Part 5
God Geometrizes
The Triangle, the Torus and Polyhedra
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Figure 5-1
The small icosahedrons are mouse interactive.
(The large icosahedron surrounding the central moving star is more difficult to control with the mouse.)
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"Every stage of development in Raja Yoga is symbolized by a geometric symbol"
— H.P. Blavatsky, The Voice of the Silence
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W
e have earlier discussed the relationship of triangles to any evolving inner spiritual life and the need to strengthen its many points of light into an impenetrable grid to deter inimical forces from entry. Triangles, which are the basic form of all the different types of polyhedra, should be studied to see how they create endless geometric patterns and rotating inner bodies. This study of the polyhedra and Platonic solids forms the basis of all sacred geometry.
All sheaths of a torus/toroid are made up of energy and force that are in constant motion. Because their outer sheaths or shells are composed primarily of the third aspect of deity, they can also become tessellated over their surfaces with many intricate and detailed designs when their vibrational frequencies are slowed down. This geometric solidification is similar to how free-flowing water when exposed to extreme cold will begin to tessellate into an endless variety of geometric patterns of ice. All rotating bands of energy that slow down to a certain speed and begin to occultly cool will begin to exhibit some type of polyhedral form and crystallize into certain types of geometric shapes. These shapes can be transparent and profoundly ethereal compared to their physical plane counterparts and can still be capable of shifting or morphing into other shapes depending upon the purpose of the indwelling life. The hidden geometry of any spiritual organism is important because it has a direct relationship to the advancement of that life and its light-bearing capacity and the rays with which it works. DK has this to say about geometry under the heading "Microcosmic and Macrocosmic Alignment."
"This morning, I would touch on the matter of egoic alignment again, showing you, under the Law of Correspondences, the universal application. It lies based in geometry, or in figures and numbers….
"Each life that the Personality leads is, at the close, represented by some geometrical figure, some utilisation of the lines of the cube, and their demonstration in a form of some kind. Intricate and uncertain in outline and crude in design are the forms of the earlier lives; definite and clear in outline are the forms built by the average advanced man of this generation. But when he steps upon the Path of Discipleship, the purpose consists in merging all these many lines into one line, and gradually this consummation is achieved. The Master is He Who has blended all the lines of fivefold development first into the three, and then into the one. The six-pointed star becomes the five-pointed star, the cube becomes the triangle, and the triangle becomes the one; whilst the one (at the end of the greater cycle) becomes the point in the circle of manifestation." (LOM:5)
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Figure 5-2
Changing grid of triangles:
Use the arrow keys to change it.
Use the mouse to rotate.
Click on Sphere
↑ Up Arrow: Increase Triangles
↓ Down Arrow: Decrease Triangles
Move Mouse: Rotate
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Triangles form the basis for all of these different types of tessellations or geometric patterns and are drawn together by many variations of polyhedral design. Let us give some examples taken from the site of Paul Bourke of Swinburn University.
The five basic three-dimensional shapes are known as the Platonic Solids:
the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron.
Polyhedra are shapes formed by three or more planes intersecting at a common point. All architecture is built on these polyhedral principles which make up the majority of forms in the natural world. This same architecture in its many and varied forms can be found on the inner spiritual planes.
The ability of these polyhedra to nest or embed into one another is crucial in understanding their multifaceted connections. Every time an organism sends out a living thread or stream of lighted force and connects in to some other part of its overall design that organism can be considered strengthened and capable of receiving more powerful vibrations into its central life. This comprehension and conceptualization has unfortunately proven to be a difficult challenge for most students to visualize.
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Figure 5-4
Rotate geometric forms with the mouse.
"Inscribed"
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Inscribing One Polyhedron in Another
There are many ways to inscribe one regular polyhedron in another. One useful case is that when each vertex of the first polyhedron lies at the center of a face of the second, and each face of the second polyhedron contains one vertex of the first.
The cube can be inscribed in this way in the octahedron, but also reciprocally. The same reciprocity holds between the icosahedron and the dodecahedron. Such polyhedra are said to be dual. The tetrahedron is self-dual.
The following ways to inscribe regular polyhedra will be useful too: one or two tetrahedra in a cube, and cube in a dodecahedron.
Figure 5-5
Dual Polyhedra
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Figure 5-6
Inscribed Polyhedra
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Figure 5-7
Star within the Grid of the Sphere
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Figure 5-8
Dodecahedron inside icosahedron.
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Recent microwave mapping of the universe indicates that the universe may be in the shape of a dodecahedron. The torus donut shape or spherical shape have also been proposed in cosmological theory. (Read more at the National Geographic website.)
The six-pointed star, a common occult and mystical symbol, can be found in the dodeca-icosohedron by crossing the six outer points diagonally across the inner spaces. These two interlaced triangles often referred to as the Morgan David glyph or Star of David are essentially a two-dimensional rendering of a three dimensional tetrahedron.
The three dimensional tetrahedron when seen as two interfaced vortices rotating in opposition directions hints at an even deeper and more fourth dimensional life.
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Figure 5-9
Rotate cubes and tetrahedrons with the mouse.
"The Master"
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Special magical properties have been ascribed throughout history to the geometric forms of the polyhedra and Platonic solids.
The ancients believed that certain geometric shapes had the power to draw unseen life forces to them as we have earlier demonstrated in our review of triangles and spherical grids.
Gaze at the domed ceiling of the Sistine Chapel or the cathedral of Chartres and the magic of sacred geometry becomes obvious. These Platonic solids can also be seen in many naturally occurring forms such as snowflakes, crystals, and the growth of flora and fauna.
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(Refresh the page to see the animation again.)
When the image above is set in motion, a series of concentric circles wind their way around from right to left until they return to their departure point. Midway through this circular movement when the colored rings begin to overlap themselves we can see the Vesica Piscis form emerge. It is here we can see the image of the heart, a classical symbol that has been used to depict the opening of intuition and love. The Vesica Piscis form should not be envisioned as simple two linear energies or forces meeting, but rather as spiraling energies that overlap, each occupying a different dimension or different quality of force. Words of power must traverse distances by this same sinuous spiral twisting motion.
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Geometry moves into a higher realm of power and influence and becomes occulty sacred when it is seen in the context of movement. All that truly lives is in a constant and ever-changing state of motion. This idea can best be seen through time-lapse photography — in the way a pinecone forms in concentric bands or in the growth of the petals of a flower. Each conforms to a predetermined series of polyhedra and unfolding geometric spirals. Even a fifty-year-old tree reduced visually to a few seconds through time-lapse photography is seen as a momentary flash similar to an exploding torrid or a spiraling fractal. Here we see nature's perfect patterns repeating themselves on infinitely larger or smaller scales. What can be said of natural forms can equally be applied to the inner worlds of energy and force.
Fractals and polyhedra geometric designs in their many and varied combinations and facets have an important relationship that can be seen in some of the images included in this section. They are a visual feast to the mind as well as the intuition when seen in the contexts of sacred geometry and how the different polyhedra can face any sphere with exquisite exactitude and sweeping grace.
Polyhedra are the visual archetypal principles and building blocks of all organisms as they transform themselves from the mundane to the sacred. They should be studied by all serious students who approach the divine initiatory experience from the second ray, which we know is the Divine Builder.
Why is the vesica piscis so invocative in many world religions and why does it hold such a high place in the pantheon of sacred geometry and golden ratios? It is the magical cross over or bisecting point where two or more vortices of energy overlap and meet. It is also a symbol of the mysterious cave of initiation spoken of by the Tibetan. Each organism in the hidden world of energy and force exceeds beyond its normal parameters of self-awareness by forcing itself into the streams of those lives greater and more advanced than itself. Where these two lives meet and crossover a new and more dynamic vortex or toroid is created. This is also the rationale for why the Will must be highly developed during the later part of this work and is emphasized in the DK's outlines of the third stage of building the antahkarana. Here we see the act of "projection", in which the disciple is asked to drive his personality and soul rays up onto triadic levels. The Will supplies the initial driving force and holds the creative tension for the second and third aspects to complete their important building work.
In the creation of a vesica piscis we see a vacuum or a vortex forming. It is through the process of invocation and evocation, which is governed by the law of sympathetic vibration, that one will learn the secret of all occult and esoteric bridge building.
Several examples of the vesica piscis:
Figure 5-12
Click to see larger image.
The center or crossover point beween two interfaced spheres or circles is the toroid or ovid seen here. Eventually these two organisms that share a common point or place of exchange will fuse or blend together making one entity.
Figure 5-13
The Flower of Life Patterns
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Click to see larger image.
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Multiple vesica piscis-s create the "Flower of Life", a key visual image as it shows how crossover points of different forces create endless patterns of opportunity, growth and interaction — multiple organisms that cross each other's path and move irresistibly into each other's periphery, each trying to learn the mysterious secrets of the other until the two or the many become the One. To the untrained eye these many vesica piscis images will simply reveal beautiful patterns. To the esotericist trained in the recognition of color, geometry, and sound they help reveal the underpinnings of the body of God and his magnificent tapestry of interconnected energies and forces.
On a macrocosmic level and in relationship to the Solar Logos and the seven Planetary Logoi we might see an image similar to the ones below. Like several other images that have been presented here its value lies in showing the transition or cross-over point between what we see as distinct petal formation and the whirling vortex.
Figure 5-14
Click to see larger image.
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Figure 5-15
Click to see larger image.
The seven divine rays seen as circulating vortices or toroids.
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"The solar system is (from the higher cosmic planes), seen as a vast blue lotus and so on down the scale; even the tiny atom of substance can be so considered.... The solar system is a twelve-petalled lotus, each petal being formed of forty-nine lesser petals. The planetary lotuses differ in each scheme and one of the secrets of initiation is revealed when the number of petals of:
a. Our earth planet
b. Our planetary polar opposite
c. Our complementary equilibrising planet
is committed to the initiate." (CF:1018)
The petals described here by DK are essentially spherical and rotating energies that reduced to a stationary image become petals. Technically the petals of any center or chakra symbolize the number of energies that are active in the vortex or spherical life. Without this somewhat simple and stationary description of the petals it would be impossible for the average student to have even a basic understanding of these hidden and elusive lives. On a more holistic level the petal formation of any occult entity should be visualized as a "whirl" that is in constant motion. In Figure 5-15 above, which should be viewed in a large format under high resolution, we have superimposed the vortex whirl idea with the petal formation which gives the viewer a more synthetic grasp of these great lives as they exist in their own natural state.
If the curser is positioned to the far left bottom corner of the square in Figure 5-16 and slowly drawn horizontally to the right, some idea visually can be grasped through animation how the initiating rod of fohat, symbolized by the electrical shaft, could create many variations and divisions of its own one essential life. The multiple figures appear to get smaller the further away from the original parent we move the cursor. Reverse this process and the disks appear to get larger until they all fuse back into the original parent.
Trying to put this idea of many diverse forces emerging from out of one primal parent into a chart or image form prior to the development of modern graphic animation techniques and you will begin to understand the difficulty for those in past centuries or cycles.
"Never does the geometrical pattern, the numerical progression or the Law of Correspondences break down in the understanding of the purpose and the plans of the planetary Logos — established before the worlds were created and finding their prototypes upon the cosmic mental planes." (TEV:90)
"We must resign ourselves to the fact that the only way in which we can find the clue to the mystery of the rays, systems, and hierarchies, lies in the study of the law of correspondences or analogy. It is the one thread by which we can find our way through the labyrinth, and the one ray of light that shines through the darkness of the surrounding ignorance." (IHS:6-7)
Correspondence and Analogy
Figure 5-17
Click to see larger image.
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