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
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
The triangle is a universal symbol signifying a wide variety of trinities, triads and tetrads. The
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
The Christian Trinity in the
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
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
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
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
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
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
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
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
The Dialogues of Plato are replete with distinctions regarding the realm of the noumenon and that of phenomena. The
Plato taught that
The
As Philo Judaeus put it, those who are not initiated into the great mysteries argue |