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Sacred Geometry Home Page by Bruce
Rawles
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 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 CircleThe circle is
a twodimensional 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 PointAt 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
& 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 everincreasing literature on socalled "neardeath
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 matrixseed conjugation. Our dictionary refers to the word matrix both as a
womb and an array (or grid lattice). Our language has some wonderful builtin
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 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 = (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
 ratio of segments in 5pointed star (pentagram)
considered sacred to Plato & 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 selfgenerating 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
, & Pythagoras),
the intentional use of these natural proportions in art of various forms expands
our sense of beauty, balance & 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 3dimensional case, 7 colors are needed, meaning that the torus
has a high degree of "communication" across its surface. The image shown is a
"birdseye" view.
DimensionalityThe
progression from point (0dimensional) to line (1dimensional) to plane
(2dimensional) to space (3dimensional) 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 3dimensional space introduce
crucial distortions in our view of reality that a higherdimensional 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
socalled "King's Chamber" of the Great Pyramid, along with the 2/3/root5 and
5/root5/2root5 triangles, utilizing the various diagonals and sides.
Here
are LOTS of math details and images of the Platonic Solids and Archimedean
Solids
The Platonic Solids
The 5 Platonic solids (Tetrahedron, Cube or (Hexahedron),
Octahedron, Dodecahedron & 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.
Here are some animations of counterrotating
polyhedra and images of the Platonic solids showing their relationships as
duals
Here are foldup patterns for the Platonic
Solids.
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.
Here are foldup patterns for the
Archimedean Solids.
Stellations of The Platonic & 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 3dimensional 5pointed star.
Click
here for more images of polyhedra (Platonic & Archimedean Solids)
Metatron's Cube
Metatron's Cube contains 2dimensional 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 background graphic for
this page is a repetitive hexagonal grid based on the Flower of Life.
http://www.intent.com/sg
Updated: Tue, 26 Apr 2005.
