Only in recent years has sacred geometry engaged the attention of an ever-widening esoteric audience. This comes as something of a surprise, given HPB’s statement about: “Geometry, the fifth divine Science (“fifth” – because it is the fifth key in the series of the Seven Keys to the Universal esoteric language and symbology)…” (The Secret Doctrine II, page 471) and her identification of this mother lode of wisdom: “…the evolution and correlation of the mysteries of Kosmos, of its growth and development – spiritual and physical, abstract and concrete – were first recorded in geometrical changes of shape.” (The Secret Doctrine I, page 321)
“Perhaps the time has come….to the fifth Root Race entering the Age of Aquarius, which sign is the transmitter of only one ray, the fifth.” Of no small significance is the Tibetan’s emphasis on the importance of mystical geometry. (A Treatise on Cosmic Fire, Page 396)
For ease of entry into the geometry, it may help to frame the discussion along lines familiar to all esotericists, i.e., the fourth kingdom in nature, and the seven rays. “All these Rays of Cosmic Mind, Primordial Activity, and Divine Love-Wisdom are but essential quality demonstrating through the agency of some one factor.” (A Treatise on Cosmic Fire, Page 73) The solitary expression of the second cosmic ray is buddhic energy, “the sumtotal of the life force of Vishnu or the Son…” (A Treatise on Cosmic Fire, Page 1115), and
“Humanity is that evolution through which the Son aspect is to express itself most perfectly in this cosmic incarnation.” (A Treatise on Cosmic Fire, Page 241)
Ray four on which the soul of humanity is manifesting provides the closest alliance with the Divine Ray, the former being the optimal vehicle for moving the Second Logos toward his goal.
Hence, the three cosmic rays mentioned above correspond to our third, fourth and fifth rays, the numbers indicating the second phase of logoic development in our threefold solar system (for examples of this 3-4-5 process, see A Treatise on Cosmic Fire, Pages 687-8, 695-6, 376-7). Although other kingdoms bear some of this load, assignment for the lion’s share of Vishnu’s progress rests with humanity, the lowest level of self-conscious beings. We, as cells in His body, are tasked to fulfill the logoic requirements of number 5 (the manasic principle), number 4 (“the intuitive omniscience of the Spiritual divine soul – Buddhi” (The Secret Docrine I, Page 279)), and number 3 (the planetary throat centre; intelligent activity).
Operating in these ways takes us far beyond arithmetic’s simple numerals and quantitative values. It demands a quantum leap into geometry, variously preceded or qualified by an adjective such as sacred, mystical, contemplative or philosophical. Rooted, literally and figuratively, in the forms and force familiar to us all, this discipline is a cross between the spiritual and the mathematical. At the most basic level, numbers themselves reveal this hybrid characteristic. Two examples will make this point.
First, the historical veneration of the number 4:
“It is said that it expressed a living being, I, bearer of the Triangle… the emblem of God, i.e., man bearing within himself a Divine Principle.” (Morals and Dogma, Page 633)
Second,
“The Tetrad is esteemed….the most perfect, or rather sacred number, because it emanated from the one, the first manifested Unit, or rather the three in one.” [The Quaternary] “…the perfect number, was the noumenal, ideal root of all numbers and things on the physical plane.” (The Secret Doctrine II, Pages 599, 582)
Moving to a higher level of abstraction to witness the connect between the sacred and the geometric, the measurable and the incommensurable, we come up against irrational numbers such as pi (3.14159…). Most of us can recall its relationship to a circle, symbol of ONE or Unity. If the diameter of a circle has a value of one, the circumference will always be of the incommensurable type: 3.14159…x one (or any other number). This ratio of known to unknown inheres in virtually all three-dimensional or volumetric forms in nature. As we will soon see, the incommensurable ratios, e.g., the relationship of one to p (pi), “are unique in their ability to consistently replicate through endless divisions in space, even as the identical ratio remains present in the relationship of one level of subdivision to the next.” (Proportion and The Living World, Page 42)
“The relationship between the fixed and the volatile (between proportion and progression) is a key to Sacred Geometry: everything which is manifest, be it in the physical world or in the world of mental images and conceptions, belongs to the ever-flowing progressions of constant change; it is only the non-manifest realm of Principles which is immutable.” (Sacred Geometry: Philosophy and Practice, Page 27)
The Logoic Personality
The Ancient Wisdom presented to the Western world by HPB and the Tibetan was written in a manner that is frequently more like an encryption than an elucidation. Toward the end of her life, HPB was queried by Colonel Bowen on how to study Theosophy:
“She says: If one imagines that one is going to get a satisfactory picture of the constitution of the Universe from the S.D. one will get only confusion from its study. It is not meant to give any such final verdict on existence, but to LEAD TOWARDS THE TRUTH.” (Madame Blavatsky on HOW TO STUDY THEOSOPHY, Page 8)
It is in pursuit of TRUTH that I found what I consider to be a viable hypothesis in the application of sacred geometry to one sentence in Bailey’s Esoteric Astrology. Following three brief paragraphs describing the cosmic physical, astral, and mental planes, we read:
“From these three cosmic planes (embracing the sacred personality of the Logoi, solar and planetary) come the united energies of three constellations which control and energise our solar system: the Great Bear, the Pleiades, and Sirius; these work through the medium of the seven rays and these in turn express themselves through the twelve constellations which form the great zodiacal wheel.” (Esoteric Astrology, Page 595)
The first round of brooding on this passage raised the question of what the sacred personality is. By the Law of Correspondences, and assuming a fivefold manifestation (physical to atmic planes inclusive), cosmically, I saw that one of the Brothers of Fohat – AGNI – “is the life of the logoic Personality….He is to the solar Logos, on His own plane what the coherent personality of a human being is to his Ego in the causal body….His is the coherent force that makes a unity of the triple logoic Personality.” (A Treatise on Cosmic Fire, Page 603)
This Lord of the mental plane, cosmically and systemically, funnels and unifies the energies coming from the three major constellations, situated on the third, fourth, and fifth cosmic planes. Two of these placements are made explicit in the Tibetan’s writings: the Great Bear on the cosmic buddhic (A Treatise on Cosmic Fire, Page 344) and Sirius on the cosmic mental plane (A Treatise on Cosmic Fire, Page 571). No such specificity as to which cosmic plane is home to the Pleiades!
Consequently, inference from other information on this constellation is all we have to guide us. Perhaps the most persuasive argument rests on the statement that the Great Bear and the Pleiades “are a higher correspondence to atma-buddhi.” (The Rays and the Initiations, Page 405) Added weight can be found in the fact that the Pleiades are “the feminine opposite of Brahma” (A Treatise on Cosmic Fire, Page 156); and the third cosmic ether literally embodies Brahma (A Treatise on Cosmic Fire, Page 313). Selections from The Secret Doctrine are à propos. Occupying nearly half a page discussion of the Pleiades, we find these phrases:
“…the central point around which our Universe of fixed stars revolves.”
“…the central group of the system of sidereal symbology.”
“…the sidereal septenate born from the first manifested side of the upper triangle, the concealed r.” (The Secret Doctrine II, Page 551)
In this vein, a fourth indicator of prominence appears in Esoteric Astrology on page 416: the Pleiades, Cancer and Venus are referred to as the ‘First Triangle’, with triangles headed by the Great Bear and by Sirius, designated as second and third. Finally, the Pleiades “are in reality and literally the expression of the Principle of Life which we call vitality, prana in its various stages or degrees, ether or substance.” (Esoteric Astrology, Page 504)
Complications arise exponentially as we endeavor to wrap our minds around the lower three cosmic planes, having little or no direct experience thereof. Nevertheless, reassurance and encouragement are provided repeatedly in the Tibetan’s writings, mainly in the form of incentives available through the Law of Correspondences or Analogy. Two instances will suffice to highlight His strong suggestion that we expand our reach upward, periscopically:
“Until our students extend their concept to include in their calculations the logoic astral and mental bodies as the Logos seeks to express emotion and mind on the physical plane (through His physical body, a solar system) they will not progress far towards the heart of the solar system.” (A Treatise on Cosmic Fire, Page 834)
“[the student] should endeavour to ascertain the relationship between the universal mind (or the systemic mind) and cosmic mind…” (A Treatise on Cosmic Fire, Page 1098)
Returning to the logoic Personality, when viewed from the cosmic mental plane, it is constituted of the three Logoi (Brahma, Vishnu, Shiva) who “are seen functioning as one”. (A Treatise on Cosmic Fire, Page 64) Just in case we might miss this point, The Tibetan punches it up: “…these Three constitute the PERSONALITY OF THE LOGOS…” (Ibid.) Of course, the Heavenly Men “Who in Their totality form the Body of the threefold Logos” are the sevenfold manifestation of the triple Logos. ‘Body’ is one descriptor of the lowest aspect of threefold Being, e.g., Body, Soul, Spirit; another term is Personality, as in Monad, Ego, Personality. (Esoteric Astrology, Page 80) Hence, the logoic composite nature of the lowest 18 cosmic subplanes whose cognomen is “sacred personality”.
Greater detail is given on ways the three constellations work within our planetary scheme (A Treatise on Cosmic Fire, Pages 1162-3):
- Great Bear….Spiritual energy….three higher planes….the Monads;
- Pleiades….Buddhic force….fourth plane….the solar Angels;
- Sirius….Manasic force….two lower planes….the four kingdoms of nature.
Whether this particular planar allocation of forces and energies obtains for all planetary schemes is, for me, an open question.
One last point about the singular “sacred personality of the Logoi, solar and planetary” (note the plural) is that it accords grammatically with a similar construction regarding the planetary Logoi. “The causal body of the Heavenly Men is upon the third subplane of the cosmic mental plane…” (A Treatise on Cosmic Fire, Page 532 #10). [Emphasis added.] Neither is a typo. Rather, both exhibit the same pattern. We have one personality comprised of Logoi acting in concert, leading to a more perfect Unity. Offering conjecture as to the dynamic functions of each aspect – personality and causal body – would necessitate venturing too far afield of the topic in hand.
Triangles
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 polyhedral) 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 millennium 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: Philosophy 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’ centres), 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).
Philosophic Geometry
Since we’re traveling 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)
Lest readers think we’ve been flying too high for too long, let’s get grounded with a pencil and paper exercise to show how “the triple triangle, a figure of five lines uniting five points” (Morals and Dogma, Page 634) is drawn – a geometric proof that three triangles do align to make a pentagon. First, construct a pentagon and mark the vertices A, B, C, D, E. Take a red pen and connect points ACD, inscribing a triangle. Using a green pen, draw triangle BDE. With a blue pen, trace triangle BCE. Voila! Three triangles make a pentagon.
One glance at any pentagon inscribed within a pentagram indicates to the eye and mind that these figures are self-replicating and self-accumulating: five triangles projecting outward from the sides of the pentagon…which is divisible into five smaller triangles. Connect the five points of the pentagram, and you haves pentagon that’s larger than the one within the original pentagram. The open secret of this unending progression is the relationship of the side of the pentagon to its diagonal, the ratio of one to F (phi) – another of those pesky incommensurables involving the square root of 5, an irrational, supra-rational or transcendent number.
According to one author, the “Pentad draws us into the most widespread shape in nature – the spiral.” (A Beginner’s Guide to Constructing the Universe, Page 138) He describes spirals as “the purest expression of moving energy”, adding that “wherever energy is left to move on its own it resolves into spirals.” (Ibid., page 139) No surprise, there, for esotericists! Fohat is “THE FIERY WHIRLWIND” who “TRACES SPIRALLINES”. (The Secret Doctrine I, Page 31) Later in the evolutionary process, the Second Logos: “Availing Himself of the rotary motion of all atoms, He adds to that His own form of motion or of spiraling periodical movement, and by circulation along an orbit or spheroidal path (which circles around a central focal point in an ever ascending spiral)…” (A Treatise on Cosmic Fire, Page 143).
Many are the Teaching’s references to the hand-in-glove relationship between Three and Five. Whether it’s about triangles and pentagons producing each other, or the inversion of the third and fifth planes (A Treatise on Cosmic Fire, Page 605), or their close alliance in the first and second solar systems (Letters on Occult Meditation, Pages 216-8), we come upon ever more layers to be peeled away. But this very interplay offers, indeed invites ‘play’. The Sanskrit word for it is Lila. Through play, delight, sport, amusement (The Secret Doctrine II, Pages 53, 126), creation eventuates, as Vishnu “sports like a playful boy in frolic”.
I am persuaded that we might benefit from following His lead. My inclination to move in this direction occasionally finds me on a new thought-train. One example emerged as I was pondering the three vis-a-vis the five.
The ciphers themselves, their morphology engaged my attention, and posed questions. Did the number three acquire its shape by dividing a circle (symbol of the ONE) into two hemispheres, opening like a hinged bracelet (3)? Was a similar process responsible for the five? Were the two Greek accents(‘,) that gave us the number 5 (The Secret Doctrine II, Page 576) but representations of a bisected circle, with the right hemispheric arc being rotated sagittally upward? Could the imaginative eye see, in these two scissions, the suggestion of a Spiral (3) and a Serpent (5), the latter being a visual portrayal of hissing (5 shaped and sounded like the letter S)?
As students of esotericism, we are continually stretching our minds to grasp the significance of numbers enumerative values via numerology. However, as denizens of the higher manasic realms, we extend our reach even further by attempting to engulf, digest and assimilate the principles and conceptual values of numbers.
Adopting this approach permits us to move past arithmetical values to enigmas aplenty. The Sacred Triangle confronts us with its 4-line at cross-purposes to the 3-line. Four, the perfect number in manifestation, wars against the first perfect number, Three, at spiritual levels. Contained within the numeral 4 is a triangle, two sides (3-line and 4-line) of which are shared by two right-angle arms of a cross. Paradoxically, it is the challenges of the three Crosses that lead the successful to the promised land of At-ONE-ment on the fourth plane.
While there’s much to be gained from geometrizing, from drawing geometric figures on two-dimensional sheets of paper, each such drawing is but a frozen nanosecond of cosmic reality. The mind scoops up these figures, puts them on the dumbwaiter, and sends them Upstairs for restoration to their roles in an ever changing, dynamic world. Motion continuously swallows sand castles, shifts our degree of well-being, turns night into day, changes our seasons, and brings solar flares to glitch our electronics. And we must ride these waves to understand what they’re about, and how we are rivulets running and rippling into the Greater Flow-through.
Some readers will have been disappointed by this piece, as they were looking for a tit-for-tat correspondence of the major constellations, three cosmic planes, and side-lengths of the Sacred Triangle. My inner compass pointed in quite another direction: fanning the embers of faint orange to flame in minds eager to board a new mode of conveyance to the Heart of things.
BIBLIOGRAPHY
Cited Works
Bailey, Alice A.
Treatise on Cosmic Fire. Lucis Publishing Company, New York, 1977.
Esoteric Astrology. Lucis Publishing Company, New York, 1974.
Letters on Occult Meditation. Lucis Pubulishing Company, New York, 1979.
The Rays and The Initiations. Lucis Publishing Company, New York, 1972.
Blavatsky, H. P. Isis Unveiled, Vols. I and II The Theosophical Publishing House, Wheaton, IL, 1972.
The Secret Doctrine, Vols I and II (Original Facsimile Edn.). The Theosophy Company, Los Angeles, CA, 1974.
Bowen, Robert Madame Blavatsky on HOW TO STUDY THEOSOPHY. The Theosophical Publishing House, Adyar, India, 1973.
Fletcher,Rachel “Proportion and The Living World”, Parabola, Vol. XIII, Number 1. The Society for Myth and Tradition, Inc., New York, 1988.
Lawlor, Robert Sacred Geometry: Philosophy and Practice. Thames and Hudson, New York, 1998.
Pike, Albert Morals and Dogma. L. H.Jenkins, Inc., Richmond, VA, 1946.
Row, T. Subba Esoteric Writings. The Theosophical Publishing House, Adyar, India, 1980.
Schneider, Michael P. A Beginner’s Guide to Constructing the Universe. Harper Collins Publishers, Inc., New York, 1995.
