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click to open   3. Dimensions
.    Yantras
.    The Kala Yantra
.    Quaternity
.    Double Tetrahedron
.    Fractals
.    Fractal Prisms
.    Transdimensional
.    Three Dimensions
.    Fourth Dimension
.    Fibonacci
.    Fib. Dimension
Kalarhythms & Sacred Geometry
Sacred Geometry - Double Tetrahedron - small

Yantras, as described by Alain Danielou in The Gods of India (Inner Traditions Ltd, NY 1985): "are geometrical figures made from linear elements intended to represent, in a synthetic form, the basic energies of the natural world, which are the deities. They are the visual equivalents of the mantras or thought-forms. The yantra has the mantra as its soul. The deity is the soul of the mantra [sacred thought-form expressed in sound]. The difference between the yantra and the deity is similar to that between a body and its soul." (Kaulavaltyan, quoted in Sabda-kalp-drum. [596]) "Yantras are believed to be the basis of the natural, or true, ideograms from which writing is derived. ... Yantras can express the inner aspect of any form of creation. The nature of men and animals as well as that of gods can be expressed through yantras."

The center space and dot symbolize the zero-point limit(1) of manifestation, Bindu, the extensionless mathematical point which is the origin of manifestation. It is from Bindu that creation must emerge to transcend non-being into being.

The Star of David, or Mogen David, is formed by the merger of to equilateral triangles, or, trikonas. It became a Jewish symbol as the shield of God, protector of David, and the guard against evil. Theologian Franz Rosenzweig interpreted the symbol to mean God, world, and man; creation, revelation, and redemption. The Star of David comes from antiquity, and the Yantra.

Kala Yantra title

Kala Yantra - 1
The Kala Yantra represents a fractal prism of energy. The Physical, Emotional and Intellectual triad are depicted with two over laping triangles, (the Shiva/Shakti trikonas, or the star of David). The double triangle has three axes. The double triangles of Hindu mythology, however, are said to be a two dimensional representation of a three dimensional reality. The significant energy field inside of planets, according to astronomer Richard Hogland, and all life forms, according to Drunvalo Melchizedek, are overlapping tetrahedrons, one pointing up, and one pointing down. This is the double tetrahedron, or star tetrahedron. The double tetrahedron has four axes. What then is the fourth axis beyond Physical, Emotional and Intellectual? It is time.

The Kala Yantra depicted here is conceived by the author with the suggestion that is can provide a model for our triad of energies and various combinations. All energies influence all things with a trirhythmic effect of varying degrees. The Triangles symbolize our basic energies, Physical, Emotional and Intellectual. Three Circles symbolize our three Rhythmic Cycles. The Squares symbolize how each of the three Cycles are divided into four quarters by the Universal Pattern.

Quaternity is revealed when we look our makeup and the subtle level of creation. Mort Gale identified a fourth Biorhythm in his book Biorhythm Compatibility (Warner Books, New York, 1978). Beyond the 23-day Physical Cycle, the 28-day Emotional Cycle and the 33-day Intellectual Cycle, Gale holds that we have a 38-day Intuitive Cycle. Carl Gustav Jung identified four functions of the human psyche, the "sensory-function," the "feeling-function," the "thinking function" and the "intuitive function." Einstein and other scientists theorize that beyond the dimensions of Height, Width and Depth, we may consider the fourth dimension to be Time. In the same way that individuals grow through physical, emotional, intellectual and spiritual phases, the collective of humanity has transcended through stages that relate to what Ken Wilbur calls the physiosphere, biosphere, nooshpere and the theosphere. In each case, perceiving the fourth aspect means going beyond the corporeal and looking into the subtle realms. In the subtle fourth may be aspects of higher realms that intersect with our triadic universe.
Whereas the duality of Yin and Yang is part of, and encompassed within, a Triadic Universe,
so too may Trinity be encompassed by Quaternity.
Double Tetrahedron title

The Double Tetrahedron, or star tetrahedron, is a three dimensional object. It consists of a four faced pyramid, called a tetrahedron, that faces up, combined with a tetrahedron facing down. It is the three dimensional counterpart to the Yantra.

The Four Axes of a Double Tetrahedron

Polarities - Physical Emotional Intellectual Spiritual

In the geometric form known as the Double Tetrahedron are four pairs of opposing vortices with four axes connecting them. Modeled here is the intersection of four axes that suggest a fourth energy beyond the triad of Physical, Emotional and Intellectual. That fourth energy could be Spiritual. The Spiritual axis is a transcendental awareness that is metaphysical, beyond emotional, and not arrived at through logic. It is our mystical side. This too can move in Rhythm.

The Double Tetrahedron, or Star Tetrahedron, is a three-dimensional object. It consists of one tetrahedron (four-faced pyramid) that facing upward, combined with another tetrahedron facing down. This three-dimensional energy is known as the Flower of Life that is found throughout creation. The Double Tetrahedron is the shape of an energy field around matter and living things. The original eight cells of a developing fetus, known as the stem cells, arrange their spheres in this pattern, whereas each vortex is centered in the middle of each cell. This significant energy field has been found to exist inside every planet by according to astronomic researcher, Richard Hoagland, who helped NASA with Pioneer space probe project. One axis connects planetary poles with two vortexes while the remaining axes bring their vortexes to planet surfaces with three at 19 1/2> degrees north and three at 19 1/2> degrees south. This manifests energetically as volcanoes on earth vortexes or other disturbances like the great red spot on Jupiter. A Double Tetrahedron also models the energy fields around all living things is the basis for counter-rotating fields of energy known as the Merkaba (aspects of which are taught by Drunvalo Melchizedek).

Double Tetrahedron

In that the Double Tetrahedron has four axes in a three-dimensional field, it may serve as a model for a quaternity of energies, Physical, Emotional, Intellectual and Spiritual. A person, a group of people, a nation-state, or the global collective has a body, heart, mind and spirit in its collective conscious. Physical, Emotional, Intellectual and Spiritual aspects recur with the Double Tetrahedron of energy fields in a fractal universe.

. The Math Provides the Form, The Form Provides the Energy,
and the Energy Shows Up in the Now of a System.
The Double Tetrahedron is realized through the Yantra, or Star of David. As noted by Danielou:
Yantras, though drawn on the flat, must be conceived as solids. The flat drawing is a mere suggestion of the three-dimensional figure which the yantra is. The three-dimensional yantra is itself but a static image of the moving, living combination of forces represented in a particular divinity.
The Yantra, therefore is like a shadow of the three-dimensional Double Tetrahedron. Polarities - time space

Any business or organization that functions inside a nation also has a body, heart and mind, like the nation. It is a microcosm. Its Physical, Emotional and Intellectual aspects become a fractal lens of energy through which the Physical, Emotional and Intellectual energies of the nation will flow. The double triangle model depicting the three energies, however, does not show durations of the Cycles. It does not show anything for time. Time is the fourth axis. That is how the two dimensional model of triangles becomes the three dimensional model of the double tetrahedron.

Absoulute time is the fourth axis. The idea of time being as an axis may be expressed in Hindu mythology as akhanda dandyamana, the indivisible rod of time. As the fractal lens of the Physical, Emotional and Intellectual triad moves through time, it rides on the fourth axis. As the triad intersects with absolute time, it creates relative time. It is from this that we get the undulation of Rhythm. As there are many microcosms, great and small, fractal lenses differ with their own relative time. That is why time is not the same everywhere. One relative time is different then another. This is the geometry of time.


Fractal, as a term comes from Bernoit Mandelbrot who indentified fractal geometry. Mandelbrot is a mathematician and IBM researcher who became fascinated with the study of self-repeating patterns. A shape that repeats its pattern is a fractal. With this in mind, Mandelbrot developed a mathematical formula for plotting dots on a plane. It is a simple formula that produces complex and beautiful shapes. What makes it a fractal is that it produces infinite self-similar patterns within patterns. It has been plotted on computers with startling results that are striking to behold. One can zoom in on any part of the pattern with computer magnification to find intricate similarities on different scales. To zoom in far enough anywhere in the pattern is to go through a myriad of patterns that eventually reveals a shape resembling the starting point. This is the Mandelbrot Set.

Mandelbrot created the name "fractal" in 1975 from the Latin adjective fractus, or frangere in verb form, which means to break. Fractals are everywhere.(2) Elements of the macrocosm and the microcosm were seen throughout time from ancient mystics, and medieval visionaries, to modern day scientists. Seventeenth century scientist and philosopher Wilhelm Leibniz expounded on the notion of a universe made up of monads that contain miniature universes. Modern author Michael Talbot described a "Holographic Universe" in his book by the same name. A hologram proves to be a significant analogy. By shining a light through any piece of a hologram picture, one can reproduce the entire picture from which it came. The light of the spirit shines through all things in creation. It too reveals the Oneness of it all through everything within it, including us.(3) Our average number of respirations per day, and our number of heartbeats every six hours, each equal the number of years in the precession of the equinoxes, (a 25,920-year Cycle of where the earth is in its orbit on the first day of Spring). We find ourselves as part of it all.

Fractal Prism title

Continually we find correlations between things that may have once seemed separate. Religious symbols reveal parts of our being. Changes in a person's life give clues to the passing of the ages. Ancient scriptures may seem more up to date than tomorrow's newspaper. The Mayan Calendar affirms the timing of Spiritual movements. Our number of heartbeats in one day correlate with the precession of the equinoxes over thousands of years. Dimensions in our bodies hold formulas found throughout nature. Through these, and all others things, basic forms reappear in isomorphic (changed form) variations to reveal how simplicity breeds complexity with a universal DNA. How is it that we can find a common denominator, again and again, between so many things? The answer comes in what they all have in common. The energy fields of existence.

Fractal Kala Yantra

As we discover ourselves and the world we are part of, we realize that we are star seeds to a holographic universe that progenerates through the microcosm and the macrocosm with never ending reflections of the Creator .(4) Time and again, the Double Tetrahedron, depicted above, reappears as a Fractal Lens of Energy. Physical, Emotional, Intellectual and Spiritual aspects of our being recur in living systems from a person, a nation-state, to the world collective, and more. If these energies recur from tetrahedral form, then the Double Tetrahedron is transcending scales of Space to encompass living systems great and small. When we note the Cycles of Change in living systems that move with Rhythms and Fractalrhythms, the Double Tetrahedron holds a key to the transdimensional recurrence of Time.(5) In looking at the tetrahedral shapes from mineral growths to planetary energy fields, the Double Tetrahedron may be a basis for the manifestations of Matter.

The two major Tetrahedrons that form the Double Tetrahedron may represent the duality of the universe with yin and yang. Triangular shapes throughout the Double Tetrahedron may represent the triadic universe. Its four axes between vortices may represent the Quaternity of energy. The two major Tetrahedrons join to make eight smaller tetrahedrons in a fractal relationship that may represent transcending patterns between the microcosm and the macrocosm, whereas, as above, so below. In the Unified Field of self-referencing super-strings in creation, things go forth and multiply by making complex variations of simple elements. The Fractal Prism of Energy may therefore be an isomorphic form of a universal principle for time, space and matter.

Transdimensional Recurrence is the fractal relation between the microcosm and the macrocosm. Rhythms as part of the whole in a universe of Rhythms are addressed in Chapter III: More Parts of the Pattern under Fractalrhythms. When a pattern within a life, or thing, repeats on smaller scales within itself, or larger scales around itself, it is a fractal. Fractals are one form of Transdimensional Recurrence. Patterns of self similarity on smaller and larger scales might exist ad infinitum in a myriad of ways. Repeating patterns can relate to Rhythms of time, contours of space, and geometric shapes of matter.

Nature is replete with Transdimensional Recurrence. Specialists have observed this and given it different names in their own fields of expertise. Physicists see Doubling, economists note Scaling, philosophy cites Monads, (miniature universes), biology knows Biogenetic Law and DNA; mathematicians have Fractals, Mandelbrot set, Cantor sets, and the Koch curve; other fields have more. All of these have Transdimensional Recurrence. This term unites many partitioned observations of the same principle and gives them a single name that applies to all, Transdimensional Recurrence.
Perfect Transdimensional Recurrence occurs when a smaller scale of self repeating patterns is evenly divisible into a larger one. A minute with sixty seconds divides evenly into an hour with sixty minutes, for example.

Imperfect Transdimensional Recurrence occurs when a smaller scale of self repeating patterns is not evenly divisible into a larger one. The number of lunar orbits around the Earth is not evenly divisible into the Earth's orbit of one year around the sun, for example.

Transdimensional Recurrence happens when any discernable pattern repeats on different scales of size or time, even if in similar or varied form. A self-repeating pattern is a fractal that transcends a microcosm and a macrocosm with patterns that can relate to Time, Space or Matter. Researchers have given this different names in their respective fields. Physicists see Doubling, economists note Scaling, philosophy cites Monads, biology knows Biogenetic Law and DNA, mathematicians have the Mandelbrot set, Cantor sets, and the Koch curve; other fields have more. All of these are fractals. Other forms of Transdimensional Recurrence come as self-referencing patterns.

Higher and lower dimensions still intersect. When I was a school boy, my teacher read us a story by a mathematician who created a tale of traveling to lower dimensions. First, he passed through the second dimension where all beings were polygons on a plane. His head was first to go through the plane as creatures saw him suddenly appear as a circle. As he transcended the plane from the tip of his head to his ears, the circle got bigger and developed wings. The circle became smaller for his neck, then larger for his shoulders. As he continued to ascend, two more circles formed for his arms only to be replaced by five circles each for his fingers. Two dimensional beings in this world were surprised by the emergence of other still more polygons, then, as he passed through the plane, the two polygons that were his feet disappeared. No one knew where these circles came from, or where they went. They could not understand that the separate circles were part of the same system in a higher dimension. Next, he passed through the one dimensional world where flat beings lived on a line. Beings there saw him emerge as a short line for his head that got longer for his ears, shorter for his neck, longer again for his shoulders and so on. Additional lines appeared for his limbs, then mysteriously disappeared. Again, to the mystery of all.

torusIn the mathematician"s story we can conceive how lower dimensions are contained in our world and how we intersect with higher dimensions. Duality of the two-dimensional word where Yin and Yang were born exists within our triadic world of three dimensions. At the point where higher dimensions intersect with us, we experience them as they cast their "flat" shadows in our world. That is why we experience linear time while the next dimension experiences past, present and future simultaneously in universal time. Clues to Transdimensional Recurrence are around us.

three dimensions title

Three Dimensions by which we define space were described by Euclid as height, width and depth. Our three-dimensional world also contains two-dimensional worlds and one dimensional worlds within it. The zero dimensional point-limit, Bhindu, is without beginning or end. It is from Bhindu that all dimensions spring forth.

polarity - one-dimensional

One-dimensional existence came first.(6) A rough model of this can be envisaged as a simple line. A point on the line has one axis of motion, back and forth. This line intersects with no other and has no other direction. A point on this line cannot go around another point or jump over it. Only one dimension exists.

Two-dimensional existence came as the first dimension intersected another dimension with a 90-degree axis to create a bipolar world.(7) This formed new directions that exponentially increase motion. A rough model of this can be envisaged as a plane. Any point on the plane has 360 degrees of movement. Points can move around one another, but they cannot jump over each other. A two-dimensional world contains one-dimensional worlds within it, but there are still no up and down.

polarities - two-dimensional
polarities - three-dimensional

Our three-dimensional world was created when two dimensions intersected a third dimension at another 90-degree axis. New directions exponentially increased motion again. The three-dimensional world expounds with lower dimensions within it and allows greater manifestation of the divine triads.(8)

The one dimensional stratum holds a key between the unmanifest (avakya) and manifest nature (Prakriti). Something is, or it is not. That which is manifest must cross the first dimension. From the two-dimensional stratum comes bipolar duality, such as, on and off, left and right, in and out, and more. Duality of opposites is a basic principle of yin and yang. Yin and yang bring motion between the two, and motion over time equals Rhythm. In 3-D strata, Rhythms runs in triads of energy. We see this in the three parts of the self that mirror creation.(9)

We exist, and have made it through the first dimension. We have bipolar duality in the form of symmetrical bodies and live as male as female because we have made it through the second dimension. We manifest a triad with a mind, heart and body because we live a third dimension. What is next?

fourth dimension title

siddha loka, akhanda dandayamana
[definition pending]

Fibonacci Sequence title

The Fibonacci Sequence is a recursive number sequence named after Leonardo Fibonacci, an Italian mathematician who published Liber abaci in 1202. He calculated a response to "How many pairs of rabbits can be produced from a single pair in one year if it is assumed that every month each pair begets a new pair which from the second month becomes productive?" Fibonacci#39;s response was a sequence wherein each number, except the first, is attained by adding the two previous numbers together. Hence his self-referencing sequence (from the formula Xn = Xn-1 + Xn-2):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . . .

What makes this pattern phenomenal is that it is replete in nature. Branch growth in many plants exemplify this (with 1 branch, for example, that yields 1 more branch which splits into 2 branches with 3 stems each that have 5 smaller stems and so on). The number of counter-rotating rows in pine cones and sunflowers exemplify this (with perhaps 8 spirals going left and 13 going right, or 21 spiral spirals going one way and 34 spirals going the other way and so on). The relationship between any two consecutive numbers approximates the phi ratio (1.618) developed by Eighteenth century mathematician Leonard Euler. This ratio is in the Golden Mean Spiral found in plant and animal life like Nautilus shells and more. The Fibonacci Sequence is therefore a basic underlying principle in nature.

Fibonacci Dimensions

Polarities of Various Dimensions?

Fibonacci Dimensions

For something to come into existence it must first transcend the boundary between being and non-being and emerge from Bhindu. Therein lies the first axis, 1. It enters the first dimension where is has the single axis of linear motion, 1. After that, another axis intersects at a 90-degree angle to create the second dimension with duality, 2. Then comes our third dimension with another axis intersected at a 90-degree angle with Euclidean geometry, 3. Next comes the 4th dimension with two additional axes of universal time and omnipresence, 5. This may be followed by the 5th dimension with eight poles of expanded dimension, 8, and so on. The Fibonacci Sequence in nature may be a shadow from how polarities change dimension. This is theory presented to provoke thought and productive discourse for where it may lead.


We are part of forces that go beyond us. As we go beyond ourselves, we return with self realization.(10) R. Buckminster Fuller wrote, When we approach our problems on a universal, general systems basis and progressively eliminate the irrelevancies...We gradually uncover you and me in the heart of now. The Now we are in is a system of change that locates time as a subset of eternity. We are living eternity now and will therefore transcend all systems within it until we realize our individuation as synonymous with all. "We are not human beings having a heavenly experience, we are heavenly beings having a human experience," said one. Seek the God within.

K. David Katzmire
Om Namah Shivayah
1. From the Works of Chuang Tzu of Taoist Scripture: "For man's intellect, however, keen, face to face with the countless evolutions of things, their death and birth, their squareness and roundness, - can never reach the root. There creation is, and there it has ever been."

2. Vivasara Tantrain: "What is here is elsewhere."

3. Hsi T"u: "All movements of the world are constantly subject to one and the same rule."

4. The Flower Garden Sutra of Buddhism, also the Avatmasaka Sutra in Sanskrit, and the Chinese Hua Yen Sutra: "In the heaven of Indra there is said to be a network of pearls, so arranged that if you look at one you see all the others reflected in it. In the same way each object in the world is not merely itself but involves every other object and in fact is everything else."

5. From the poet Wu-Men of 13th century China: "One instant is eternity, eternity is the now. When you see through this one instant, you see through the one who sees."

6. Chandogya Upanishad 6.2.1-2. [6]: "In the beginning, my dear, this world was just nondual Being (sat). To be sure, some people say that in the beginning this world was just nondual non-Being (a-sat), and that Being arose from non-Being. But how could that be? How could Being be produced from non-Being? In the beginning this world must have been pure Being, one and without a second."

7. This gives us the bipolar duality of the Universe as represented in Yin & Yang.

8. Lao Tzu, Chapt. XLII: "Out of Tao, One is Born; Out of the One, Two; Out of Two, Three; Out of three, the myriad things."

9. From Chineses mythology: "Yet these three Sages of Taoism are but one First Cause, that is, the One Indivisible Monad, called Tae-keih [the Supreme Self]."

10. Taittiriya Upanishad 2.1 [11]: "He who realizes the sphere of space hidden in the cavern of his heart grasps all that my be desired and comes into contact with the Immensity."