# Introduction to

Sacred Geometry

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

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 3.141592653589793238462643383279
50288419716939937511...) 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 1.618033988749894848204586834365

638117720309180...) 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.

The outline of the Parthenon at the Acropolis near Athens,
Greece is enclosed by a Golden Rectangle by design. Phidias
directed and supervised the construction of the Parthenon in 434
BC, as well as designing the sculptural decoration. The
mathematical golden ratio is represented by the Greek letter
'phi', taken from Phidias' name because Phidias employed the ratio
in making the Parthenon and sculptures, which perfectly exhibit
the proportions of the golden ratio.

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.

This most aesthetically pleasing proportion, phi.

# 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 which follows a law
everywhere in evidence.

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

A fractal is a natural phenomenon or a mathematical set that
exhibits a repeating pattern that displays at every scale. If the
replication is exactly the same at every scale, it is called a
self-similar pattern. 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 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

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 Archimediean 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

These are stellations of a dodecahedrons 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 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.