**
**

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Mathematics is the language of the Universe.**

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All things resonate to certain geometric frequencies - sometimes referred to as the Sacred Geometry.
**

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The doctrine of Kabbalah existed at the core of both the Hebrew
and the old Arabic religions. It was a traditional (oral) method of
handing down moral and occult truths. It was/is a metaphysical
system that defines and illustrates the nexus between God and
Man by means of symbolism and the alphabetical/numerical system
known as Gematria.
**

**
Within this ancient and complex system was an understanding of
the levels of 'existence' or rather of spiritual forms and forces,
of Intelligences that ascended from the microcosmic world of
man through a hierarchy of angels and archangels and principalities,
of powers, virtues and dominations, of thrones and cherubim and
seraphim to God, the omnipotent life force. Within this system
was also accounted a Qlipoth of demonic intelligences.
**

**
Kabbalism contains an 'experience' of many lifetimes of study and
meditation and spiritual development that is still pursued today
both within the Jewish faith and in the Western Mystical Tradition
of Qabalah. The disciplines of astrology and the Tarot are closely
tied to this tradition.
**

**
Within this great system of study works the Gematria that links
the lineage of number to the letters of the written alphabet(s) and
to the language itself. Likewise, each Arabic letter has a numerical
value and each number has significance. The names of and words for
all things contain numeralogical meaning , and thence comes a magical
linking between all things acrosstime and space.
**

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Groups of letters had significance and were used in amuletic inscriptions.
As they also had numeralogical significance the concept of Magical Squares
came about. When and where the signs of the entities or the spirits of
the planets and the invention of magical squares came about is not known,
but it is probable that they are of Sumerian or Indian origin.
**

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Great significance was attributed to the observed planets moving
around us and significant names were given to the planets connecting
them with entities and metals and qualities, founding both the science
of astronomy and the discipline of astrology. The knowledge of
arithmetic required by those who constructed magic squares was
considerable and it seems to have formed the foundation of the
Arab science of mathematics.
**

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The Kabbalists called the magical squares "Kamea", which translates
as "Amulet". They were worn as pendants and the word "Cameo" derives
from it.
**

**
**

The role of number in formulating the language of the Universe was perhaps most eloquently described by the Greek mathematician, Pythagoras, and his followers. It was he who tied together the fundamental concepts of ratios, geometries and harmonics. If the concepts of ratios, geometries and harmonics strike one as a little too abstract, they may be substituted with the more familiar terms of number, form and vibration.

Ratios simply describe the relationship between two or more numbers, such as the familiar 2:1 ratio that we associate with the octave in music. In fact, any single number may be represented by a ratio. Even the so-called irrational numbers like p and � 2 that are, by definition, infinite may be approximated by a ratio between two whole numbers. Our Universe is described by ratios that are incorporated into the forms (geometries) and vibrations (harmonics), comprising everything that we see, feel and hear. In essence, geometries and harmonics are simply expressions or symbols of number. So monumental was this Pythagorean understanding that it serves as the basis for everything from music theory to subatomic physics.

Whether symbols are spoken in the form of words (vibrations) or written in the form of characters (geometries), they are indeed based on number (ratios). It was the belief of the ancient Greeks, as well as a myriad of other ancient cultures, that number represents the sacred language of God or Spirit or the Creator. Ancient and modern philosophers have repeatedly raised the question of whether "God is number." Ratios between two or more lengths, areas, volumes, angles, spirals or tones were considered to represent a numeric code by which creation is manifested from the infinite unmanifested potential of the Universe that I will refer to as the Absolute.

The well-known Greek concept of logos, which is often used to denote the manifesting potential of the Creator, is actually rooted in the word meaning "proportion." Proportion refers to the relationship among two or more ratios. For instance, 1:2:3:6 or one is to two as three is to six indicates that the same proportion underlies both ratios. One could think of the relationship among numbers, ratios and proportions as being analogous to the relationship among letters, words and language.

Vibration

In many ancient traditions, numbers were considered divine in their roles as changeless principles underlying a world of matter that was constantly changing. In the Pythagorean tradition, numbers were based on harmonic motion (e.g., swinging of a pendulum or vibration of a string) and an entire cosmology was built on the laws of sound or tonal frequencies inherent in vibrating strings.1 Just in case one were to consider the Pythagorean ideas to be out of touch with modern science, the latest theory that physics has put forth to explain the nature of both matter and spacetime is known as superstring. Superstring theory supposes that particles are represented by vibrating strings that are about 1020 (100 billion billion) times smaller than a proton; such that each subatomic particle is represented by a single string vibrating at a distinct frequency. 2 In this theory, particles of the Universe are not fundamental, but are simply the harmonies created by an almost infinite number of these tiny vibrating strings. Astrophysicists have furthered this notion by postulating the existence of cosmic strings to account for the observed pattern of galaxies. So whether we are talking about quarks or quasars, it all seems to come down to vibration.

Vibration exits as waves that are present in a vacuum (e.g., outer space) as well as in a medium such as air and water. What we refer to as sound or tone is just waves traveling through a medium such as air or water, permitting the vibration to be detected by our senses of hearing and touch. Science tells us that visible light and audible sound are just two examples of a much wider electromagnetic spectrum (including microwaves, X-rays, radio, etc.) that may be distinguished by their unique vibration or wavelength. For the purposes of this discussion, vibration and waves are interchangeable inasmuch as they create and are created by each other. While air and water are ubiquitous media through which waves travel on Earth, this is not the case for most of the Universe. One might inquire as to the medium through which waves travel in the vacuum or void of "outer space." The ancient Greeks would have answered this question by saying that waves travel through the mysterious substance known as aether (i.e., akasha in the Hindu tradition). This aether has been redefined by modern physics as higher dimensions of space that are hidden within the world we observe with our five senses.

Vibration may be defined simply as an oscillation about a reference position (e.g., a pendulum or a pinwheel). Each oscillation is considered a cycle or a change from one state to the opposite state and then back through a common reference point. The number of times that a system moves through this oscillatory cycle per unit time is referred to as its frequency. We commonly use the time unit of one second to quantify frequencies, which are expressed as the familiar "cycles per second" or hertz. In this manner, we have a common reference period for comparing one or more vibrations.

For instance, the tone associated with middle A on the piano has a frequency of about 440 hertz, while red light has a frequency of about 1014 hertz and the rotation of the Milky Way Galaxy has a frequency of about 10-14 hertz. As such, the time required to complete one cycle ranges from less than a trillionth of a second for visible light to millions of years for a galaxy. Understand that there is nothing magical about the units of cycles per second (hertz) and that we could just as easily define our unit of measure as cycles per year or cycles per millennium or cycles per revolution of the planet Jupiter --- the choice is arbitrary.

Any vibration in the Universe may be described by its characteristic frequency or by its wavelength and energy. Therefore, vibration and waves and energy are simply different descriptions of the same thing. That "thing" is what comprises our world and, for the sake of consistency, I will refer to it by vibrational frequency rather than by wavelength or energy. However, keep in mind that vibrations, waves and energy are interchangeable. Every unique expression of that "thing" has a distinctive vibration known as its natural frequency. All natural frequencies include a unique fundamental vibration, as well as a set of harmonics or overtone vibrations. While it is beyond the scope of this article to explain how the harmonics are generated, it is enough to understand that there are no pure tones (i.e., those without harmonics) in nature.

Pythagoras was the first to explain why two or more vibrations or tones, referring to frequencies within the audible range, may result in a sound that is either consonant (i.e., harmonious or pleasing to the ear) or dissonant (i.e., disharmonious or disagreeable to the ear). In essence, the degree of consonance or dissonance is a function of whole number ratios. If whole numbers in the ratio are small, such as 1:1 (unison), 1:2 (octave), 2:3 (fifth) or 3:4 (fourth), then the resulting musical sound is consonant. If the whole numbers in the ratio are relatively large, such as 8:15 (major seventh) or 15:16 (minor second), then the resulting musical sound is dissonant. This rule holds whether we are talking about two different fundamental tones or a fundamental tone and one or more of its overtones (i.e., harmonics). The common thread is number, which underlies all natural frequencies and the countless harmonics that are inherent in them.

Form

Ancient philosophers considered that the orderly principles or Laws of Nature giving rise to the physical world were not only knowable, but describable using numbers. Born from the aether or akasha, the four Elements (i.e., fire, air, water and earth) were considered to be vibratory in nature and represented by number in the form of frequencies or tonal ratios. These same ratios were used to describe the relationship among angles, which give rise to the characteristics (e.g., vertices, faces, edges) of a very special class of 3-dimensional (3-D) geometries known as the Platonic solids. The Platonic solids were believed to make symbolically visible the orderly movement from the Absolute to the endless array of interconnected forms that we know as matter or the manifested world. According to Robert Lawlor, 1 "the Platonic solids serve as essential forms and numbers that interface between the higher and lower realms and that possess, through their analogues with the Elements, the power to shape the material world." In other words, matter is created through the number or ratios contained within these Platonic solids.

The five Platonic solids were considered essential because they are the only volumes which have all interior angles equal and because their rotation or spin generates a sphere. Sacred geometry associates the sphere with the unmanifest and, as such, the Platonic solids are the only angular 3-D geometries that form a perfect interface with the Absolute. All other volumes or 3-D geometries in the Universe are created from these five, which include the tetrahedron (fire), octahedron (air), cube (earth), icosahedron (water) and the dodecahedron (aether). Because they symbolize the creation of matter itself, the Platonic solids are the likely precursors of many familiar written symbols or glyphs. For example, each Platonic solid is composed of faces representing the triangle, square and pentagon, which constitute the template for many glyphs. Similarly, a 2-D representation of the star tetrahedron yields the six-pointed Star of David and symbolizes the yin-yang.

It is through these spinning or rotating Platonic solids that our material world is connected to the Absolute and that time is intrinsically linked with movement in space. Not surprisingly, the same ratios that describe consonant tones in music can be derived from the ratios of angles, faces, edges, vertices and other parameters describing the Platonic solids. For instance, just the ratio of vertices to faces yield the unison (1:1 in the tetrahedron), the fourth (3:4 in the octahedron) and the major sixth (3:5 in the icosahedron). Similarly, the ratio of vertices to edges yield the octave (1:2 in the octahedron) and the fifth (2:3 in the tetrahedron, cube and dodecahedron). Physics has shown us that matter is created, not from fundamental particles, but from waves or energy or vibrations that are brought to focus at a point in spacetime. As such, matter is simply a special manifestation of energy and the two may be converted according to Albert Einstein�s famous equation, E=mc2.

Form is no different than vibration, and both are underlain by a sacred code that we refer to as number. The famous British scientist and inventor, Buckminster Fuller, considered there to be only two possible covariables in the design of the Universe.3 The first of these is frequency (for modulating waves or vibrations or energy) and the second is angle (for modulating forms or geometries or matter). But angle and frequency are just different expressions for the same variable --- number. Herein lies the root of the ageless question asking whether God is number. Is God (a.k.a. Creator, Absolute, Spirit, Source) actually number or does number represent a more fundamental expression of nature?

� Cycles

To answer the question of whether number represents something more fundamental, we need to return to our discussion of vibration. Cycles are fundamental to the Universe inasmuch as everything is a vibration. Recall that vibrations are defined as cycles per unit time, but time itself represents just another cycle or portion of a cycle. In essence, the only way in which we are able to describe or quantify a vibration is in terms of a ratio of cycles; otherwise, the concept of a cycle�s "duration" is utterly meaningless. Let�s examine why this is true. If we use the familiar units of hertz, then all cycles are referenced to the time increment of one second. But what is a second? A second is simply a cycle described by one 86,400th of the Earth�s rotation on its axis. Our most accurate measure of a second is the periodic ejection of a particle from the nucleus of a cesium atom. In both cases, a second is nothing more than a cycle or a portion thereof.

Cycles repeat infinitely and are independent of the relative measure we use to quantify them. Whatever we select as our reference cycle will simply change the ratio and thus alter the numeric value of frequency; however, the cycles themselves are obviously unchanged. Hence, ratios are necessary to describe all vibrations because one cycle can only be described in relation to another. Stated most simply, vibrations express the ratio of one cycle to another. While this notion may seem odd to us, many ancient cultures were well aware that all phenomena are cyclic and that cycles are the basis of spirals, vortices and a myriad of other natural forms. If the four Elements, which were believed to be the basis of music, are represented as a repeating cycle of four ascending musical notes, an expanding spiral is created.4 The proportion of each Element used to create matter is described by tonal ratios or musical chords, thus illustrating that it is the ratio of cycles which are coded by the numbers describing vibration and matter.

One of the ancient cultures that left behind the most detailed description of cycles (e.g., seasonal, lunar, solar and galactic) was the ancient Maya of Central America. Their calendrics and understandings of how seemingly unrelated cycles were synchronized to one another is so precise that only now are we able to recognize their significance. The Mayan calendar is derived from a more fundamental 260-unit radial matrix known as the Tzolkin, which codes cycles as number (e.g., ratios, harmonics, geometries) and accurately describes things as diverse as planetary orbits and the sequencing of DNA. These descriptions are facilitated by calculating the relationship among cycles that otherwise appear to be independent.

For example, the Mayan system tracks the waxing and waning of sunspots on the surface of Sun such that shorter cycles (tens of years) are nested within longer cycles (hundreds of years) that are nested within even longer cycles (thousands of years). These shorter cycles accurately predict the solar activity that routinely disrupts Earth�s upper atmosphere, affecting telecommunication systems and satellite orbits. The longer periods predict changes in tropical ocean temperatures, global climate trends and human behavior patterns.

The so-called Great Cycle of the Maya is 5,125 years in length. The current Great Cycle is believed to have begun in 3113 bc and will end in 2012 ad, at which time we will enter yet another Great Cycle --- that�s the thing about cycles! Interestingly, archeological and geological records dating back to the beginning of this Great Cycle (~3000 bc) suggest a dramatic cultural shift that occurred simultaneously all over the world. 5 Furthermore, it appears that the modern version of the El Nino phenomenon began about the same time, as perhaps did the movement of our Solar System into an interstellar cloud or bubble. It is fascinating to note that science is discovering actual links between these cultural, climatic, solar and galactic events --- all of which are cyclic.

The tabulation of cycles as rhythms gives rise to our perception of time and space. In other words, counting cycles gives rise to 3-D space and linear time and thus relates number (quantity) to vibration and form (qualities). Since both vibration and form are cycles, number is simply a coding of cycles. The idea that our perception of "reality" is based on counting cycles is one that is supported by more than the ancient Greeks and Maya. Recent neurological research has indicated that the common rhythmic pulsing of synchronized nerve cells throughout the body generate the brain wave states (e.g., alpha, theta, delta) that have been recognized and quantified for many years.6 The rhythmic electric output of synchronized neurons throughout the body give rise to our visual perception and to brain functions such as memory and attention.

According to the Mayan understanding of a Universe as cycles within cycles, it is ridiculous to separate time from space when, in fact, both are cyclic. The illusion of our senses tells us that space, as a rather arbitrary 3-D cross section of spacetime, is simply pushed along the axis of linear time. There are those who believe that recent popular interest in the Mayan calendar signals a change in the collective perception of time as cyclic, rather than linear. This may actually permit us to experience a 4-D world rather than a temporally constrained 3-D one.

This is not to say that time is exclusively the fourth dimension because we could have four or more dimensions of space (n) folded upon each other, always leaving time as the n+1 dimension. Astrophysicists postulate that, immediately following the Big Bang, ours was a 10-D Universe (9 space + 1 time). The energy of this new Universe was so intense that this arrangement could not be sustained and, through the breaking of symmetry, quickly split into our familiar 4-D (3 space + 1 time) Universe. The remaining 6 dimensions of space were folded or rolled-up into our 4-D reality, where they may comprise what the mystics referred to as aether. The mathematics of superstrings suggest that unification of physical forces occurs in both 10 and 26 dimensions, the latter of which may characterize other Universes

Perhaps the most poignant of ancient texts regarding the importance of cycles is the so-called Emerald Tablets, which are believed to have been left behind by the great master of ancient Egyptian writing known as Thoth. In his translation of the tenth Emerald Tablet, Doreal writes, "Chanted they the songs of the cycles, the words that open the path to beyond. Aye, I saw the great path opened and looked for an instant into the beyond. Saw I the movements of the cycles, vast as the thought of the Source could convey."7 If the nature of the Absolute is indeed cycles, then number appears to be the sacred code.

- D.L. Marrin

FIBONACCI Born: 1170 in Pisa, Italy The "greatest European mathematician of the middle ages" He changed the face of mathematics Wrote a number of important texts which played an important role in reviving ancient mathematical skills The mathematics of creation... sequences... 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. Fibonacci and His Numbers Life and Numbers of Fibonacci Fibonacci Numbers and Nature ## MATH ARTICLES

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