Sacred Mathematics, by way of definition, refers to the concepts which, according to ancient (and a few modern) scholars, encompassed all of creation in mathematical terms.  It includes numerical ratios of the most profound significance (literally Transcendental Numbers), the nature of numbers themselves (Numerology), their mutual relationship (Astrology), strange beasts such as Magic Squares and Infinite Series, those astounding aspects of geometrical relationships in Sacred Geometry which seem to defy logical or rational explanation, and the manner in which all of these aspects describe the universe.  

Sacred Mathematics represents a wholly contained and internally consistent philosophy -- a philosophy which describes physical reality, its cosmogony and every aspect of its science.  The Greek Philosopher, Plato, has said: “Geometry is knowledge of the eternally existent.  Numbers are the highest degree of knowledge.  It is knowledge itself.”           

This profoundly philosophical interpretation or Mathematical Theory is echoed in many spiritual traditions.  Such references are not always direct, however, and in fact may have been intentionally obscure in order to prevent the “uninitiated” from gaining access to the power inherent in such knowledge.  The School of Pythagoras, for example, was known to limit the dissemination of the mathematical and philosophical understandings they achieved.  The so-called Mystery Schools of ancient Egypt, Sumeria, Greece, and Judaea were also very much into keeping their own counsel and limiting access by the uninitiated.  The biblical prohibition from eating of the Tree of Life or Tree of Knowledge of Good and Evil may represent another case of a little knowledge being a dangerous thing -- at least to someone's thinking.  

The same can be said of the modern world, where mathematics are taught in such a dismal and inadequate manner, that mathematical anxiety has become a psychological condition semi-officially recognized by the public school systems, and in some respects encouraged as a means of convincing the majority to avoid like the plague the very idea of numbers and geometries.  Accordingly, the odds are very good that those who think of themselves as mathematically untalented (those who might quickly refer to A Non-Mathematical Digression, for example), may find they know far more mathematics than they might otherwise have expected.  Music, for example, is a primary illustration of the ability of individuals to unknowingly appreciate geometry and mathematics, even when they’re not serious left brain thinkers.  

The key in this regard [pardon the pun] is that Music is basically about ratios, frequencies (geometrical sine waves), and timing (a very mathematical kind of thingamagig).  There is also a strong geometrical connection, in that, if one takes the unique 3-4-5 right triangle (with sides of 3, 4, and 5 equal units -- the only triangular shape utilizing a combination of single digit whole numbers which result in one angle of the triangle equaling 90 degrees --however, larger numbers such as 5, 12, and 13 also work), and strings a continuous fine wire to each of the three points of the triangle, it is then possible to tune one of the sides to a particular note, and have the other two sides be in a tuned harmony.  The three sides of the triangle form a series of tones that are equivalent to the first three strings of a tuned guitar.  (One might also note that geometrically, one only has to tune one side of the triangle, while the guitar requires all three strings to be individually tuned!)  

Other examples of geometrical knowledge not always appreciated include:  The well known Vesica Pisces (at least in one of its interpretations), The Great Pyramids (which are readily visualized and understood), and diamonds of various cuts (i.e. geometries). The latter are often fully understood and greatly appreciated by the mathematically disinclined (if not, in fact, a set of geometries quickly and accurately appraised from a financial view point by the truly discerning individual -- i.e. most of the females on the planet).  Initially, one may not be familiar with the distinction between a dodecahedron and an icosohedron (or a Rose cut), but once enlightened as to their definition and practical aspects, examples of the geometries suddenly begin showing up most everywhere.  

Sacred Mathematics is not exclusively about geometry, however.  There are, for example, Magic Squares , Fibonacci Numbers, numbers in general (e.g. Nines), Numerology, and a whole host of other strange relationships among the very real aspects of the universe.  

Pythagoras (of Pythagorean Theorem fame) taught that the motions of the planets, the Moon and Sun (as well as an invisible Anti-Earth on the other side of the Sun), created the Harmony of the Spheres, but which ordinary people could not hear because they were too accustomed to it.  [Either that, or an ancient Napster got caught in the act!]   

Pythagoras and his followers made no distinction between music, mathematics and magic. Magic and music, for example, were found to be based upon mathematical laws (albeit, sometimes very subtle ones).  Furthermore, everything in the universe followed these mathematical laws and the universe was created out of the geometrical relationships of the numbers, and thus constituted the true basis of reality.  Which says a lot.  

Pythagoras believed that each number was holy and had its own powers. One was the indivisible monad, creating everything out of itself.  Two was the pure duality, perfect balance between opposites. Three was the number of the gods, while four was the number of the material world (hence the four elements). And so on. This became the foundation of the Numerology that the Order of Hermes ultimately adopted.  The Tarot is also strongly reminiscent of Pythagorean thought in this regard.  

Plato -- in addition to being a geometry nut (the Platonic Solids are named after him; see also A Graphics Description) -- was also into the numbers. So to speak.  After Socrates had been executed -- allegedly for the horrendous crime of accepting money from those he taught!  (Imagine!) -- Plato left Athens and traveled to Egypt, Sicily and Italy.  At the latter pit stop, Plato learned of Pythagoras, and quickly came to appreciate the value of mathematics.  Based on ideas Plato gained from Pythagoras’ disciples, Plato decided:  

            “...that the reality which scientific thought is seeking must be expressible in mathematical terms, mathematics being the most precise and definite kind of thinking of which we are capable.”  <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html>  

Plato even came to believe strongly in the importance of the numbers 216 and 12,960,000.  There was no indication in Plato's writings as to why these two numbers were important, but we can note that both reduce to Nine.  (See Numerology for an explanation of reducing numbers.) However, as one reader was quick to point out, 216 = 6 cubed), while 12,960,000 is 60 to the fourth power. This is all very Sumerian, in that 6 and 60 were part of their segesimal mathematics. What is probably not particularly Sumerian, however, is the curious fact that one could plausibly write: 216 = 6x6x6, or dropping the multiplier symbols (a common practice in mathematics and science), one obtains 666. This is, of course, the number of the beast inRevelations -- as well as the number of gold talents received in a year by King Solomon. Apparently, Revelations was not enamored with either Plato, King Solomon, or the Sumerians. (Alternatively, the "beast" in Revelations is man, and man being an animal... perhaps the most dangerous beast is simply man.)

As for 60606060... Perhaps it has something to do with an equilateral quad-angle in four dimensions with 60 degree angles are all corners.

Theon of Smyrna put together a handbook for philosophy students of Plato which showed how prime numbers, geometrical numbers such as squares, progressions, astronomy, and music were interrelated.  With respect to numbers Theon went Pythagorean, and discussed odd numbers, even numbers, prime numbers, composite numbers, square numbers, oblong numbers, triangular numbers, polygonal numbers, circular numbers, spherical numbers, solid numbers with three factors, pyramidal numbers, perfect numbers, deficient numbers and abundant numbers.  Clearly, Theon’s days were numbered.  Link to: <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Theon_of_Smyrna.html> for more information.

More recently, Stan Tenan <www.meru.org> in his treatise, “The God of Abraham, A Mathematician’s View”, notes that, “Each and every culture has made accurate and effective models of the cyclic, self-propagating and self-referential nature of all life in terms appropriate to its needs and experience.”  In other words, the idols of sophisticated people are the cultural, social, and scientific Paradigms of their societies.  On this basis,Abraham “acts as a mathematician: he postulates a meaningful and functional definition of Unity.”  [emphasis added]  

Mathematically, any wave shape can be derived from an Infinite Series of simple sine waves.  Typically, an approximation is accomplished by using only the larger terms of the series, but even in the case of a “square wave”, a finite number of sine waves can provide a close mathematical approximation.  It follows by extrapolation, according to Tenan, that an infinitely loud, short, sharp pulse (an “Om” or “Big Bang”, which might be compared to a musical or mathematical singularity) would produce the harmonic spectrum of all tones.  The latter could be considered to be equivalent to All-That-Is.  

To add a bit of scientific credibility to all of this, we might note also that quantum physics -- the state-of-the-art version -- is now fully supportive of Sacred Mathematics.  Lothar Schäfer, for example, in discussing the concept of Causality, has noted that, “Epistemic principles are transcendental because they are neither derived by a process of reasoning, nor my operations performed on physical reality.  They are, simply, principles of the human mind.  Thus, identity, object permanence, causality, external reality -- all the requisites for a resonable and enlightened life, albeit uncertain to experience and reason -- are valid because they are transcendental principles provided by the human mind.  By producing these principles, it is as though the mind remembered a higher order than can be found in the laws of logic or the visible patterns of physical reality.  Thus, it is a valid question whether evidence can be found from physical science of this transcendent part of physical reality, where such a higher order might have its roots.” [1]  

More specifically, quantum waves are a third type of wave (in addition to matter waves and electromagnetic waves -- sound and light and so forth). These waves are not only non-material -- i.e., needing no material medium in which to propagate -- but they are empty.  “Lightwaves can travel in empty space, but they carry energy.  Quantum waves also exist in empty space, but carry no energy or any other mechanical quantity.” [1]  

In other words, quantum waves are simply numbers, numerical relationships.  “Because they are empty, evidence of their existence is circumstantial;we must think that the universe is a network of quantum waves because the observable order appears as a manifestation of their interference.”  Also, “The reality of quantum waves is inferred from the expression of their interference in the observable patterns of reality.” [1]  

Forms, patterns, geometries, ratios...  

What an electron does is based on probabilities, which are in turn dimensionless numbers.  “Probability waves are empty in that they carry no energy or mass.  Numerical relations are their exclusive contents.”  “At the foundation of reality, we find numerical relations -- non-material principles -- on which the order of the universe is based.” [1]  

But it doesn’t end there.  Quantum “stuff” is basically “mind-stuff” -- see Wave-Particle Duality and/or Quantum Knowing.  “The mind-like properties of the background of reality are also suggested by the fact that its order is determined by principles of symmetry [aka geometry], abstract mathematical patterns, to which the constituents of the material world have to conform.” [1]  

Schäfer also references Werner Heisenberg -- one of the most known world class physicist (for whom the Heisenberg Uncertainty Principle is named). “The elementary particles in Plato’s Timaeus are not substance but mathematical forms.  ‘All things are numbers’ is a sentence attributed to Pythagoras.  The only mathematical forms available at that time were such geometric forms as the regular solids or the triangles which form their surface.  In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms, but of a much more complicated nature.” [1] [emphasis added]  And maybe, just maybe, not that much more complicated!  

In sum: “In the quantum phenomena, we have discovered that reality is different from what we thought it was.  Visible order and permanence are based on chaos and transitory entities.  Mental principles -- numerical relations, mathematical forms, principles of symmetry -- are the foundations of order in the universe, whose mind-like properties are further established by the fact that changes in information can act, without any direct physical intervention, as causal agents in observable changes in quantum states.” [1]  

Finally,

“Philosophy is written in this grand book --

I mean the universe --

Which stands continually open to our gaze.

But it cannot be understood

Unless one first learns to comprehend the language

And interpret the characters in which it is written.

It is written in the language of mathematics,

And its characters are triangles, circles, and other geometric figures,

Without which it is humanly impossible to understand a single word of it.”

                                                                                -- Galileo Galilei, Il Saggiatore (1623)

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