Divine Proportion:
By Priya Hemenway

Phi = 1.61803

Divine Proportion:  The whole is to the larger in exactly the same proportion as the larger is to the smaller.

The Divine Proportion is found far back in the stories of the Old Testament.  In Exodus 25:10, God commanded
Moses to build the Ark of the Covenant:
Have them make a chest of acacia wood
Two and a half cubits long,
A cubit and a half wide,
And a cubit and a half high.
These measurements render a shape that is perfectly proportioned according to the Divine Proportion.

Fibonacci Sequence (1,1,2,3,5,8,…):  Each number is the sum of the two preceding numbers.  The ratio of any
neighboring numbers in the series approaches the Golden Mean.


      Early Calendars

The earliest Egyptian calendar was based on the moon’s cycles, but around 3100 BCE they realized that the “Dog
Star” (which we call Sirius) rose next to the sun every 365 days, about the same time as the annual inundation of
the Nile began.  With this understanding they devised a 365-day calendar.

5000 years ago, Sumerians in the Tigris-Euphrates valley had a calendar system that divided the year into 30-
day months, the day into 12 periods, and these periods into 30 parts.


      Early Math

Of all the devices used to record calculations, the earliest discoveries are clay tablets from areas in present-day
Iran and Iraq (ca. 3200 BCE), where unbaked clay was marked with shapes assigned to numerical values.  The
tablets found in Iran were based on a counting system of a base of 10, while those found in Iraq used a base
system of 60.  Both of these systems are still used by us today- base 10 in our use of a decimal system that
counts things by tens and base 60 in our use of sixty minutes to tell time and to measure degrees in a circle.

For the development of our own systems, we look back to the mathematical foundations developed by the ancient
Hellenic Greeks.  They, in turn, traced all human inventions- from calculus, geometry, astronomy, and dice games
to writing- back to the Egyptians.


      Golden Age of Greece

The people of Athens established Greece’s most famous democracy in which the individual freedom of citizens
flourished to a degree unprecedented in the ancient world.  The emergence of this new form of governance
brought a period of political, philosophical, artistic, and scientific achievements that formed the legacy upon which
Western civilization is based.  It lasted from about 480 BCE, the time of the defeat of the Persian invaders, until
the death of Alexander the Great in 323 BCE.


      Pythagoreans

The Pythagorean philosophy was based on an attempt to describe the underlying harmony of existence and the
nature of the perfect universe in numbers.

The invention of the word ‘philosophy’ is attributed to Pythagoras.  He was once asked, “Are you wise?” and he is
said to have replied, “No, but I am a lover of wisdom.”  The Greek word ‘philo’ means love and the word ‘sophia’
means wisdom.

Essential Beliefs:
1        At its deepest level, reality is mathematical in nature.
2        Philosophy can be used as a tool for spiritual understanding.
3        The human soul can be experienced as in union with the divine.
4        Certain symbols have a real mystical significance
     
Symbols:
     
      Monad:  
The circle is the parent of all subsequent shapes.  Ancient mathematical philosophers referred to the monad as
The First, The Seed, The Essence, The Builder, and The Foundation.  They also called it Unity.  The
Pythagoreans believed that nothing exists without a center around which it revolves.  The center is the source
and it is beyond understanding, it is unknowable, but like a seed, the center will expand and will fulfill itself as a
circle.
     
     Dyad:  
Greek philosophers referred to the dyad as ’audacity’, implying a boldness in separation from the original
wholeness and ’anguish’ due to a yearning to return to oneness.  The Greeks observed a paradox about the
dyad:  While it appears to separate from unity, its opposite poles remember their source and attract each other in
an attempt to merge and return to a state of unity.  The dyad simultaneously divides and unites, repels and
attracts, separates and returns.  
     
The Pythagoreans saw in the ancient symbolic drawing of the ‘vesica piscis’ a passageway to the journey of
spiritual self-discovery.  Its vulva shape had long been associated with fertility and the Divine Feminine and their
interpretation of the symbol relates the spiritual journey to the passage of birth.
     
     Triad:  
“The triad has a special beauty and fairness beyond all numbers, primarily because it is the very first to make
actual the potentialities of the monad.”
     
Ancient mathematical philosophers referred to the triad as prudence, wisdom, piety, friendship, peace, and
harmony.  To them, the shape of the triad was a statement about relationship and balance.  The triangle is the
only polygon structure that is rigid by virtue of its geometry.  Its stability and strength are unmatched by any of its
parts which by themselves do not have these properties.  The three lines are enhanced by the efficiency,
balance, visual appeal, and symbolism of the triangle.
     
The word ‘trinity’ derives from ‘tri-unity’ or ‘three as one’ and the triangle is the world’s preeminent symbol of
divinity.  The principles of trinity appear throughout myth and religion representing a relationship of mind, body,
and spirit; birth, life, and death; past, present, and future.
     
     Tetrad:  
To the Pythagoreans the even-sided square represented justice, because it is the first number, 4, divisible every
way into equal parts.  Four is associated with wholeness and completion, the four elements, the seasons, and the
ages of man.
     
     Pentad:  
The pentad represents the next level of cosmic design by introducing the symbol of life itself.  It is a symbol of
power and invulnerability.  The pentad was so revered in early societies that its construction was kept secret.  The
Pythagoreans used it as a secret sign to recognize one another.  They had studied its principles in geometry and
nature and they also knew of its effect on the human psyche.
     
The most predominant principle of the pentad is its ability to regenerate.  Pentagram-shaped stars can continue
generating similar shaped stars in a repeating pattern of growth.  In the way that leaves and flowers emerge it is
easy to see the pentad’s principle of self-similarity, or the smaller in the larger.  The veins in a leaf reveal the
branching pattern of the whole tree.  Each seed on a dandelion’s head mimics the plant’s head in miniature.  In a
crown of broccoli or cauliflower we see a reflection of the whole vegetable.
     
The pentad is best known to us in the symbol of the pentagram, the simplest form of a star shape that can be
drawn with a continuous line.  The fivefold symmetries of the pentagram are inherent to the human body in our
five senses.  A circle around a pentagram contains and protects its core.  The circle symbolizes eternity, infinity,
and the cycles of life and nature.  The center of a pentagram implies a sixth, formative element of love that exerts
its own power from within.
     
     Decad:  
The number ten was seen as an assembly point in mathematics and became a symbol of both world and heaven.  
Like our two hands with their ten fingers and the ability that gives us, the decad with its ten-ness comprises all that
is necessary for understanding the construction of the universe.  Like the monad, any number multiplied by ten
results in essentially the same number, but the number is brought to a higher level and becomes an expanded
version of itself.
     
     
     Art
      
Art is an experience of balance, of the relationship of its parts to the whole.  Perceiving it as anything else is
missing its most fundamental component.  A fine painting, a piece of sculpture, a work of architecture, music,
prose, or poetry is organized and gracefully balanced around a hidden sense of proportion.
     
Vitruvius

Vitruvius wished to preserve the classical tradition in the design of temples and public buildings.  Throughout the
Renaissance his book was the chief authority on ancient classical architecture.
     
“Without symmetry and proportion there can be no principles in the design of any temple.”
     
“Since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it
appears that the ancients had good reason for their rule, that in perfect buildings the different members must be
in exact symmetrical relations to the whole general scheme.  Hence, while transmitting to us the proper
arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods,
buildings in which merits and faults usually last forever.”
     
“It was by employing the symmetrical proportions of the human body that the famous painters and sculptors of
antiquity attained to great and endless renown.”
     
Vitruvius’ writings inspired Leonardo da Vinci to draw his famous ‘Vitruvian Man’ in which the proportions Vitruvius
speaks of were so brilliantly translated in one of his journals.
     
     
Parthenon
      
Based on modern calculations the Parthenon appears to be built on a square-root-of-5 rectangle, that is, with its
sides the length of the irrational 5.  The front elevation is built as a Golden Rectangle.
     
     
Notre Dame
      
The west façade is beautifully designed to the Divine Proportion.  The original design called for steeples, but they
were never added to the structure.
     
     
The Great Pyramid of Giza
      
The oldest of the Seven Wonders of the Ancient World and the only one that still survives.  It was built around
2500 BCE.  The maximum error between side lengths is astonishingly less than .1%.  The relationship of the
pyramid’s faces to its height is a relationship of Divine Proportion.
     
     
The Renaissance
      
In place of the medieval ideal of a life of penance as the highest and noblest form of human activity, the
Humanists looked forward to a rebirth of a lost human spirit and wisdom.  Its effect was to help people break free
from the mental strictures imposed by religious orthodoxy and to inspire free inquiry, criticism, and a new
confidence in the possibilities of human thought and works of art.
     
The spirit of the Renaissance is most readily found in painting.  Art was seen as a branch of knowledge, valuable
in its own right and capable of providing images of God and creation as well as insights into the place of humans
in the universe.  In the hands of men like Leonardo da Vinci it became a science and was a means of exploring
nature and recording discoveries.  In the works of painters such as Masaccio, Fra Angelico, Botticelli, Perugino,
Piero della Francesca, Raphael, and Titian; sculptors such as Pisano, Donatello, Verrocchio, Ghiberti, and
Michelangelo; and architects such as Alberti, Brunelleschi, and Palladio, human dignity found expression in the
arts as well as in the ancient mathematical principles of balance, harmony, and perspective.
     
Artists such as Alberti, Piero della Francesca, and Leonardo da Vinci all studied engineering, mathematics,
science, and architecture while pursuing their art.
     
Da Vinci did dissections in the hospital to broaden his understanding of the structure and function of the human
body.  He made systematic observations of the flight of birds and engaged in studies of the movement and
properties of water and air.  His investigations became increasingly driven by a conviction that force and motion
operate in accordance with orderly, harmonious laws.
     
“The Ancients… did in their Works propose to themselves chiefly the Imitation of Nature, as the greatest Artist at
all Manner of Compositions.”
     
      
     
Music
      
Plato so distrusted the emotional power of music that he thought it was necessary to impose a strong censorship
on its performance.  He concluded that the sensuous qualities of music are actually dangerous and lead to
depression and disorder.  Plato’s influence on music was to be dominant for at least a millennium.  The
conservative aspects of his philosophy, with its inherent fear, were conducive to the maintenance of order, and
the restricted role of music is nowhere more clearly illustrated than in the history of Christianity.  Melody was used
for textual illumination only, and the configurations of sound took their cue from the words.
     
     
     Nature
      
The pentagram whose segments are related to Phi can be seen in the center of an apple.
    
Spirals
      
The spiral, an essential tool in nature’s palate, has long been regarded as one of the most significant.  Look into
the sky and there are spirals there; look at water, look into the wind.  Peel off the leaves of a head of lettuce or a
cabbage, peer closely at the seeds in a daisy.  Spirals are everywhere- some obvious and some less so.  From
embryos to galaxies.
     
The eye of every spiral is a dynamic place where all opposites meet and where life and death are one
phenomenon.  All the forces that create growth and keep it in balance are at work in the eye, the source.
     
      
The Fibonacci numbers appear over and over again in certain spiral formations and show up most beautifully in
the symmetric arrangement of petals on a rose, where successive leaves grow at an angle of 137.5°- the Golden
Angle.
     
     
Inner Ear
      
The spiral cochlea that sits in the inner ear part of the ear is the part of the organ that ‘hears’.  Its shape
corresponds to how musical octaves appear when graphed as wavelengths.  Each note is identical to those
directly above and below it on the spiral, but with a difference of one octave.  Our human cochlea with its two-and-
three-quarter spiral turns allows us to hear approximately ten octaves of sound.  Other mammals have different
numbers of turns, and this explains why they can perceive different frequencies than humans do.
     
When musical notes are graphed they form the shape of the Golden Spiral.  When it is stretched into three
dimensions it becomes the shape of the cochlea.
     
     
DNA
      
The DNA molecule, the basis of all life, has some remarkable measurements.  It measures 34 angstroms long by
21 angstroms wide for each full cycle of its double helix spiral.  Both 34 and 21 are numbers in the Fibonacci
series and their ratio, 1.6190476... closely approximates Phi, which is 1.6180339...
     
The DNA in the cell appears as a double-stranded helix referred to as B-DNA.  This form of DNA has two grooves
in its spirals, with a ratio of Phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms
to 13 angstroms.
     
Furthermore, a cross-sectional view from the top of the DNA double helix forms a decagon- in which each spiral of
the double helix traces out the shape of a pentagon.
     
     
     Science
      
Space
      
In October 2001, NASA began collecting data on cosmic background radiation, which can tell scientists a lot about
the physical nature of space.  The wavelength of the radiation is remarkably pure, but like a musical note it has
harmonies associated with it.  These harmonies reflect the shape of the object in which the waves were
generated.  In February of 2003, NASA released the first data from the probe, and in October a team of scientists
used this data to develop a model for the shape of the universe.  The study has revealed that if the Observatoire
data is correct, it would predict a universe that is finite and is shaped like a dodecahedron.  This closed universe
should be about 30 billion light years across.
     
One of the astounding things about this discovery is its relation to Plato’s proposition 2500 years earlier that the
universe had finite limits and was in the shape of a dodecahedron.
     
      
     
Platonic Solids
      
Each of the five Platonic Solids fits perfectly within a sphere presenting an identical view in all directions and their
surfaces all have the same shape.  There are only five volumes that fulfill the requirements of equality by
repeating identical corner angles, edge lengths, and surface shapes within a sphere.
     
     Cosmos - Dodecahedron - Pentagon -12
     Fire - Tetrahedron - Triangle - 4
     Air - Octahedron - Triangle - 8
     Water - Icosahedron - Triangle - 20
     Earth - Hexahedron - Square - 6
     
These five shapes were looked upon with a sense of awe in ancient times.  The construction and study of their
forms were considered the ultimate goal toward which those who studied numbers would arrive and represented a
pinnacle of ancient geometric and esoteric knowledge.
     
     
     Mysticism
      
Divine Proportion represents a significant relationship of parts to the whole, and concealed within its encoded
teachings are laws of sacred creativity, and hints of an experience that transcends the serious world of restrained
thoughtfulness and thrusts its participants into an uncharted journey toward the ultimate reward of self-revelation.
     
Many people experience what they describe as a divine state when they walk into sacred spaces like cathedrals,
and this is not accidental.  Acoustics, colors, light, and many hidden elements contribute to an experience that
has been refined by centuries of experimentation.  For long periods of time the designs are carefully defined by
religious law, then time passes and the laws are abandoned.  People forget how to build these sacred spaces,
and were it not for their permanence, very specialized knowledge would be lost.  But the keys are there, locked
into stone and embedded in shape.
     
One of the great paradoxes of Western tradition is a belief in the oneness of God and the unity of existence- and
yet the multiplicity in the world we experience.  The problematic result of this is the awkward belief that we are
somehow separate from the rest of life.  
     
Greek mystical philosophers such as Pythagoras, Heraclites, Socrates, and his disciple Plato had spoken of a
universe in which all things are one.  Discovery after scientific discovery has led us to understand what all the
world’s spiritual teachers have always known and taught:  Truth is an inner phenomenon.
     
      
Meditation manuals have taught methods by which different yantras were to be envisioned deep within one’s
being.  Accessing deeply embedded creative powers, meditators were taught to externalize their consciousness,
or bring it forth into the world.  This practice enabled them to identify so deeply with an external object that the
dichotomy of subject and object dissolved.  When this happened, and everything ‘fit together’ into a harmony
where opposites meet and disappear, where polarities do not exist, where all paradoxes are resolved and all
contradictions disappear, these meditators would then have the internal power to transform the objective world by
changing the inner subject.
     
All the planes of existence are involved in it.  It is not only that the soul is full of music:  once you have heard the
inner music, your mind is full of it; all your layers of being are full of it.  Once known, not only do you hear it inside
of you, it is outside too.  In the song of birds you hear it, and in the wind passing through the trees you hear it,
and in the waves striking on the rocks you hear it.  In sound you hear it, in silence you hear it.  In fact the greatest
music in the world is nothing but an echo of the inner music.
     
The flower is a timeless symbol of truths that are mysteriously conveyed.  The scent of its perfume carries an
unspoken message, its transient beauty conveys a sense of the transitory phases of our lives, the flowering
phenomena itself speaks of what is possible when we allow the divine to enter.
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