One of the sacred meanings associated with the symbolism of the triangle is the fiery aspiration of the human soul towards higher unity. Psychologically it represents the urge to escape from duality or the extension represented by the base of the triangle, a movement towards an origin or an irradiating point at the apex. The triangle is the geometric image of the ternary and the myriad concepts of an archetypal trinity which crowns the mystical endeavour. Nicholas of Cusa designated the upthrust triangle as a symbol of fire, the downward pointing a symbol of water, the upward pointing truncated triangle a symbol of air and its opposite form, the symbol of earth. If one sees the elements of air and earth as alchemically subordinate to transformations revealing something of the essential nature of fire and water, then one might be able to fathom the meaning of the truncated triangles. The meanings of the two complete triangles are subtly intimated in several traditions. The upward pointing triangle is solar. It is the symbol of fire and the masculine principle in nature which is linked up with love, truth and wisdom. The downward pointing triangle is lunar and feminine. It is the emblem of the shakti or consort, and also of the maternal. Together, the two are androgynous and signify evolution and involution, as suggested by the Hermetic axiom "As above, so below."
In Hindu symbology the double triangle is the sign of Vishnu as the god of the moist principle, Narayana ('moving principle in water'), joined together with the fire of Shiva, who dances with the triple flame in his hand. These interlocked triangles produce the septenary (with the point in the centre) and the triad with ten points representing the Pythagorean decad. Pythagoras also drew attention to the hexagonal nature of the double triangle and its allusion to the six limbs of the Microprosopus. He made the number six sacred to Venus because "the union of the two sexes, and the spagyrisation of matter by triads are necessary to develop the generative force, that prolific virtue and tendency to reproduction which is inherent in all bodies". This corresponds with the description in The Secret Doctrine of the initial emergence of the triangle and of how the two opposites in cosmic nature (fire and water, heat and cold) began their metrographical manifestations, one through a trimetric mode, the other through a hexagonal system.
The triangle is a universal symbol signifying a wide variety of trinities, triads and tetrads. The Tetrad of the Kabbalah or the Pythagorean School was considered a sacred number because it emanated from the one, the first manifested Unit . . . "or rather the three in one". Though manifest, it was considered to be ever impersonal, sexless and incomprehensible to the finite mind. Triads are distinguished from tetrads and trinities in that they represent three distinct entities, such as fire-light-ether, spoken of by Zoroaster. The Trinity is a confusing label, as it is used to depict such personalized entities as Father, Son and Holy Ghost in the Christian system, as well as the Egyptian Osiris-Isis-Horus and the Graeco-Roman Zeus-Poseidon-Hades. Brahmā-Vishnu-Shiva of the Hindu Trimurti are shown in the Mahabharata as three different states of Prajapati, who is Brahmā, the progenitor and synthesis of all creative powers. This is not to be taken as a simple identification of the progenitor with the apex of the triangle. Maha Vishnu is said to have sprung from Brahman, the Absolute Unborn, and to have emanated Brahmā as the crucial factor in the process of world creation. Shiva, associated with the destruction of form, is also the lord of regeneration, whose nature draws the mind beyond the limits of form and cycle to an absolute motionless point. Each aspect of the Trimurti could be placed at the apex of an equilateral triangle, which is doubtless the basis of rival sectarian claims within the Hindu tradition. However, in contemplating the Trimurti, the emphasis is so strongly focussed upon an eternal cycle of all three conditions that the symbol seems to spin within the mind of the thinker and the triangle is drawn back within the vortex of a circle.
Turning to Western theology, man is said to be a Christian when he believes that God lives in Three Persons. Leaving aside the sectarian claim that the mystery of the Trinity was revealed only by Jesus, there is the theological concern with the protracted debate of the third century on the Trinity, which eventually led to the schism between the Roman Catholic and the Eastern Orthodox Churches. Basic to Christian dogma is the belief that of the Three Persons, the First is God, the source of all, the Second is the Son, the agent of grace, and the Third is the Holy Spirit, who gives life to our souls. During the great debate the issue at stake was whether God is one divine substance which had three parts. This is reminiscent of the idea of the Trimurti as signifying three states of Prajapati or of the Pythagorean concept of the Tetrad, but with essential differences. In the Trimurti there is no pretence to absolute Deity nor are the three gods ranked in a hierarchy, while in the case of the Tetrad the notion of the Three-in-One is completely impersonal and abstract.
The Christian Trinity in the Gospel According to St. John suggests that Jesus, who does the work of the Father, is equally important yet dependent upon the Father who sent him. Thus the Son is to the Father as the Holy Spirit is to the Son. Theologians have proceeded to claim that the Spirit is the soul of the church and sent to bear witness for Jesus. This would seem to complete a triangular baseline which is firmly cemented to the world's domain. With such a constriction placed upon the individual realization of an unmanifest source of proliferating triads, it is hardly surprising that Christians have tended in general to become caught in an endless series of philosophical dichotomies which propel the seeker back and forth along a line between two points. There is a persisting difficulty in encompassing in one inclusive line of thought the entire triangular formation. In practical affairs, however, the Trinity is quite logically represented in both the Greek and Latin marriage ceremony where the priest, at the apex of a triangle, presides over the bride and the groom who stand at its lower vertices and whose union is cemented by a line of witnesses standing along its baseline. Through its ritual, Christianity reflects something of the Pythagorean notion of the spagyrisation of matter by triads, while in doctrine there is a great metaphysical gap between its concepts and those of ancient Greek and Eastern thought. With characteristic decisiveness, H.P. Blavatsky stated that it was because it rejected the Pythagorean monad and its cognate geometrical figures that Christian theology evolved "its self-created human and personal God, the monstrous Head from whence flow in two streams the dogmas of Salvation and Damnation". She proceeded to state that "those unable to seize the difference between . . . the ever hidden and the revealed Logos . . . ought never to meddle in philosophy".
In Pythagorean thought the first point is the monad from which essence all proceeds. Beneath this is the duad of two points creating a line. It is a line which separates the Unmanifest from that which will manifest and which is symbolized in the three points underlying the duad. These three points represent the surface or superficies and they, in turn, are supported by the four points that mark the bottom line of the Pythagorean decad and the manifestation of solids. It is suggested that the three sides of the triangular decad are the barriers of noumenal matter that separate it from the world of thought. Hence we may imagine a point momentarily emerging in a great darkness (or light) from which, even as it withdraws, angled boundaries emerge marking the two, the three and the four. Pure subjectivity has been divided and then made capable of multiplication and, finally, solidification.
As the triangle is the first of all rectilineal figures, it was considered by Pythagoras to be the creator of all sublunary things. Plato also taught that the triangle is the first plane figure and that all "surface is composed of triangles". To put this in the language of the Timaeus, in the three-dimensional realm there is depth. All depth must be bound by surface and every surface that is rectilinear is composed of triangles. Even a curved surface can be said to be composed of triangles, their magnitude decreasing in size in proportion to the degree of curvature. Plato identified two triangles by dividing a perfect square and an equilateral triangle. From the first he obtained an isosceles triangle, which has two equal sides, and from the second a scalene triangle, whose sides are all unequal and of which type there can be an endless number. These triangles are like keys of evolution because all four Platonic solids can transform into one another by being broken down into their elementary triangles and regrouped. Thus the pentagonal faces of the dodecahedron, which is the form whose combined geometric superficies represent the manifest universe, are constructed by means of an isosceles triangle having each of its base angles double that of the vertical angle. These twelve pentagons can be broken down into triangles which themselves can be divided and reassembled into other figures. That which seems rest in nature is only the change of one form into another and the change of substance goes hand in hand with that of form.
Of the primary bodies which Plato attributes to the work of the Demiurge, the tetrahedron, octahedron and icosahedron are composed of equilateral triangles, while the cube involves six square faces made up of twelve isosceles triangles. All of these are divisible and capable of infinite combinations but always within the larger framework archetypally emanated along the equilateral lines of the decad. Ten is the perfect and sacred number of the decad because it applies to the totality of creators synthesized in the one Protogonos-Purusha who, through sacrificial manifestation, is splintered into infinite numbers of triangular fragments. He is symbolized in the great bird Kala Hansa, which is the vehicle of Time's cycle and which is constructed by hundreds of triangular bricks in the sacrificial altar of the ancient Vedic fire ritual. The power of the sacrificial ritual is focussed in all the strength within the form of the whole and perfect Purusha, and when the bricks are destroyed and scattered at the end of the ritual, the power of archetypal sacrifice is once again released into the world. In the same way, the perfect ten of the decad manifests and undergoes a separation out into form. But the upper triangle remains always in the invisible and metaphysical world, while the three points of the decad become the first cause of centre, space and circumference found in all planes and solids. Thus the second triangle hovers over the square and potential cube of the three-dimensional world, forming the septenate division of manifest life overbrooded by the Seven Sephiroth, who themselves are Rays emitted by the Second Logos.
The highest aspect of the decad has to do with the circle and the point over which Pythagoras threw a veil and, instead, laid the origin of differentiated cosmic matter at the base of the first triangle. Sacred teachings have revealed "to such men as Heaven favours" the mystery of how the point appears in the circle and emanates the first three points and connects them with lines, thus forming the first noumenal basis of the second triangle in the manifest world. It is only from this second triangle that the creator god emerges as a third-level emanation. Through the unification of the lower vertices symbolizing the dual masculine and feminine potencies, the Logoic 'Creator' generates his essence in a vertical line which cuts through the baseline of the universal plane of potential manifestation. In effect, this line cuts through a paradigmatic equilateral triangle to form two scalene triangles, which Plato showed to be subsequently so vital in the progressive unfolding of the universe. With the dropping of this line, the triangle became the Tetraktis, parent of the perfect square and the six-faced cube on earth. It is thus that the Macroprosopus became the Microprosopus, according to the Pythagoreans, and because of the connection between the manifest and divine worlds provided by the vertical Logoic projection, they placed their most sacred vow upon the Tetraktis as the highest Self. Owing to Plato's awareness of the sacredness and potency of such divine geometric expression, he was determined to preserve its investigation and understanding on an abstract level of thought. According to Plutarch, "When Endoxus and Archytas applied geometry to mechanics in order to demonstrate proofs to the satisfaction of the senses, Plato was enormously indignant. He said it was a mere corruption of and annihilation of the one good of geometry, which was being made to shamefully turn its back upon the unembodied objects of pure intellect to recur to sensation and to ask help from matter."
The sum of the angles of any triangle is one hundred eighty degrees, which is one-half of a full circle. Imagining that the half-circle represents the phenomenal world which lies beneath the noumenal half of the circle, one obtains a clue as to why measurement of physical magnitudes proceeds upon a trigonometric basis. There is a dependable similarity in that if all respective angles are equal, the corresponding sides will be proportional. This is the basis of all scale drawings and models made by triangulation. If a side of one triangle is accurately represented on a chosen scale, the map will be correct throughout if the triangles are drawn with accurate angles. In units and subunits, then, they must add up to a whole that corresponds to a multiple of one hundred eighty degrees, or the half-circle.
The Pythagorean theorem is critical to the understanding of the triangular unfoldment and the measurement of the universe. In modern mathematics it has come to be used as a mere definition but it can be shown to have been repeatedly used as a guiding principle in the development of new branches of mathematics. One writer on the subject referred to this as "the life cycle of the Pythagorean theorem". In general, everything in physics involves the use of vectors to describe forces, velocities, directions, etc., and the value of all vectors is established by the use of the Pythagorean theorem. Vectors denote some sort of magnitude. In geometry it is the magnitude of distance from point A to B, while the entire subject of trigonometry is based upon right triangles. It is a science of Vector spaces' which, though a term used consistently on an abstract level, is based upon a breaking down of variables of force, direction, etc., through the use of the Pythagorean theorem.
Fundamental to all this is the fact that in the triangle one is dealing with a magnitude having to do with the relationship of the sides to the hypotenuse. One is trying to add one magnitude to another magnitude to find a third magnitude, and this can be reversed through the use of the theorem so as to break things down in order to analyse them. This always gets us back to components related to vector space which are, by analogy, the sides of triangles. These components do not describe what appears to be going on, but they are the means of breaking down phenomena so that we can actually understand what is going on. The Pythagorean relationship is intimately linked up with the whole world of vector analysis which involves time, light speed, etc., leading eventually to E = mc2, the theory of relativity. In the mathematics to determine this the keys are vector calculus and the postulate that the velocity of light is a constant (determined through empirical observation). They demonstrate the relationship between energy, mass and the speed of light and they illustrate how "the life cycle of the Pythagorean theorem" has progressed through a focus on primitive right angles to Einsteinian physics.
When in one's mind one attempts to abstract back to a primordial beginning, one must 'map' back the concept of magnitudes (all that can be potentially deduced in the external world) to an archetype that synthesizes all these possible magnitudes. Manifestation necessarily involves two lines emanating from a point, establishing separation and duality. Why are the lines pointing down at an angle and not extending out from the point in opposite directions? Intuitively one perceives that if they did extend sideways they would, in some sense, neutralize one another. They would not be capable of establishing the dual tension necessary to ensure the development and generation of higher, recapitulating creative units. This capability freely occurs within an orthogonal activity matrix where both legs neither tend too far apart towards the horizontal cardinal points, nor do they fall together. Therefore, the first generative cosmic triangle should be an orthogonal or right triangle. Following this, the right triangle can be seen as fundamental to emanation (generation and process) and to the 'mechanism' of figuring out anything in subsequent manifestation . . . involving breaking down elements and moving back towards a more abstract appreciation of what is going on.
The equilateral triangle, on the other hand, seems to have to do with the very structure of reality. It is a symbol of completion. This can be demonstrated in the Pythagorean decad, as well as the fact that if you take the radius of any circle, it becomes the basis of six equilateral triangles forming a hexagon within that circle. The basic formation of the world is suggested by such a hexagon, and one is reminded of Fohat guiding the triangular 'Sparks' and placing them in the six directions. This archetype is reflected faithfully in the creation myths of many people, like that of the Wintu Indians, who passed down through generations the sacred story of a triad which was headed by Olelbis in the Heaven World of Olepanti. They told of the two old women who completed the triad and who helped Olelbis build a six-pillared lodge which grew until it encompassed the world and would last until the end of time.
The Pythagorean ideal triangle has been described as "an imaginary figure constructed of three mathematical lines" which symbolizes the subjective spheres, the line enclosing a mathematical space which is equal to "nothing enclosing nothing". With that promising clue, let us consider the question of the Protyle, which scientists seek and which occultism locates at the basic line of this ideal triangle. To do this is to attempt a closer examination of the mysterious border drawn between the noumenal and phenomenal worlds. As suggested earlier, the point that appears in the circle and emanates the first three points forms the first noumenal basis of the second triangle of the manifest world. After this has occurred, however, the point 'retires into' and merges with the circle. By omitting this point emerging momentarily in the circle, Aristotle disregarded the true metaphysical apex of the unmanifest triangle and focussed instead upon a limited triad characterized by line, surface and body. To this degradation of Platonic thought can be traced the subsequent problems regarding concretizations and personalizations of the Christian Trinity. The true Protyle can only be found beyond the baseline of the Pythagorean triangle. Below it is Vishvanara, the objective world; above it are the three principles into which Mulaprakriti is differentiated . . . the three aspects of the Pythagorean triangle.
The Dialogues of Plato are replete with distinctions regarding the realm of the noumenon and that of phenomena. The Philebus includes an examination of the concept of the Three-in-One which is discussed together with the idea of a trinity of Divine Hypostases in the Parmenides, wherein a carefully developed dialogue investigates the metaphysical process whereby Unity, partaking of existence, becomes the many. Plato established a strong distinction between νοητά and αίσθητά (noeta and aistheta), the 'Intelligibles' and the 'sensibles'. The νοητά are single in their several kinds and are considered to be only real existences and objects of knowledge. The αίσθητά are the many and are therefore the foundations of opinion only. The νοητά include spiritual substances and abstract qualities as opposed to things which merely manifested these qualities. They are, therefore, a direct reflection of the essential nature of the Final Cause, Το 'Αγάθόν (the Agathon), or what Plato designated as the Good. In the Philebus one learns that νοϋς (nous) or mind "is cognate with the Cause of all things". Therefore, the point which momentarily appears and then retires may be identified with Το 'Αγάθόν, which projects a Logoic Ray that acts as the First Cause, the apex of a triangle which includes νοϋς and ψυχη (psyche) or soul.
Plato taught that νοϋς was γενούστης (genoitstes) or 'cognate' and 'generated with' the First Cause. Another term used to describe νοϋς is συγγενης (sygenes), which means 'congenital' or 'born with' as kindred. ψυχη or the soul, represented by the opposite vertice of the triangle, is the creative substance-power which is capable of reflecting the order and harmony of νοϋς. When they become merged together, they are εννους ψυχη (enous-psyche), the intelligent or rational soul. Plato taught that this was true at the microcosmic as well as macrocosmic level and, like Pythagoras, perceived the importance of bringing into balance these masculine and feminine aspects of the androgynous Logoic apex. One is reminded of the whimsical contribution of one of the participants in the Symposium who spoke of man's original spherical and androgynous state and went on to caricature the painful condition of the two severed halves who, split from one another, had fallen into the separated masculine and feminine states and wandered about endlessly hoping to locate their 'other half. This condition, though amusingly described, is poignantly common to the human race. Plato's stress upon the identification of άγάπη (selfless love) with the unification of νοϋς and ψυχη lays the rational basis for the strengthening of a stronger connection with the higher Logoic Triad which alone can return the wanderer to a state of original wholeness.
The Logos in Platonic teaching is thus the superior and apprehending spirit that unites νοϋς to ψυχη, which union engenders the comprehending Demiurgos who is the creator of our world. Thus the archetypal pattern and the image are distinguished. The ίδεα (idea) and ειδος (eidos or 'specific form') compose the paradigm which is, then, like light and shadow, repeatedly impressed upon substance in a downward dialectic involving generational triads and culminating in the degrees in which intelligence manifests in the physical forms of the world. That dialectic process itself can be described as triangular, a progressive projection from the apex down through the merging (formerly separated) vertices of unfolded triangles. This would involve a continual breakdown and reassembling of triangular forms analogous to the process whereby the Platonic solids are capable of transmutation into one another. Thus out of the potential latency in the One Reality the supraconscious thought of the Logos infuses itself into the Primordial Substance to produce the Tetraktis or Demiurgos, who extends forth as a line capable of generating subsequent triads which become the basis for all the geometric forms of which Nature is composed. From the level of molecular structure, to plane contours, to the shapes and patterns of human thought, this geometry pertains. Like molecular configurations and solid forms, the patterns of human thought can also be reduced to triangles. They can be broken down and measured with a metaphysical trigonometry in order to work back towards a clear synthesis of νοϋς and ψυχη, the intelligent soul.
As Philo Judaeus put it, those who are not initiated into the great mysteries argue a posteriori. "They ascend to the summit of the Intelligibles and there contemplate Images of the power and goodness or the First Cause impressed upon his works." The perfect Initiate rising above these works receives a manifest representation of the Logos and so derives from within himself a conception of the Logos as well as his shadow, which is his own Self and this world. This is possible only because in the process of triadic manifestation, the living spiritual fire of the Logos was carried forth in the possession of the Triangles, the Kumaras or Agnishwatha Pitris. Incarnating in man, they became the pentagon which is the symbol of the human mind, whose concentrated focus can engender a heat capable of inspiring an upward dialectical process. When this synthesizing process begins, thought can move beyond one set of dichotomies after another, continually transcending the pairs of opposites and establishing an ever higher apex of focus in consciousness. As one does this, one is progressively breaking down myriad formations of thought and karmic action into basic triangular constituencies which can be examined, synthesized and transcended. Gradually, in this way, a line of pure Buddhi-Manasic perception is established. As these two triadic aspects fuse into perfect union with one another, the sacred invisible line of the Tetraktis, which long ago radiated downwards from a loftier apex into the Trinity of the higher Self, now points the way to the peak of enlightenment, the mountain top upon which the Mahayogin sits in a perfectly triangular position meditating ceaselessly upon the One. As above, so below.