A
Sacred Geometry Primer
Grahame Gardner
Sacred
Geometry is one of the three main foundations of western
geomancy (the other two being dowsing and astronomy
– astrology). Whenever we wish to commune with our Gods,
meditate, worship, contact our higher selves, whatever
name we give to the practice; we tend to disassociate
ourselves from the everyday mundane world by going to
a place that has some spiritual meaning for us. Whether
this is a church, temple, stone ring or simply a corner
of our bedroom, the concept is the same. Most of these
places (with the possible exception of the last) are
sacred spaces constructed using sacred geometry.
Sacred
Geometry depends on the use of what are called irrational
numbers. These are numbers like Pi ( )
and Phi (Φ), where the decimal part is infinitely
long and non-repeating. Pi is also a special kind of
irrational number called a transcendental number,
but we won’t go into that here because it’s defined
more by what it isn’t than what it is, and in
any case it’s not relevant to this discussion
[1] .
Some
irrational numbers are so commonly known that they have
names; Pi, Phi, Feigenbaum’s constant and the Comma
of Pythagoras are just a few examples. Others are known
only by their function;  ,
 , 
for instance. They are found to be an inherent component
of many natural processes and structures, and form an
integral part in the design of sacred spaces. From Stonehenge
to Chartres cathedral, from Maes Howe to the Great Pyramid,
you will find irrational numbers underpinning the layout
of the space. I say underpinning, because they are never
openly manifest. They appear only in the relationship
of one quantifiably measurable part to another (another
name for irrationals is incommensurate numbers).
To understand the importance of these numbers, we need
to delve deeper into the noble and ancient art of Sacred
Geometry.
Sacred
Geometry deals with our perception and definition of
space. It is the Universal framework whereby the spiritual
manifests into the material. Spaces constructed using
the principles of sacred geometry act as a bridge between
the worlds, and sacred geometric forms naturally produce
dowseable energy fields. It makes no difference if it’s
a chalk circle on your living-room carpet or the dizzying
architecture of a gothic cathedral; the principles are
exactly the same. To the masons who raised the great
gothic cathedrals and other places of worship, sacred
geometry was of paramount importance to the construction;
indeed no religious establishment could be expected
to function properly without it. The spaces are designed
to be uplifting to the spirit, to resonate on
a subconscious level in such a way that the possibility
of a spiritual connection is maximised. An impressive
side effect of sacred geometry is that many of these
structures possess extraordinary acoustic properties,
a result of the inter-related harmonic structure behind
both music and geometry. Certain types of music (e.g.
Bach) were designed to be sacred geometry you can hear,
so it is little wonder the buildings resonate with it.
In
ancient times it was believed that numbers are the underlying
reality behind all things. All things were linked through
number and could be manifested through number. Number
expressed in time is music, number expressed in space
is geometry, and number expressed in space-time is astrology.
Just as astrology can provide us with a map of consciousness,
so can geometry. These concepts underlie the entire
world-view of the ancients but have largely been forgotten
in our modern hyper-rational scientific orthodoxy.
We’ve lost sight of the spiritual qualities of number
and shape by our emphasis on brute quantity and hard
computation. We’re taught to see numbers as mere quantities,
instead of possessing qualities and characters with
distinct personalities, resonating with each other and
Universe in a harmonious cosmic dance (it used to be
called Music of the Spheres). And this is why irrational
numbers are so fascinating to the geometer; they cannot
ever be expressed as a quantity, they can only
be appreciated as qualities. They are the inner,
esoteric face of number, becoming visible only in the
relationships between parts. I think of them as the
cracks between the paving stones of reality. The Pythagoreans
of ancient Greece, who are credited with discovering
the infinite nature of these numbers in the sixth century
B.C., were so puzzled and awe-struck by their discovery
that they tried to keep it a secret by proscribing the
death penalty for those who would divulge it.
[2]
Before
we go any further, we’re going to have to take a brief
mathematical foray to look at numbers. Now don’t panic;
things are really not as complicated as they seem. There
are only a small handful of basic concepts that you
need to grasp. If you can remember
Pythagoras’ theorem on right-angled triangles that states
that the square on the hypotenuse (the long side) is
equal to the sum of the squares on the other two sides
(a2+b2=c2), you’ll
be fine with sacred geometry.
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However
it is important to note that this is a formula,
which is part of algebra. Algebra has
nothing whatsoever to do with sacred geometry,
and in fact was a much later invention of
the thirteenth century. The original theorem
comes from geometry, and can be proven by
geometry. The key is in the wording.
Let’s look at the most basic of the right-angled
triangles, what is known as the 3-4-5 triangle
(Fig 1). The corner with the little square
in it is a right angle (90o). The
lengths of the sides are shown as 3,4 and
5 units. The theorem says that the square
on the long side is equal to the sum of the
squares on the other two sides. It
literally means just that. So if we draw squares
on those sides, suddenly we see things in
a new light (Fig. 2).
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Here
we’ve drawn square grids on each side of the
triangle according to the unit length; so
3x3, 4x4 and 5x5 squares. We can easily see
that the number of squares on the long
side is 25. The theorem states that this is
equal to the sum of the squares on
the other two sides. On those we have grids
of 9 squares and 16 squares. 9 + 16 = 25 squares.
Hence the length of the long side of the triangle
is the square root of 25, which is
of course 5.
As
long as we know the length of any two of the
sides in a right-angled triangle, we can work
out the length of the remaining side using
this method.
Now
that wasn’t too bad, was it? And that’s about
as complicated as things need to get in sacred
geometry.
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We
have seen that Sacred Geometry is interested in irrational
numbers. These numbers are the cornerstone of sacred
geometry because they effectively manifest the infinite
in normal space. For a practical example of this, let’s
take a look at an ancient example of sacred space, the
King’s Chamber in the Great Pyramid. The shape of this
is a form known as a double cube. That is, the
long sides are twice the length of the short sides.
Consider just the floor area for the moment – long side
twice the length of the short side remember – the floor
would be a double-square rectangle (Fig. 3). The important
thing about the double-square is the diagonal. If we
say that the short side of the square has a length of
one unit, then the long side has a length of 2 units
(these can be anything you like - sacred geometry is
only interested in proportions and ratios, not actual
measurements. It doesn’t matter if the units are inches,
metres, or aardvarks). That means that a diagonal of
this rectangle will have a length equal to the square
root of five (by Pythagoras’ theorem: hypotenuse2
= 22 + 12 = 4 + 1 = 5, therefore
hypotenuse = ).
Now
the square root of five (2.236…) is another one of those
irrational numbers that can’t be calculated precisely.
It goes on forever, never repeating, always changing.
So it can be quite accurately said that in a sense,
you cannot ever measure this diagonal exactly. It represents
the infinite. Both diagonals of this rectangle are root
5, so if you were to stand exactly at the centre of
this double-square rectangle, you stand in the centre
of a harmoniously proportioned space, but you also stand
at the crossing point of two diagonals of incommensurable
length
[3] .
What better space to commune with the One?
Let’s
take a look at some other basic shapes and ratios as
we take a tour of the numbers One through Five. Along
with Pi (Π)
and Phi (Φ), the main irrationals used are the
square roots of 2, 3 and 5, and combinations thereof,
like the :
proportions prevalent in Chartres cathedral
[4] . It’s too large a subject to cover fully
here, but this will give you a basic grasp of the important
concepts involved.
The
Circle: Simplest
shape of all, and manifestation of the One. Draw a circle
around yourself and you immediately distinguish your
personal space from your surroundings. That’s really
the power of the circle. It is the first expression
of Universe; a horizon, a boundary between Self and
Other. With no beginning and no end, the circle’s circumference
is a profound statement about the transcendental nature
of reality. Defined as ‘an infinite number of points
equidistant from a centre’, the circle in its manifestation
implies the divine generation of shape and form from
nothing to everything.
The
irrational number that the circle generates is, of course,
Pi (Π=3.1415926….). The circumference is calculated
from the radius by the formula 2Πr, where
r is the radius of the circle. Because of the
use of Pi, we cannot ever know the value of both the
radius and the circumference in whole-number units.
If either the radius or circumference is measurable
in whole, rational units, then the other will always
be an endless, irrational decimal. Thus the circle represents
the limited and the limitless in one body.
Examples
of sacred spaces based upon the circle are Stonehenge,
Ring of Brodgar in Orkney, and the Merry Maidens circle
in Cornwall. Most of what we call ‘stone circles’ are
not true circles, but more complex geometrical forms.
A more accurate term would be ‘stone rings’, and you
do see this being used in some literature today. Some
researchers think that the complicated compound outlines
of the rings were attempts to produce whole-number circumferences
and radii in a quest for sacred geometric perfection;
however research by the EEG seems to indicate that the
ring will have greater beneficial earth energy effects
the closer the perimeter divided by the sum of the two
axes approximates the Golden Mean of 1.61803…
[5] .
The
Vesica Piscis:
The circle divides, the One becomes Two. Something generates
Other. The circle replicates by contemplating itself,
reflecting its light, and casting an identical shadow.
Now we have two separate qualities, but they
are still in the Void, distinct and without interaction.
Nothing can happen until they merge, and then we have
a concept of Three-ness and true manifestation can begin.
To the ancients, Three was the first proper number;
One and Two were abstract principles, unmanifest. The
shape formed by two overlapping circles is called the
vesica piscis (Fig. 4). The vesica piscis
embodies all the concepts of duality that you can think
of; the two circles both attract and repel each other,
giving polarity and tension, whilst the overlapping
area gives us a portal of manifestation.
 Any
two circles that overlap will produce a vesica
(the overlapping area), but only two circles of identical
size whose centres are located on the circumference
of the other produce a vesica piscis (it’s Latin
for ‘bladder of the fish’). This ancient symbol is significant
in esoteric Christian lore, and forms the basis for
the ‘fish’ sigil that was used initially as a secret
sign between early Christians and is now seen most often
decorating car bumpers. But the symbol is much older
than Christianity. The vesica doesn’t just represent
a fish bladder; it’s really about the birth portal,
the cosmic Yoni of the Goddess. Imagine looking up from
between her outstretched thighs and you’ll get the picture.
All subsequent numbers and geometric shapes can be produced
through the portal of the vesica using the geometer’s
tools, but regrettably that is beyond the scope of this
article.
The
irrational number produced by the vesica piscis is the
square root of 3. If the radius of each of the circles
is 1 unit, then a vertical line drawn down the centre
of the vesica has a length of root 3 (=1.7320…).
The
vesica piscis crops up a lot in sacred architecture,
but perhaps its most obvious manifestation is the Gothic
arch, bastion of so many of our spiritual buildings.
A Gothic arch is basically the vesica part with vertical
extensions from the centre of the sides. Less obvious
manifestations of the vesica are in the floor plans
of many churches and chapels where the vesica defines
the dimensions of the rectangular layout. The Mary chapel
in Glastonbury Abbey is said to be the most perfect
example of this
[6] . The vesica is also representative of
the shape of the human aura, and is used as such in
many religious paintings. This has an interesting symbolism;
if the two circles of the vesica piscis are seen as
Heaven and Earth, then the vesica part represents the
bridge between the two – or in other words, Jesus and
the established Church. This is why you often see the
figure of Christ enclosed in a vesica-shaped aura on
the front of churches
[7] .
The
Square:
The Square symbolises the imposition of structure upon
the earth. Instead of the limitless circle of the One,
we now have the orientation and implied directions of
the Four. Four quarters. Four directions. Four winds.
Four elements… and so on. We’re perhaps more familiar
with this shape than any other, since it permeates our
lives in practically everything we build or make (Fig
5).
 In
working with sacred space, the important bit of the
square is the diagonal. If the square has sides of 1
unit in length, then the length of the diagonal is the
square root of 2 (=1.41421…).
You can easily prove this for yourself using Pythagoras’
theorem. Again, this is an irrational number. Stand
in the centre of the square and you stand at the centre
of two theoretically infinite lines. A classical example
of this was the Holy of Holies in Solomon’s Temple.
That’s the bit that nobody was allowed into, where they
kept the Ark. A more contemporary incarnation is the
Kaaba in Mecca.
The
combination of the square and the circle represents
the fusion of heaven and earth, and ‘squaring the circle’
is regarded as the pinnacle of the sacred geometer’s
art. This means producing a circle overlaying a square
such that either the circumference of the circle
equals the perimeter of the square, or the area
of the circle equals that of the square. John Michell
has produced possibly the most aesthetic example of
this using the dimensions of the earth and moon,
[8] but space precludes us going into that
here.
Many
early religious buildings were designed using this squared-circle
geometry, perhaps most famously the Hagia Sophia in
Istanbul, one of the oldest Christian churches in existence,
commissioned by the Emperor Justinian and completed
in 532 c.e. Its square form with staggering 180-foot
high hemispherical dome is still a wonder to behold.
It was the largest enclosed space on the planet
for over a thousand years. Interestingly, when Constantinople
fell to the Turks, the building was converted into a
mosque and the beautiful gold mosaics of the interior
painted over. But its form was considered so perfect
that all subsequent mosques in Istanbul were modelled
on it, and so the squared circle form was introduced
to Islam. Many years later, the West took the design
of the Gothic arch from Islamic architecture, so it
seems a fair trade!
The
Double Square: As
it sounds, two squares side by side. A rectangle
with short side of 1 unit and long sides of 2. This
represents the 2:1 ratio of the octave in musical terms.
We looked at this shape earlier as the basis of the
King’s Chamber in the Great Pyramid (Fig. 3); it’s also
the shape of ‘The Holy Place’ in Solomon’s Temple. That
was the main part of the temple, not quite as sacred
as the Holy of Holies.
The
Pentagram and the Golden Proportion: The
four terms of the square are enough to account for the
idea of matter, or substance. But when we get to Five
we introduce the concept of Spirit, or governing intelligence;
or in other words Life. Five is seen as the union of
Two (female) and Three (male). Thus the five-sided pentagon
and in particular the pentagram star within it symbolises
life and regeneration. It also manifests one of nature’s
most startling proportional relationships, the Golden
Mean.
[9]
The
Golden Proportion, Golden Section or Golden Mean is
one of Nature’s universal constants, perhaps the most
difficult to get your head around, but also the most
cosmic. It is a proportion that is found all around
us, in the growth patterns of all living things, the
proportions of our own bodies, and in classical architecture
to name but a few instances; and yet goes largely unnoticed
by most. It’s hard to understand by definition, but
relatively easy to grasp once you see some examples
of it. It seems to be programmed into our very minds,
in that we tend to pick out items embodying Golden Mean
principles as being the most ‘visually pleasing’ to
us, in the same way that a major fifth is the most ‘aurally
pleasing’ subdivision of the octave in music. Indeed
our entire concept of beauty is determined by how closely
the facial features of others approach Golden Mean proportions.
By definition, it’s a way of dividing something into
two unequal parts, such that: whole/large part = large
part/small part = Phi (Φ). Numerically, the ratio
is 1:1.61803… and of course this is another irrational
number.
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But
we can understand the Phi proportion better
if we see some instances of it; so let’s look
at the Golden Proportion as it manifests itself
geometrically. In sacred geometry, where all
forms (and therefore numbers) are generated
through the cosmic birth-portal of the vesica
piscis, the pentagon/pentagram is the third
such form to emerge after the triangle and
square, and is the first in which the Golden
Proportion has to be ‘invoked’ in order to
draw it (Fig. 6). It’s relatively difficult
to draw accurately geometrically, and this
is partly why it has developed the occult
associations that it has today (if you would
like to learn how to construct one, click
here). It was worn as a hidden sign of
recognition by advanced initiates of the Pythagorean
mystery school around 500 BCE and, one thousand
years later, the secrets of its construction
were kept in the oral tradition, revealed
only to initiates of the Craft Guilds and
Masons that built the great gothic cathedrals.
It wasn't until 1509 that the monk Fr. Luca
Pacioli, who was the mathematics teacher of
Leonardo da Vinci, let the cat out of the
bag when he published the secret in his book
'De Divina Porportione'.
[10]
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The
Pentagram is interesting to the sacred geometer
because it embodies the Golden Proportion
in every single part of it (Fig. 7). For example:
look at the top horizontal crossing leg of
the figure. From one point to where it crosses
the next line, call that one unit. From where
it crosses the line to the opposite point
is 1.618… or Phi. The relationship or proportion
of the first part to the larger
part
is the same as the larger part is to the
whole line. The smaller is to the larger
as the larger is to the whole. The same proportion
is repeated throughout the Pentagram. Every
part of it is in some sort of Phi relationship
to every other part. It is a truly remarkable
figure.
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Because
if its self-symmetry and use of the Golden Proportion,
the pentagram contains within itself the seed of its
own replication in progressively smaller or larger scales
(Fig. 8). Today we call this fractal geometry.
It is the governing template of nature.
 Archaeologists
have found pentagrams on Mesopotamian potsherds dating
back to 3500 BCE. Pentagrams also appear in ancient
Egyptian, Greek and Roman art.
The
use of the pentagram in Christian sacred geometry is
due in no small part to the writings of Hildegard of
Bingen, the twelfth century Benedictine nun and abbess.
For her, the pentagram was the central symbol of the
microcosm, the reflection on Earth of the divine plan
and the divine image. Hildegard saw the pentagram as
representing the human form because we have five senses
– sight, smell, hearing, taste, and touch; and five
extremities – two legs, two arms and a head. And, because
humankind was made in God’s image, she also saw the
pentagram as representing God.
Other
Christians saw the symbol as representing the five wounds
of Christ and, as such, it was considered a potent protection
against evil. Earlier Hebrew tradition associated the
pentagram with the Pentateuch, the first five books
of the bible.
In
the late Middle Ages, the pentagram became a symbol
of knightly virtues. In the poem Sir Gawain and the
Green Knight, the points of the pentagram symbolise
chastity, chivalry, courtesy, generosity, and piety.
During
the Inquisition, the inverted pentagram became associated
with evil and the Devil. The pentagram was thought to
represent the head of a goat, the devil, or a witch’s
foot. Regrettably the pentagram still carries these
associations today for many people, but it is a gross
debasement of such a truly Divine symbol.
In
architecture, the Golden Mean has been used for millennia
to design buildings, and can be easily spotted in many
a classical façade, usually as a series of Phi rectangles
(Fig. 9).
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To
make a Phi Rectangle, we begin with a square.
Now divide the square into two equal parts
by drawing a vertical line exactly down the
middle. Place your compasses at the bottom
point of this line (i.e. in the centre of
the base line of the square), and set the
radius to one of the top corners of the square.
Draw an arc down to where the base line of
the square would be if it was extended, and
then do that very thing until the base line
cuts the arc. Do the same thing from the top
point of the vertical line, and extend the
top side of the square outwards until it cuts
that arc. Connect those two new points with
a vertical line, and there you have your Phi
rectangle. You can spot this Phi rectangle
in buildings ancient and modern, from the
Parthenon of Athens to the United Nations
building in New York.
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Let’s
delve a little deeper. If you were to subdivide
this second rectangle by making a square within
it (Fig. 10), thus making a smaller rectangle,
then the relationship of the smaller rectangle
to the larger rectangle will be the same as
the larger rectangle is to the whole figure.
You now effectively have two rectangles with
a Phi relationship, and they are both Golden
rectangles. If you keep on doing this sequence
of square, golden rectangle, smaller square
and so on, you would pretty quickly produce
a Golden Spiral. This is the governing form
of growth, and you see this pattern in mollusc
shells, in the arrangement of leaves on a
plant, and the way flies spiral in towards
a light source.
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Where else can we
find this proportion? Almost everywhere in nature. In
the human body, the navel divides the whole body into
a Phi section. In the face, the brow divides the face
into Phi proportion. The lengths of the bones in the
fingers relate to each other in the same way, and so
on, right down to the spacing of protein molecules in
our DNA. So working with the Golden Proportion is very
harmonious to the human body. The Golden Proportion
also manifests in Nature as the spiral of the nautilus
shell, the orbital spacing of the planets, the way plants
grow, and many other processes. There is a mathematical
example known as the Fibonacci sequence that demonstrates
this. The Fibonacci sequence is a specific number series
in which each term is the sum of the two terms preceding
it. It begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55….
and so on. Can you see the progression there? You add
two terms together to get the next one up. Now if we
were to divide each term by the one before it, and plot
the results on a graph, we would get a wildly up-and-down
squiggle that very quickly settles into a slight oscillation
around the number 1.61803… Phi. It never gets there
exactly (it can’t – this is an irrational number remember).
And
finally, did you know that the planet Venus traces out
a pentagram in the skies as it moves along its orbit?
If the positions of the planet are plotted along the
ecliptic (as on an astrological chart, for instance),
then over the course of eight years it will appear to
reverse direction or go retrograde five times and will
trace out a pretty good pentagram! Note the numbers
involved here: five and eight. Both adjacent terms in
the Fibonacci sequence: another Phi relationship.
The
most obvious example of space constructed on pentagonal
principles is, of course – the Pentagon in Washington
D.C. Here the form is utilised more for its defensive
and protective aspects. A much better example is the
Universal Hall of the Findhorn Foundation in Scotland,
designed by George Ripley. But pentagonal geometry can
be found lurking just beneath the surface of many sacred
spaces, perhaps defining the layout of a church apse
or the pattern of a rose window.
So
the next time you visit a gothic cathedral, pause a
while and reflect upon the supreme skills of the Master
who designed and laid out the entire space using little
more than the straightedge, compasses and druid’s cord
of the sacred geometer.
That
concludes this introduction to Sacred Geometry. We have
barely scratched the surface of an entire philosophy,
but I hope this has given you a taster. There are plenty
of books out there, and best one I know of for getting
people ‘into’ Sacred Geometry is Michael S. Schneider’s
“A Beginner’s Guide to Understanding the Universe –
The Mathematical Archetypes of Nature, Art and Science.
A Voyage from 1 to 10.” Don’t be put off by the title;
it’s a terrific book with lots of pictures, and is very
easy to understand. If you’d like to try something more
advanced, try Gyorgy Doczi’s “The Power of Limits: Proportional
Harmonies in Nature, Art and Architecture.” This is
especially good on the Golden Proportion.
Š
Grahame Gardner 2003
Please
contact us to find
a Geomancy Group Member in your area who is available
for workshops and talks.
[1]
An irrational number cannot be defined
as a fraction p/q for any integers p
and q, i.e. it cannot be written as one whole
number divided by another. A number is called algebraic
if it is the root of a polynomial (of any degree)
with rational coefficients. Any number that is non-algebraic
is called transcendental.
[2]
György Doczi: “The Power of Limits” p5
[3]
It doesn’t matter what the actual measurements
of the sides are. As long as they are in 1:2 proportion,
the length of the diagonal will still be an irrational
number.
[4]
John James: “The Master Masons of Chartres”
p39
[6]
Gordon Strachan: “Jesus the Master Builder”
p201
[7]
Robert Ferre: “Sacred Geometry” (lecture
tape)
[8]
John Michell: “The Dimensions of Paradise”
p33
[10]
Michael S. Schneider: “A Beginner’s Guide
to Constructing the Universe” p104
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