Sacred Geometry

Posted: August 20, 2012 by noxprognatus in Occult

And See the Wonderfully Patterned Beauty of Creation

The strands of our DNA, the cornea of our eye, snow flakes, pine cones, flower petals, diamond crystals, the branching of trees, a nautilus shell, the star we spin around, the galaxy we spiral within, the air we breathe, and all life forms as we know them emerge out of timeless geometric codes.

The designs of exalted holy places from the prehistoric monuments at Stonehenge and the Pyramid of Khufu at Giza, to the world’s great cathedrals, mosques, and temples are based on these same principles of sacred geometry.

As far back as Greek Mystery schools 2500 years ago it was taught that there are five perfect 3-dimensional forms – the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron … collectively known as the Platonic Solids; and that these form the foundation of everything in the physical world.

Modern scholars ridiculed this idea until the 1980’s, when Professor Robert Moon at the University of Chicago demonstrated that the entire Periodic Table of Elements – literally everything in the physical world – truly is based on these same five geometric forms. In fact, throughout modern physics, chemistry, and biology, the sacred geometric patterns of creation are today being rediscovered.

INTRODUCTION.

In nature, we find patterns, designs and structures from the most  minuscule particles, to expressions of life discernible by human eyes,  to the greater cosmos. These inevitably follow geometrical archetypes,  which reveal to us the nature of each form and its vibrational  resonances. They are also symbolic of the underlying metaphysical  principle of the inseparable relationship of the part to the whole. It  is this principle of oneness underlying all geometry that permeates the  architecture of all form in its myriad diversity. This principle of  interconnectedness, inseparability and union provides us with a  continuous reminder of our relationship to the whole, a blueprint for  the mind to the sacred foundation of all things created.

The Sphere

                (charcoal sketch of a sphere by Nancy Bolton-Rawles)

Starting with what may be the simplest and most perfect of forms, the  sphere is an ultimate expression of unity, completeness, and integrity.  There is no point of view given greater or lesser importance, and all  points on the surface are equally accessible and regarded by the center  from which all originate. Atoms, cells, seeds, planets, and globular  star systems all echo the spherical paradigm of total inclusion,  acceptance, simultaneous potential and fruition, the macrocosm and  microcosm.

The Circle

The circle is a two-dimensional shadow of the sphere which is regarded  throughout cultural history as an icon of the ineffable oneness; the  indivisible fulfillment of the Universe. All other symbols and  geometries reflect various aspects of the profound and consummate  perfection of the circle, sphere and other higher dimensional forms of  these we might imagine.

The ratio of the circumference of a circle to its  diameter, Pi, is the original transcendental and irrational number. (Pi  equals about 3.14159265358979323846264338327950288419716939937511…)  It cannot be expressed in terms of the ratio of two whole numbers, or  in the language of sacred symbolism, the essence of the circle exists  in a dimension that transcends the linear rationality that it contains.  Our holistic perspectives, feelings and intuitions encompass the finite  elements of the ideas that are within them, yet have a greater wisdom  than can be expressed by those ideas alone.

The Point

At the center  of a circle or a sphere is always an infinitesimal point. The point  needs no dimension, yet embraces all dimension. Transcendence of the  illusions of time and space result in the point of here and now, our  most primal light of consciousness. The proverbial “light at the end of  the tunnel” is being validated by the ever-increasing literature on  so-called “near-death experiences”. If our essence is truly spiritual  omnipresence, then perhaps the “point” of our being “here” is to  recognize the oneness we share, validating all “individuals” as equally  precious and sacred aspects of that one.

Life itself as we know it is inextricably interwoven  with geometric forms, from the angles of atomic bonds in the molecules  of the amino acids, to the helical spirals of DNA, to the spherical  prototype of the cell, to the first few cells of an organism which  assume vesical, tetrahedral, and star (double) tetrahedral forms prior  to the diversification of tissues for different physiological  functions. Our human bodies on this planet all developed with a common  geometric progression from one to two to four to eight primal cells and  beyond.

Almost everywhere we look, the mineral intelligence  embodied within crystalline structures follows a geometry unfaltering  in its exactitude. The lattice patterns of crystals all express the  principles of mathematical perfection and repetition of a fundamental  essence, each with a characteristic spectrum of resonances defined by  the angles, lengths and relational orientations of its atomic  components.

The Square Root of Two

The square root of 2 embodies a profound principle of the whole being  more than the sum of its parts. (The square root of two equals about  1.414213562…) The orthogonal dimensions (axes at right angles) form  the conjugal union of the horizontal and vertical which give birth to  the greater offspring of the hypotenuse. The new generation possesses  the capacity for synthesis, growth, integration and reconciliation of  polarities by spanning both perspectives equally. The root of two  originating from the square leads to a greater unity, a higher  expression of its essential truth, faithful to its lineage.

The fact that the root is irrational expresses the  concept that our higher dimensional faculties can’t always necessarily  be expressed in lower order dimensional terms – e.g. “And the light  shineth in darkness; and the darkness comprehended it not.” (from the  Gospel of St. John, Chapter 1, verse 5). By the same token, we have the  capacity to surpass the genetically programmed limitations of our  ancestors, if we can shift into a new frame of reference (i.e. neutral  with respect to prior axes, yet formed from that matrix-seed  conjugation. Our dictionary refers to the word matrix both as a womb  and an array (or grid lattice). Our language has some wonderful  built-in metaphors if we look for them!

The Golden Ratio

The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean  a.k.a. divine proportion) is another fundamental measure that seems to  crop up almost everywhere, including crops. (The golden ratio is about  1.618033988749894848204586834365638117720309180…) The golden ratio is  the unique ratio such that the ratio of the whole to the larger portion  is the same as the ratio of the larger portion to the smaller portion.  As such, it symbolically links each new generation to its ancestors,  preserving the continuity of relationship as the means for retracing  its lineage.

The golden ratio (phi) has some unique properties and makes some interesting appearances:

    • phi = phi^2 – 1; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum.
    • phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2.
    • phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/…)))))
    • phi = 1 + square root(1 + square root(1 + square root(1 + square root(1 + square root(1 + …)))))
    • phi = (sec 72)/2 =(csc 18)/2 = 1/(2 cos 72) = 1/(2 sin 18) = 2 sin 54 =  2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you trigonometry  enthusiasts.
    • phi = the ratio of segments in a 5-pointed  star (pentagram) considered sacred to Plato and Pythagoras in their  mystery schools. Note that each larger (or smaller) section is related  by the phi ratio, so that a power series of the golden ratio raised to  successively higher (or lower) powers is automatically generated: phi,  phi^2, phi^3, phi^4, phi^5, etc.

  • phi = apothem to bisected base ratio in the Great Pyramid of Giza
  • phi = ratio of adjacent terms of the famous Fibonacci Series evaluated at infinity; the Fibonacci Series is a rather ubiquitous set  of numbers that begins with one and one and each term thereafter is the  sum of the prior two terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144…  (interesting that the 12th term is 12 “raised to a higher power”, which  appears prominently in a vast collection of metaphysical literature)

The mathematician credited with the discovery of this series is Leonardo Pisano Fibonacci and there is a publication devoted to disseminating information about its unique mathematical properties, The Fibonacci Quarterly

Fibonacci ratios appear in the ratio of the number of spiral arms in  daisies, in the chronology of rabbit populations, in the sequence of  leaf patterns as they twist around a branch, and a myriad of places in  nature where self-generating patterns are in effect. The sequence is  the rational progression towards the irrational number embodied in the  quintessential golden ratio.

This most aesthetically pleasing proportion, phi, has  been utilized by numerous artists since (and probably before!) the  construction of the Great Pyramid. As scholars and artists of eras gone  by discovered (such as Leonardo da Vinci, Plato, and Pythagoras),  the intentional use of these natural proportions in art of various  forms expands our sense of beauty, balance and harmony to optimal  effect. Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and  a decagon (which contains the golden ratio) for alignment of the  central figure of Jesus.

The outline of the Parthenon at the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.

The Square Root of 3 and the Vesica Piscis

The Vesica Piscis is formed by the intersection of two circles or spheres whose centers  exactly touch. This symbolic intersection represents the “common  ground”, “shared vision” or “mutual understanding” between equal  individuals. The shape of the human eye itself is a Vesica Piscis. The  spiritual significance of “seeing eye to eye” to the “mirror of the  soul” was highly regarded by numerous Renaissance artists who used this  form extensively in art and architecture. The ratio of the axes of the  form is the square root of 3, which alludes to the deepest nature of  the triune which cannot be adequately expressed by rational language  alone.

Spirals

This spiral generated by a recursive nest of Golden Triangles  (triangles with relative side lengths of 1, phi and phi) is the classic  shape of the Chambered Nautilus shell. The creature building this shell  uses the same proportions for each expanded chamber that is added;  growth follows a law which is everywhere the same. The outer triangle  is the same as one of the five “arms” of the pentagonal graphic above.

Toroids

Rotating a circle about a line tangent to it creates a torus, which is  similar to a donut shape where the center exactly touches all the  “rotated circles.” The surface of the torus can be covered with 7  distinct areas, all of which touch each other; an example of the  classic “map problem” where one tries to find a map where the least  number of unique colors are needed. In this 3-dimensional case, 7  colors are needed, meaning that the torus has a high degree of  “communication” across its surface. The image shown is a “birds-eye”  view.

Dimensionality

The  progression from point (0-dimensional) to line (1-dimensional) to plane  (2-dimensional) to space (3-dimensional) and beyond leads us to the  question – if mapping from higher order dimensions to lower ones loses  vital information (as we can readily observe with optical illusions  resulting from third to second dimensional mapping), does our  “fixation” with a 3-dimensional space introduce crucial distortions in  our view of reality that a higher-dimensional perspective would not  lead us to?

Fractals and Recursive Geometries

There is a wealth of good literature on this subject; it’s always  fascinating how nature propagates the same essence regardless of the  magnitude of its expression…our spirit is spaceless yet can manifest  aspects of its individuality at any scale.

Perfect Right Triangles

The 3/4/5, 5/12/13 and 7/24/25 triangles are examples of right  triangles whose sides are whole numbers. The graphic above contains  several of each of these triangles. The 3/4/5 triangle is contained  within the so-called “King’s Chamber” of the Great Pyramid, along with  the 2/3/root5 and 5/root5/2root5 triangles, utilizing the various  diagonals and sides.

The Platonic Solids

Here are

The 5 Platonic solids (Tetrahedron, Cube or (Hexahedron), Octahedron,  Dodecahedron and Icosahedron) are ideal, primal models of crystal  patterns that occur throughout the world of minerals in countless  variations. These are the only five regular polyhedra, that is, the  only five solids made from the same equilateral, equiangular polygons.  To the Greeks, these solids symbolized fire, earth, air, spirit (or  ether) and water respectively. The cube and octahedron are duals,  meaning that one can be created by connecting the midpoints of the  faces of the other. The icosahedron and dodecahedron are also duals of  each other, and three mutually perpendicular, mutually bisecting golden  rectangles can be drawn connecting their vertices and midpoints,  respectively. The tetrahedron is a dual to itself.

The Archimedean Solids

There are 13 Archimedean solids, each of which are composed of two or  more different regular polygons. Interestingly, 5 (Platonic) and 13  (Archimedean) are both Fibonacci numbers, and 5, 12 and 13 form a  perfect right triangle.

Stellations of The Platonic Solids and The Archimedean Solids

This is a stellation of a dodecahedron where each pentagonal face is  capped with a pentagonal pyramid composed of 5 golden triangles, a sort  of 3-dimensional 5-pointed star.

Metatron’s Cube

Metatron’s Cube contains 2-dimensional images of the Platonic Solids (as shown above) and many other primal forms.

The Flower of Life

Indelibly etched on the walls of temple of the Osirion at Abydos,  Egypt, the Flower of Life contains a vast Akashic system of  information, including templates for the five Platonic Solids.

The ancients knew that these patterns were codes symbolic of our own inner realm and that the experience of sacred geometry was essential to the education of the soul. Viewing and contemplating these forms can allow us to gaze directly at the face of deep wisdom and glimpse the inner workings of The Universal Mind.

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