Throughout the writings of HPB and the Tibetan, triangles abound: the Science of Triangles; Upper and lower Triads; the Trinity, triple Logos or Hindu Trimurti; Three Crosses, etc. And always the Law of 1-3-7 confronts us, from Unity to Triplicity to Septenary…to multiplicity. “First came three, or the triangle”, says HPB (The Secret Doctrine II, Page 594), and it is “the vehicle of the Unknown Deity.” (The Secret Doctrine II, Page 582) The Tibetan informs us that “the triangle is the basic geometric form of all manifestation….” (Esoteric Astrology, Page 429)
No wonder that the faces of all five Platonic solids (regular polyhedra) are divisible into some arrangement of triangles! Three (the tetrahedron, octahedron, and icosahedron) have triangular faces, each of which contains six right-scalene triangles; each face of a cube is comprised of four right-isosceles triangles; and each pentagonal face of a dodecahedron is made up of ten right-scalene triangles. (Proportion and The Living World, Page 41)
To get a sense of what’s so special about the number three and its geometric representation, we must understand the viewpoint of the ancient mathematical philosophers whose distance from present-day humanity is much greater than the millenium which separates us from them. For Plato, the concept of Number was not primarily enumerative, but “Form”-al and relational—a quantitative and exoteric aspect plus a qualitative and esoteric aspect. The former is characterized by a definite, finite, particularizing power, and the latter, a universal synthesizing power. (Sacred Geometry: Philosphy and Practice, Page 10ff)
Reality seen by Plato as pure essences or archetypal “Ideas”, “cannot be perceived by the senses, but by pure reason alone. Geometry was the language recommended…as the clearest model by which to describe this metaphysical realm.” (Ibid., page 9)
From the ancients’ perspective, One, Unity, Monad, and two, Duality, Dyad, were not numbers but the parents of numbers. Their reasoning was that in and of themselves, one (a point) and two (a line) are incapable of configuring into a form. The One is beyond all human conception—the symbol of wholeness, completeness. But the Dyad or twoness is the spoiler, the audacious break-up of Unity and Harmony. Thus do conflict, polarity, difference, and separativeness arise.
Paradoxically, this Law of Attraction and Repulsion cuts both ways: it divides and unites. “It might strictly be called the Law of Adjustment or of Balance.” (A Treatise on Cosmic Fire, Page 1166) The mating and fusion of the principles of one and two, “point and line, …is all that is required to construct the world’s geometric patterns.” So the “Dyad is the doorway or birth portal between the One and the Many.” (A Beginner’s Guide to Constructing the Universe, Page 30)
It takes three points to bring forth a two-dimensional geometric figure, the first to appear being the triangle, as noted previously. Simply visualize or draw a circle with a point at partially the center, being partially overlapped by a second circle of the same size and radius; the center of each circle lies on the circumference of the other circle. Therein are three points to be connected: one line between the two center points, and one line drawn from each center point to the apex formed by the intersection of the two circumferences. The shared, almond-shaped area that gives birth to the triangle is called the vesica piscis (bladder of the fish)—the early Christian reference to Christ as the inaugurator of the Piscean Age, the Word made flesh.
No better symbol than the triangle can be found to typify the endless process of two oppositional points (terminii of the line that joins the two circles’ centers), such as spirit and matter, being resolved or mediated through the third point (Proportion and The Living World, Page 46). Harmony (Greek for “joining” or “fitting together”, taken from carpentry) means resolving conflict…as in ray four which is associated with the buddhic plane, buddhic energy being the “sumtotal of the life force of Vishnu” (A Treatise on Cosmic Fire, Page 1115), as we’ve seen, the “unifying principle of groups” (A Treatise on Cosmic Fire, Page 383), “the sixth or Christ principle” (A Treatise on Cosmic Fire, Page 704).
Since we’re travelling through lofty regions of the Logoi’s personality, our map and compass must be of universal magnitude, yet be calibrated so as to facilitate verification from the occult classics. My survey of this area has afforded the following decoding for one corner of the cosmoscape:
The sacred personality of the Logoi (solar and planetary) can be explained by the Pythagorean triangle, also called the Diaphantine or sacred triangle—the 3-4-5 right-triangle.
Bearing in mind the classic proportion, 3-4-5, that defines the Pythagorean triangle, several passages outline these number-concepts, and suggest their application to the logoic personality. First, the Pentad, since “Manas is, really, WILL working itself out the physical plane.” (A Treatise on Cosmic Fire, Page 353)
“…the figure of the universe is bounded by pentagons… a Dodecahedron, the geometrical model imitated by the Demiurgos in constructing the material universe.” (Esoteric Writings, Page 15)
“[Five] is the second and third Logos united in evolution.” (A Treatise on Cosmic Fire, Page 396n)
“… the 5 is the universal quintessence which spreads in every direction and forms all matter.” (The Secret Doctrine II, Page 583)
Because we live “inside the box”, literally, i.e., fourth globe, chain, scheme, Round, Hierarchy, etc., It’s hard to get outside it. But with a little help from the fourth plane, we “budding buddhis” may see a glimmer in the following statements, mentioned earlier.
“…the Quaternary, the perfect number, was the noumenal, ideal root of all numbers and things on the physical plane…. [It] was with the ancients only a secondary ‘perfection’, so to speak, because it related only to the manifested planes.” (The Secret Doctrine II, Page 582)
“[To the Pythagoreans, the Tetrad was] the most perfect number, because it emanated from the one, the first manifested Unit, or rather the three in one.” (The Secret Doctrine II, Page 599)
Getting to the four of form via the triangle becomes evident if we subdivide the triangle into four smaller triangles. If you cut out the larger triangle, score the three partitioning lines, and fold the three points upward, you will have your very own 3-D tetrahedron.
Moving up from the fourth to the third level puts us in a very different atmosphere—that of Tri-Unity. At the beginning of creation, Brahma (masculine; the initial circle with central point) projects outward a perfect reflection of himself (second circle with central point; the female aspect, Vach) which latter produces the Son, Viraj—a recreation of Brahma himself from within the vesica piscis. (The Secret Doctrine I, Page 89; Isis Unveiled II, Pages 170, 268, 270) This almond-shaped area, the “mouth” symbol from the Egyptians, also represents Hiranyagarba—the womb (radiant or golden egg), the portal through which the universe is formed.
The commencement of manifestation, marked by the emanation of Father-Mother-Son, takes a new twist, described by HPB:
“By mystic transmutation they became the Quaternary—the triangle became the TETRAKTIS.” (The Secret Doctrine I, Page 614)
On this point, the Tibetan adds:
“The Sacred Four are…[Father, Mother, Son] and their united manifestation making. The active evolving solar system, the logoic ring-pass-not. This is the Macrocosm.” (A Treatise on Cosmic Fire, Page 536n)
So the Threeness, the triangle, becomes the Fourness, the Quaternary. Then there’s one more conversion to note:
“The triangle becomes a Pentagon (fivefold) on Earth.” (The Secret Doctrine II, Page 79n)