11

THE COSMIC FLOWER

THE SUN GENERATES ENERGY AT ITS HIDDEN CORE and manifests it as light that streams forth from its blazing surface. This is the universal radiant gesture. In like manner each being has its own expression of the energy at its core: the slithering of the snake, the delicate walk of the deer, the swooping dance of the swallow, the rippling of wheat in the wind, the sparkling geometry of the crystal. The unique signature of each type of being has its source in the species soul, the devic energy that animates it. Their bodily expressions are their ways of radiating, or unfolding, their life energies as a bud unfolds into a flower. It may be said that all Earth life-forms knowingly or unknowingly pay homage to the Sun in their radiant unfoldings; it is in the flower, however, where this is most clear. And the polygonal shapes of many flowers that unfold from vesical buds can serve as geometric metaphors for the unfolding of time and life.

Figure 11A. The radiant gesture of the Sun is mirrored in the forms of life that expand out from a center to a bodily periphery and even beyond to spheres of sound and fragrance and awareness. Bees on a globe thistle, Echinops ritro.

THE SYMBOL IN THE SKY

For our understanding of the principles of creation to transcend mental concepts and language we require symbolic imagery that opens pathways to deeper modes of knowing, with feeling and with intuition. This is what the geometric art can provide to us. A great symbol forms in the sky when the new moon passes in front of the Sun—the solar eclipse. The equality of their apparent sizes then becomes obvious—a synchronicity beyond all random coincidence that naturally demands our attention. Like all symbols there are several or many interpretations.

Figure 11B. The new moon in the process of eclipsing the Sun.

The two—the bright masculine Sun and the dark feminine Moon—become one. Then the separation begins as the Moon continues to move across the Sun’s surface. The image is one of scission similar to the movement between the daughter somatic cells during mitosis. This is a gesture of creation, of Unity dividing in two. As geometers we examine the symbol in its details. We reduce Sun and Moon to simple circles and witness in drawing 11.1 the series of vesicas formed by the tonal nodes that the edge of the Moon silently sounds as it moves along the musical string of the Sun’s diameter. The yin and the yang join and are parted again, and during this process much has happened. The “ten-thousand things” have arisen in the form of the innumerable vesica shapes with their associated ratios and their tones that have symbolically sounded. Before the parting of the circles, however, we note an important position in the progression: when the circumference of the Moon encounters the center of the Sun, and, conversely, the circumference of the Sun touches the center of the Moon, they form the vesica known as the vesica piscis.

Drawing 11.1. A symbolic meaning of the geometry of the eclipse of the Sun by the Moon.

At totality darkness completely dominates, a surprising and shocking development to the ancients who perceived heavenly messages and warnings in the sky. Darkness could be interpreted as the eradication of consciousness or of life. Such events are possible and have nearly happened during the cataclysmic history of the Earth filled with meteorite and comet strikes, mass extinctions and post-glacial floods. Yet the message is not necessarily so dire. The darkness is not evil. It is a part of who we are. The eclipse reminds us to consider the dark hidden parts of ourselves and to integrate them into our consciousness, our light.

Drawing 11.2. The primal unfolding. Part (1) is a gestural drawing showing the unfolding, or flowering, that is abstracted from the classical construction in (2) in which this dual unfolding symbol is placed within the matrix of intersecting circles representing the Akashic Field, a section of the Flower of Life symbol (see drawing 4.7). In this context we can construct the polygons (here: triangle, square, pentagon, hexagon) using the inherent points and lines of the figure. The sides of the central triangle connecting equator ends and poles of the vesica can be seen to be folding out and around the vesica-forming circles like a flower blooming from the vesica bud. Next the sides of the square unfold. They are verticals drawn perpendicular to the baseline that intersect the vesica-forming circle and are then connected with a horizontal line. Next, the crossed lines intersecting points on the field as shown are extended to the vesica-forming circles to establish pentagon vertices from which two sides of the pentagon can be drawn. With our side lengths known we can, with compass, cross arcs on the extensions of the vertical axis to establish the point with which we can complete the pentagon. For the hexagon the right and left field-vesica poles provide points to which the lower sides can be drawn. The points where the extensions of the sides of triangle and square intersect provide points to which the upper sides of the hexagon can be drawn. The ends are connected, and the hexagon is complete.

The vesica piscis is the figure from which polygonal geometry unfolds outward from the triangular inner structure of the half vesica, like a flower unfolding from a bud. Drawing 11.2 illustrates the geometry that gives rise to this opening gesture. In placing this geometry within a matrix of vesica pisces we are able to locate the construction points from which many regular polygons can be generated.

SACRED COSMOLOGY

“As above, so below.” This is the ancient esoteric guide to those seeking to understand the workings of the world. The seeker who accepts this dictum looks for “the world in a grain of sand”—for clues to the whole contained within the part, for eternity within the moment. The stillness that we can connect with inside ourselves, like the quantum vacuum of deep space, contains tremendous power. The light within is our inner sun that we are drawn to as a sunflower is drawn to the Sun in the sky. The sound of the wind in the trees is continuous with the inner beehive hum of our neurons. The outer reflects the inner. The macrocosm is reciprocal to the microcosm.

Let us now create a symbolic geometric map of cosmic evolution with the assistance of the powerful tools of geometry, number, and the musical scale. These ancient tools have not been displaced by modern science and technology. In fact, they have been enhanced by them. On a practical level computer drawing programs have removed much of the drudgery of calculation and the attainment of accuracy while still allowing a classical approach to construction. On a deeper level the resurgence of the harmonic vision of the world owes much to the discoveries of modern science that have made available accurate measures of time and space, which enhance the accuracy of harmonic symbology. Harmonics, in turn, has brought a breath of fresh air to a science that is in need of simplification so that the ordinary person can partake of our common birthright: some understanding of our living Universe.

Musical geometry is able to accomplish this by creating symbols that awaken parts of us that purely mental constructs cannot reach. These are the realms of the sacred that were, until modern times, the province not only of mystics but also of those who studied the natural world. This study was called sacred science because it did not limit itself to the material world but entered the realms of soul and spirit that were also considered parts of the natural world and thus worthy of study.

In the previous chapter we presented a possible harmonic time line behind the evolution of the Universe as portrayed in the Mayan calendar. It is interesting that the period before Creation, which we have called the Dreamtime, had a duration of 2.73 BY, which is 1/6 the duration of the whole calendar (16.4 BY) and 1/5 the duration of the period (13.66 BY) since the Primordial Expansion (big bang). These ratios suggest that there might be hexagonal and pentagonal properties in universal creation. Both polygonal structures (see drawing 11.2) unfold from the vesica piscis, which itself is the result of the scission of circular unity into two circles. Drawing 11.2 also reveals these circles to be part of a matrix of intersecting circles whose finished symbol is the Flower of Life illustrated in drawing 4.7. As we have noted, this symbol can represent the universal energy field known as the quantum vacuum, or, as Laszlo calls it, the Akashic Field, a sea of colossal energy and information from which Creation is believed to emanate. In our present context we can imagine the polygonal structures of time and life arising from this field.

Drawing 11.3. From the universal field. shown on the left, the triangle emerges. This is Plato's demiurge, the Creative Spirit who takes on the task of creation. As the sides of the triangle unfold outward in an opening gesture (center), the square is generated. This is the symbol of consciousness, the awareness of what needs to be accomplished before the main purpose at hand, the creation of life, can unfold. When the ground is prepared the pentagon (right), symbol of life, unfolds from the square. The construction lines that form the symbols also produce the outline of a waterdrop, below, the substance that is essential to the evolution of life.

Our construction of the polygons that blossom forth from the bud of the vesica piscis have the following properties and interpretations.

(1) The generating form is the triangle, the underlying structure of the vesica piscis, which unfolds from the heart of the vesica bud and sweeps out arcs in both directions locating the construction points for square, pentagon, and hexagon. Musically, the number 3 is represented by the interval of the fifth.

 

Figure 11C. Natural hexagons from microcosm to macrocosm: (a) micrograph of cross section of fibers in a fly’s flight muscles; (b) cells in a beehive; (c) the great hexagonal cloud structure at the north pole of the planet Saturn

(2) The second regular polygon to be constructed is the square, the symbol of reason and consciousness, the organizing principles behind Creation. The square is also the traditional symbol of Earth and provides a unifying symbol of our fourfold perceptions of space and time: the four directions, the four seasons, the four phases of the Moon, and the four daily positions of the Sun—dawn, noon, sundown and midnight. The fourfold square also represents the four organizing forces of nature: gravity, electromagnetism, the strong and weak nuclear forces; the four letters of the DNA code; and the four main sectors of the human brain—reptilian, limbic, neocortex, and prefrontal lobes. The inner structure of the square is a matrix that can generate innumerable geometric constructions. We have seen throughout the book a few examples of this in the intersections of two of the many kinds of diagonals that crisscross the square and whose intersection points, when projected to the string/diameter of the enclosed circle, give us the tonal nodes from which to produce vesica constructions. Musically, 4 refers to the octave.

(3) The next polygon is the pentagon, the embodiment of the number 5, the number of the major-third interval, which, as noted in chapter 10, plays an important role in the Mayan calendar and is associated with the substance of water. The inner structure of the pentagon is dominated by ϕ-family ratios that are involved with biological life-forms. (See appendix drawing A2.16 for a demonstration of this property.)

(4) The next polygon generated by the triangle within the vesica bud is the hexagon, symbol of primal matter, of the atomic and crystalline realms that make up material creation. In our time analysis of the Mayan calendar the hexagon describes the entire calendar, including the pre-Creation Dreamtime. The inner structure of the hexagon is filled with relations based on √3 (1.732), which is the height-to-width ratio of the vesica piscis (see appendix drawing A2.17). Musically, the number 6, like the number 3, relates to the interval of the fifth.

“The symbol awakens presentiment; language can only explain. The symbol plucks all the strings of the human spirit simultaneously; language is always obliged to devote itself to a single thought. The symbol’s roots reach into the most secret depths of the soul; language only touches the surface of understanding like a slight breath of wind. One is directed inward, the other outward. Only the symbol can connect the most varied things into a unified collective expression.”

J. J. BACHOFEN¹

Drawing 11.4. A hexagonal time structure representing the evolution of the Universe from the pre-Creation period we have called the Dreamtime through the Primordial Expansion (big bang) until the present, a duration of 16.4 billion years, the proposed time span of the Mayan calendar. The sides of the hexagon are 2.73 billion years in duration, which is also the length of the pre-Creation Dreamtime (bottom) period that establishes the rhythm of the calendar. The progression proceeds clockwise in 2.73 BY increments and gives a rough idea of the timing of events, like star formation, which took place over billions of years. The diagram is meant to give a general picture of evolutionary time. The outer heptagon showing the stages of plant growth and their correspondences with Creation and evolution is more in the spirit of the Mayan calendar and in the spirit of this book, which seeks correspondences between above and below, and here, between the growth of the Universe and the growth of a plant.

Figure 11D. The number 5 manifests often in life-forms: (a) starfish, (b) flower, (c) apple cross section, (d) hand, (e) blackberry leaves.

We now return to the generating equilateral triangle at the heart of the construction. It is intimately related to the hexagon, being a sixth part of its structure. In the drawing it occupies the lower part of the hexagon, the area that we have referred to as the Dreamtime, the 2.73 BY period before Creation. The position and function of this triangle recall a feature of the Great Pyramid in which the bedrock of the base area was mostly leveled except for a section that was allowed to protrude into part of the lower section of the pyramid. This provided a stabilizing anchor for the structure as well as a connection to the Earth and its energies. A similar feature is found in the Dome of the Rock, the temple in the Old City of Jerusalem, in which the rough, exposed bedrock occupies much of the floor of the temple. In a similar way the triangle in drawing 11.3 represents the Creative Spirit emanating from the Dreamtime, the eternal matrix of existence from which it generates consciousness, life, and time continues to inform evolution as a universal collective unconscious.

Figure 11E. Six polygonal diatoms embodying the numbers 3 to 8. From left, (a) Triceratium, (b) Perissonoe, (c) Actinoptychus, (d) Aulacodiscus, (e) Asterolampra, (f) Asterionella.

“None of the hundreds of Aboriginal languages contain a word for time, nor do the Aborigines have a concept of time. As with creation, the Aborigines conceive the passage of time and history not as a movement from past to future but as a passage from a subjective state to an objective expression. The first step in entering into the Aboriginal world is to abandon the conventional abstraction of time and replace it with the movement of consciousness from dream to reality as a model that describes the universal activity of creation.”

ROBERT LAWLOR²

The property of threeness inherent in this period relates to many of the Creation myths of humanity that have a triune creator or a creator, preserver, destroyer, representing the most universal trinity of existence—birth, life, and death—that are intertwined with the three dimensions of space and the three properties of time: past, present, and future.

THE SYMBOL SPEAKS

Our nested polygons might be interpreted as follows: The field of intersecting waves, the Akashic Field, produces the vesica piscis when two waves intersect each other’s center. The inner structure of the half vesica is the equilateral triangle, the Creative Spirit, who assumes the task of creation. Its first creation is consciousness or psyche, represented by the square that mediates between spirit and life.

Life is the next emanation and is represented by the pentagon, the embodiment of the golden ratio and of the fivefold patterns found throughout the biological world, the more advanced realm of life. The next emanation is time, which is represented by the hexagon. Perhaps time serves as a container and regulator of evolution or a tool that life uses to perfect itself. Or time might be compared to a shell that contains a sea creature, life, which precedes time as the pentagon precedes the hexagon in the geometrical symbol.

It seems that while life appears contained in time it is, in some ways, independent of time. Thus, “timeless” moments can occur spontaneously in our lives. Or we can choose to enter into timeless states of mind. The dream state and deep meditation are portals into these timeless regions. The triangle represents such a place. It is the place of dreaming where existence was dreamed into being. According to the symbol, it is embedded within the ring of time, consciousness, and life, as night dreams are embedded within our daily lives. This place is a part of us and thus is forever accessible to the seeker, to the dreamer.

CEREMONY

We have seen that with a foundation of sacred number and geometry we can enter confidently into regions that are inaccessible to ordinary reason. The traditional purpose of these tools is the creation of symbols that invoke spiritual energies for personal and social benefit. A practical application of this purpose is the building of temples proportioned by cosmic principles. George Michell writes of the philosophy behind the building of Hindu temples:

Drawing 11.5. A wheel of tones and polygons, a construction, generated from drawing 11.2, in which the polygonal angles are framed in terms of tonal intervals. The semicircle with angle of 180° is considered to be a musical string equal to an open string that sounds the fundamental tone. The difference is that this musical string is curved instead of straight. The left half sounds the realm of harmonics from 1/∞ to 1/2. The right half sounds the tones of the main octave. At the 1/3 point of the semicircle is our first regular polygon, the equilateral triangle, whose base angle of 60° is 1/3 of the fundamental angle of 180°, which designates it as an octave fifth. The semicircular string (a logical impossibility in the physical world but a conceptual tool in the realm of harmonics) can be compared to a curved guitar string. When you stop it at 1/3 its length and pluck it, as you would with a guitar, you will sound an octave fifth. The 1/3 length in the present conception not only sounds an octave fifth but, being part of a circle, also creates an angle. Next, at 90°, the octave, the next regular polygon, the square, is generated. The pentagon with its 108° angle (108°/180° = 3/5) is the major-sixth ratio. At the fifth ratio, at 2/3 of the perimeter, the hexagon is formed from the 120° angle. We have also included several other tones that arise from the angles of regular polygons: octagon, the fourth; decagon, the minor third; and an 18-gon, which represents the major whole tone with ratio of 8/9. Between the 8/9 tone at 160° and the fundamental at 180° are innumerable tones and their associated polygons whose perimeters gradually approach the perfect half-circle of the fundamental. (The number of sides of each superparticular-ratio-derived [superparticular ratios are those with the numerator and denominator separated by 1] polygon is twice the number of the denominator of the tonal ratio.)

At the other end of the fundamental semicircle we see the realm of the harmonics. These increasingly smaller intervals represent the higher and higher frequencies of the harmonic series. Near the limit, which is at an infinitely high vibration, are the vibrations of the quantum world.

Only if the temple is constructed correctly according to a mathematical system can it be expected to function in harmony with the mathematical basis of the universe. The inverse of this belief is also held; an architectural text, the Mayamata, adds “if the measurement of the temple is in every way perfect, that there will be perfection in the universe as well.” Thus the welfare of the community and the happiness of its members depend upon the correctly proportioned temple, and the architectural texts stress that only work that is completed “according to the rules” will gain the desired merit for its builder.³

Figure 11F. Itzhak Bentov wrote, “The seed carries not only the information about the shape of the tree but also its unfoldment in time. . . . The seed is a unique structure because in it space-time has been condensed and stored, awaiting the proper objective time for its unfoldment. . . . It is the representation of the tree in an altered and higher state of consciousness.”⁴

The seed contains the plant in a highly compressed form, demonstrating the reciprocal relation: the larger the quantity, the smaller the reciprocal. The point is the seed of geometry. The geometric point contains the whole of geometry within itself as a seed contains the plant within itself. The point radiates out in all directions creating a sphere in which all geometric figures can be constructed. This principle gave rise to the modern creation myth of the big bang in which an infinitesimal point of infinite density expands out and becomes the Universe.

And, if we follow Bentov’s insight, the singularity/seed represents the higher consciousness from which the Universe emanates. This seed gestates within the seeding period, the Dreamtime. In our previous drawings it is represented by the triangle, the symbol of Spirit. Above, pumpkin seeds.

The beliefs that generated temple building had their origins in prehistoric shamanic cultures in which ceremonial practices were developed for the same purpose: to invoke healing energies. Like the temple builders, the shamans’ first priority was to establish sacred space within which ceremonies could unfold. A common practice was (and is) the calling in of the energies of the four directions, thereby creating a symbolic square. Within this square a circular ritual area might be scribed on the ground, as in some vision-quest ceremonies, or built, as with the round sweat lodge, or suggested, as with a central fire around which ceremonialists gather or dance. In such sanctified rings we may perceive the primordial symbol of the circle within the square (mentioned in chapter 1) representing Spirit held within the vessel of matter. In such spaces the prayers of the ceremonialists are infused with power. The temple architect and the shaman thus share a common purpose in their work—the reestablishment of harmony in the world.

Figure 11G.

Thus open out the Tao
and it envelops all of space
and yet how small it is
not enough to fill the hand!

HUAI NAN TZU