. He knew how the three forces together formed the first solid dimension. I would like to know the results of an investigation of the karmic relationship between Pythagoras and B. Fuller, either as
reincarnation or channel. Gurudas in his first book on Vibrational Healing states that B. Fuller received channeled information (from the music of the spheres, no doubt).
What Euclid was to geometry for the past two millenniums, Fuller will be to the future millenniums.
8. TETRAHEDRON AS PRIMARY ATOM-EVENT
B. Fuller further examines the tetrahedron as an energy event comprising the law of three.
An open ended triangle can be considered as one energy event consisting of our three forces of action, reaction and resultant. Two such tripartite vector spirals or open triangles, one positive and one negative, combine to form the tetrahedron.
One set of three vector open triangle groups corresponds to the proton (with its electron and anti-neutrino) and the other set of three vector triangle groups corresponds to the neutrino (with its positron and neutrino). Each of these groups is identified by nuclear physics as one-half Plank's constant or one-half spin or one-half quantum.
When brought together, these two sets of three vector groups form the six edges of the tetrahedron which is the conceptual unit of one energy quantum or atom.
Thus, Fuller establishes the Law of Three (active, passive and resultant forces) in two sets as constructing the primary atom or element as a tetrahedronal form. (See Plat no. 2 for illustration.)
9. CONCLUSION - HARDLY
Having established: (l) the ancients represented the three aspects of deity as an equilateral triangle; (2) the deity as a triangle represented the unmanifested spirit in its three aspects; (3) the law of three represented the esoteric version of the three aspects of Deity as active, passive and neutralizing forces inherent in all matter; (4) the deity as triangle transforms into the tetrahedron, as three becomes four and spirit becomes matter; (5) spirit and matter as four in tetrahedron is the first subdivision of matter; (6) Fuller established the tetrahedron as the special form of the primary atom event.
Therefore, using the isotropic law of "as above, so below," since in the microcosm, the atom is formed as tetrahedron, so in the macrocosm, the celestial sphere is formed as the celestial tetrahedron. (Thus, the name of this exercise.)
However, this symmetry does not work. We cannot replace the primary First Circle
with primary first triangle. That is why the
tetrahedron should not be thought of as universe of deity but as the First Subdivision of universe as our
familar zohar circle..
The scientific reasons for this are at least twofold. First, although the tetrahedron is the first prime volume in universe with a unique domain, it does not have a nucleus. As Fuller shows, a prime volume is different from a generalized regenerative system. Generalized regenerative systems have a nucleus and prime volumes do not. There are only three prime volumes: tetrahedron, octahedron and icosahedron. Prime volumes are characterized exclusively by external structural stability. Thus as a structure of cosmic hierarchy, the tetrahedron has the structural stability befitting its lofty status, but no center, and we cannot have a representation of deity with no center - this is blasphemy. (Actually, inside a tetrahedron comprised of four tangent spheres, there is no central sphere but there is a hole or curved convex space in the shape of an octahedron. So something could fit inside here).
Second, the tetrahedron alone, consisting of only four vectors of restraint can still move in several directions. In a nuclear domain of four close packed spheres, the tetrahedron system is stable as in Plat no. 3 . However, in a celestial domain of four vectors of restraint defining an inside-outside domain of universe: (l) each of the four corners can be pushed, thus turning the tetrahedron inside out in four directions (concave-convex) or (2) the entire tetrahedron can rotate in a 360 direction (precess locally within its celestial sphere). As shown in Plat no. 4 ,it takes a minimum of twelve vectors of restraint to immobilize the center ball of the tetrahedron , to keep it from motion to turbine, rotate or precess. Each of the four corners of the tetrahedron need three vectors of restraint to lock the four corners in place.
Thus, to have a celestial tetrahedron as permanent and immobile, locked in place, we need to have "four external star foci events" restrained by three sets of vector tensions each.
"The cohering integrity of universe is manifest to us by consideration of the celestial advantage point from which our four central restraints were mounted to prevent local in position twist and torque, each of the four corner tensional restraints will have to be multiplying replaced by three restraints, all springing from three external points at each of the four tetrahedral corners; and each of the three tensions from any one of the four corners must cross the others triangularly and be attached tangentially to the ball at the center." (B. Fuller) This concept is represented in the following diagram as a tensegrity tetrahedron .
Now, our search here has not been to find how to restrain the center of a tetrahedron or restrain the center dot of the Zohar symbol. It has been to define a celestial tetrahedron which we know somehow must exist in fact, but we became mixed up in twelve vectors of restraint as necessary to define a tetrahedron that evolved formerly with four vectors of restraint.
Now, we know that to have a celestial tetrahedron, it must be defined by four external star foci events, located somewhere in our celestial sphere, from which each of the four corners must project three sets each of tension vectors of restraint!!
Now, before we locate and define the four corners of the celestial tetrahedron, we should ask if we still have a circle
circumscribing the triangle or tetrahedron as or if now
after the analysis, the tetrahedron is outside and the circle
inside as .
Fuller also observed and defined a mathematical phenomenon known as a geodesic. It is the most economical relationship between any two events. Just as a line is the shortest distance between two points on a plane, a geodesic line is the shortest distance between any two points on the outside of a sphere. A sphere composed of geodesic lines is called a geodesic sphere and includes all the principals of tensegrity.
Each line on the perimeter of the geodesic sphere defines a great circle. A great circle is mapped when a regular polygon is rotated inside the sphere and the nodes of the polygon trace a circle on the sphere. For example, we know that our celestial tetrahedron with only four vectors of restraint can still rotate in place. If we rotate this tetrahedron, each of the four modes define four great circles on the geodesic sphere. Plat no. 5 shows the four great circles of the tetrahedron inscribed on the sphere, and the six great circles of the octahedron and the 20 circles of the icoshedron. Likewise, for illustration, Plat 6 Plat No. 6 shows the many great circles of another geodesic.
Thus, it we spin the celestial tetrahedron inside the celestial sphere, we can have it both ways. The external shape of the perimeter of universe is neither a circle nor triangle, and we do not have to address the issue of which is first. The celestial sphere is a geodesic sphere defined by the great circles of the inscribed polygon, which at a minimum is the four great circle celestial tetrahedron and a maximum, zillions of great circles tracing the rotating star events of the Milky Way Galaxy! Thus, our familar Zohar triangle prevails as unity.
Incorporated by synergy into our celestial sphere is the concept of tensegrity. Fuller created the word and concept as a contraction of tensional - integrity. As defined by Fuller, "tensegrity defines a structural relationship principal in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system. All structures, from the solar system to the atom, properly understood, are tensegrity structures. Universe is omnitensional integrity."
Thus, space is curved (after Einstein) because the celestial sphere is composed of the domain of the successive great circles. Universe is expanding (after Hubbel) because the tensegrity of the celestial sphere provides the ability to yield increasingly without ultimately breaking or coming apart.
In conclusion, the compound curvature geodesic structure of the celestial sphere employs the tensegrity principal and remember, the four corners of the celestial tetrahedron need to be replaced to form the tensegrity tetrahedron.
11. A MODEL
In the third degree of masonic initiation, a scene is related wherein King Solomon is apprehensive about the safety and whereabouts of his master builder, Hiram Abiff. To search for him, Solomon orders three workers to go each in the direction of the four cardinal points of universe. This allegory has always fascinated me because the three workers in the four directions (3 x 4) resembles the three vectors of restraint at the four corners of the celestial tetrahedron.
This allegory will be the working model to define and locate the four corners of the celestial tetrahedron which need to be replaced by four sets of three vectors of restraint. The "four external star foci events" are the four cardinal points of the world referred to by all ancients. These become fixed points on the celestial tensegrity sphere which radiate three vectors each through space down to the four fixed points (signs) located in the zodiac as Leo, Taurus, Scorpio, and Aquarius.
The center ball of the tensegrity tetrahedron is replaced by the band of the zodiac, now locked in place by the twelve vectors of restraint as was the center ball in the figure on Page l3. The four cardinal points of the "world" need to be located outside the band of the zodiac, so a distinction needs to be made between the four cardinal points and the four fixed signs of the zodiac. Within the band of the zodiac is another sphere or circle of the ZOHAR, in which deity manifests logos in our world of sun, planets, moon, comets and organic life.
Above the planetary spheres reside the domains of the Seraphim and Cherubim. And in the creation of Saturn (Thrones) form the triad of the first round. Rudolf Steiner explains as such in The Spiritual Hierarchies:
"Before a system of Thrones, Cherubim and Seraphim comes about, it must first have been a solar system. This means a sun must have reached the stage when it could unite itself again with its planets; then it can become peripheral. The sun itself becomes a zodiac. What we have come to know as the exalted beings in the zodiac are remnants that have passed over to us from a previous solar system. Seraphim, Cherubim and Thrones represent for us the highest hierarchy among divine beings because they have already accomplished this development as a solar system and have risen to an exalted rank of cosmic sacrificial service. We must picture the godhead as beyond the Seraphim. The (Seraphim) receive the plans for a new cosmic system from the divine trinity from whom they originated. The name Seraphim is always interpreted as referring to beings who receive the loftiest ideas and aims of a cosmic system from the trinity.
"The Cherubim, who belong to the next hierarchy, have the task of elaborating in wisdom the aims and ideas of received from the gods. They are capable of transporting into workable plans what is indicated by the Seraphim.
"The Thrones, the third rank of the hierarchies, have the task of putting into practice the lofty cosmic thoughts, received by the Seraphim from the gods and pondered over by the Cherubim."
The first and second threefold hierarchy is represented by Steiner as concentric circles shown in Plat No. 7.
The purpose of the first and second triad is to compress their particular sphere of influence. Cosmic bodies such as suns and planets do not simply condense, spiritual beings have to bring about the condensation, the Cherubim accomplished this task for the zodiac or animal circle.
Steiner represented this process of evolution in the following diagram Plat No. 8. The inner circle (the center ball in our tetrahedron) represents the globe of the old sun (first round). The archangels sped outward in all directions from the center globe and spread spiritually into the cosmos. The Cherubim where the beings who come to meet the archangels. "For the Cherubim, in fact, drew near, to begin with, from four sides". These four sides were the primary four sets of three vectors restraints emanating from the four corners of the celestial tetrahedron. They approached in shapes that could afterwards be represented as the four winged beasts (Ezekiel's vision) of the Bull, Eagle, Lion and Man.
For Plat No. 8, the four fixed signs of the zodiac are Leo (Lion); Scorpio (Eagle); Taurus (Bull) and Aquarius (Water-man). Notice that along side each primary Cherubim are secondary (later) zodiac signs which represents the four sets of three vectors restraints. As Steiner says:
"I have only spoken about four names of the zodiac. They represent the four principal expressions of the Cherubim, but in reality each of these cherubic beings has to the right and the left of it a kind of follower or companion. Thus we have twelvefold forces belonging to the realm of the Cherubim."
Thus, Steiner recognizes that beyond the Seraphim is the threefold trinity of deity exactly similar to our zohar model of the threefold trinity as triangle or tetrahedron. In addition, he recognizes the primary sphere, of unmanifested deity as trinity and the manifested sphere of the Seraphim as manifested deity. Or put another way, the trinity cannot manifest itself directly in our chain of worlds but can only do so in and through the manifest sphere of the Seraphim. The Seraphim and Cherubim do the work for the deity in our chain of worlds.
Steiner also recognizes the twelvefold nature of the Zodiac as a result of the work of the Cherubim. This fits very neatly with our twelvefold vectors of restraint manifested through the four principal expressions of the zodiac with its eight followers or companions of three zodiac signs.
Finally, the fixed signs of the zodiac represent the four places where the four sets of three vectors attach themselves tangently to the center sphere of the celestial tetrahedron. From
the four cardinal points or "four external star foci events" emanate four sets of three vector restraints which attach tangently to the four fixed signs of the zodiac. Thus, the four cardinal points are located (somewhere) on the surface of the celestial tensegrity geodesic sphere wholly outside the sphere of the zodiac.
As far as the order of successive chains of cosmic creation is concerned, this did not proceed progressively inward from the celestial tetrahedron. As previously stated, before a system of Seraphim, Cherubim and Thrones comes about, it must first have a solar system. The sun itself can become a zodiac, dissolve into the cosmos and become a choir of celestial beings like the Seraphim and Cherubim. For this reason, the celestial tetrahedron did not first begin as such. Through an evolution of suns and planets over four rounds only did the celestial tetrahedron become. As each successive order of spiritual hierarchies completes its assigned task, that order passes on in eternity to higher orders and is likewise replaced by similar orders evolving upwards in the spiritual hierarchy. In each of the four rounds, successive orders of the spiritual hierarchies move up and down the ladder of creation. The celestial path of the retiring Cherubim ascending as our tensional vector strings becomes the four cardinal points on the celestial geodesic sphere. The location of the four cardinal points is still a mystery to me, but I suggest the seven Rishis located in the Great Bear Constellation is one such external foci event.
13. CONCLUSION
In conclusion, our spiritual geometry evolution begins with the ZOHAR symbol of the circle with central point as here.
This circle represented the outer limits of creation (the Milky Way) by the Deity as within this circumscribing circle all worlds are manifested. The central point or monad is the universal germ or seed creating the manifest worlds through the triple aspect of Deity represented by the three inscribed circles as
Later, the three circles of the Zohar are replaced by the Pythagorean triangle, with the point or monad of the tetractys as center. Thus, (our zohar circle) becomes the occult symbol of the circumscribed cosmos of the Milky Way, inscribed by the Pythagorean triangle and the universal seed as monad of the TETRACTYS
The triangle represents the forces of the Law of Three as active, passive and neutralizing, and as the three pass into four, as spirit passes into matter, the triangle becomes the tetrahedron as primary atomic event.
With the introduction of Fuller geodesics, the celestial sphere as circle is replaced with the compound curvature spherical tetrahedron. Although a good symbol, the spherical tetrahedron is not yet perfect because it is not fixed in space, it can still rotate, turbine and twist in place. In order to immobilize the tetrahedron, it is necessary to fix each of the four corners with three vectors each of restraint. This is accomplished by replacing the center point or monad with the twelvefold band of the zodiac. With each of the twelve signs of the zodiac representing a vector of restraint, the twelve respective tangents stretch from each zodiac sign to the edge of the celestial sphere, locking the system is place as a completed spherical tensegrity tetrahedron. Each of the four corners of the spherical tetrahedron are now represented in space by four star foci events whose four primary tangents run their course from the perimeter of the celestial sphere, through the spaces of the Milky Way and attach themselves to the four fixed signs of the zodiac.
The result is the spiritual geometric evolution of: (1) the circle now as a celestial tensegrity sphere enclosing
the Milky Way; (2) the triangle now as a tensegrity tetrahedron with four fixed corners as star foci events and
(3) the central point now as the band of the zodiac with twelve vector tangents extending from the zodiac in twelve
directions through space to the edge of the celestial sphere and attached at the four star foci events, locking
the entire synergetic system in place, as shown in the frontispiece.
