Sacred Geometry
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Geomancers are interested in sacred geometry because this is the study of the way that spirit integrates into matter - by echoing and amplifying the geometry of nature and planetary movements, we help to align the resonance of body/mind/spirit with the harmonic frequencies of the above and the below.
Geomancers are interested in sacred geometry because it has been found that certain spaces, with particular ratios, enable the participant to resonate or vibrate at the appropriate rate that maximizes the possibility of connection to the One.
A violin ain't built out of a cigar box! It is built with the proper wood with the proper shape and ratios, so that it resonates correctly for the notes/frequencies it is expected to produce. These same principles are applied to sacred spaces to maximize the possibility that whatever is being done there on spiritual levels will succeed.
Definition: Two Dimensions
I've been a student of sacred geometry for over thirty years. While there has been recent interest in three-dimensional sacred geometry based on the Platonic Solids and in sacred sites themselves, most sacred geometrical documents I've read talk in only two dimensions - height and width.
Obviously there is a fourth dimension and others beyond it that are much more complex and sophisticated. But why does the record left to us from geomancers of the past come primarily in two dimensions? (Don't just tell me about paper being only two-dimensional! ;-)Two is closer to the One than three is. It's less complex. I think one of the biggest mistakes Western geomancers have made was to take something that is very simple and make it much more complex. The Chartres Labyrinth strikes me as being an example of this. This stuff is simple. If you really gnow (that is, know both rationally and intuitively) a handful of irrational ratios - pi (π), phi (Ø) and the square roots of two, three and five, you've basically got it all.
Three-dimensional sacred geometry just builds on this basic handful.
Numbers
One aspect of Sacred Geometry is that it uses both rational and irrational numbers. To go to the spiritual, one must go beyond the rational, and it appears that some of these ratios and numbers can lead us there. By being inside a sacred space that has been constructed using one of a handful of these sacred geometrical ratios, the resonance that has been set up can enhance the possibility of your making the spiritual connection you want to make.
So, what are these irrational numbers? (Thanks to Forrest Cahoon for his help with the following mathematical definitions.) Let's begin with the rational.
Rational Numbers
A rational number is a number which can be expressed as the ratio of two integers (whole numbers), such as 1/3 or 37/22. All numbers which, when represented in decimal notation, either stop after a finite number of digits or fall into a repeating pattern, are rational numbers.
Irrational Numbers
An irrational number is one that cannot be represented as a ratio of any two whole-number integers, and consequently it does not fall into a repeating pattern of any sort when written in decimal notation. IRRATIONAL JUST MEANS "NOT RATIONAL," NOT the connotations it has today of groundless, baseless, unfounded, unjustifiable; absurd, ridiculous, ludicrous, preposterous, silly, foolish, or senseless.
All of the Sacred Geometry ratios we will be working with, the square roots of two (1.414), three (1.732) and five (2.238), phi (1.618) and pi (3.1416), are all irrational numbers.
Transcendental Numbers
There are certain kinds of irrational numbers that are called transcendental numbers. Just like irrational numbers, they are defined by what they are not (they aren't rational numbers), yet transcendental numbers are so identified because they are not another sort of number,
known as an algebraic number.
Any number which is a solution to a polynomial equation is an algebraic number. A polynomial equation is a sum of one or more terms involving the same variable raised to various powers, for example:
7 (x5) + 5 (x3) + x = 137
Any x for which any such equation is true is an algebraic number. Because the square root of two is a solution to the polynomial equation
x2 = 2
it is an algebraic number.
A transcendental number requires an infinite number of terms to be defined exactly. That's one way of thinking of God/dess. There are special equations to derive transcendental numbers where the terms get smaller and smaller as you go along, so you can keep adding them together to reach any level of accuracy you need, but the true number cannot be reached exactly. That is the beauty of transcendental numbers!
pi ( 3.1416...) is such a transcendental number. It is the only one we will be using here with Sacred Geometry. One infinite equation which relates to the value of pi is this:
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π — 4 |
= 1 - (1/3)+(1/5)-(1/7)+(1/9)-(1/11)+(1/13)-(1/15)+ and so on into infinity. |
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Squaring the Circle
Using the Earth & the Moon
Now here's one that compares the Earth to the Moon! I'd like to thank John Michell for first pointing this one out to me.
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Create a square (ABCD) with (AB) = 11. |
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Construct two 3 • 4 • 5 right triangles, with the 4 • 5 angles at (A) and (D). |
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Draw line (Ee) which intersects side (AD) at (F). The smaller circle thus created is to the larger circle as the moon is to the Earth! With your compass point at (E), create a circle with radius (Ee). This creates a circle whose circumference is very close to the perimeter of square (ABCD) - another squared circle. |
The Math
(AB) = 11
(EF) = 1/2 of (AB) = 5.5; and (ab) = 3
(eF) = 5.5 + 1.5 = 7. The circumference of a circle is equal to two times the radius (the diameter) times pi (3.1416). or
C = π d. or
C= 3.1416 x 14.
or C= 43.9824 In Square (ABCD), (AB) = 11
The perimeter of a square is four times one side. 11 x 4 = 44. Very Close, what?
According to the Cambridge Encyclopedia, the equator radius of the Earth is 3963 miles.
The equator radius of the Moon is 1080. The claim is that the smaller circle (in square abcd) is to the larger circle (in square ABCD) as the Moon is to the Earth. (EF) = 5.5
(F e) = 1.5
5.5 : 1.5 :: 3963 : 1080
5.5 / 1.5 = 3.66666
3963 / 1080 = 3.6694 - (if it had been 3960, it would have been exact!)
The following page carries the reader further into Sacred Geometry. It is mostly text, but worth the effort.
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Squaring the Circle
The square represents the physical. The circle represents the spiritual. Squaring the circle is an attempt to unite/bring together the physical with the spiritual. The Great Pyramid is a physical place where one can bring one's physical body and experience the spiritual - this, of course, is the purpose of all sacred spaces. All sacred geometers have attempted the impossible: to square the circle (create a square who's perimeter is equal to the circumference of a circle.) It can not be done exactly (because we are working with irrational numbers), but we can get pretty darn close.
Here is the first of two valiant attempts:
This squaring of the circle works with a right triangle (YZE). Let's begin with a line drawn from the centre of the base of one side (Y) to the centre of the base itself (E). Then a line from the centre of the base (E) to the top of the pyramid (Z) creates the right angle (YEZ). The hypotenuse of this right triangle (line ZY) is called the "apothem." {On a pyramid, the apothem is a line that splits vertically one side or face of the pyramid (line ZY) - a line drawn from the base of the centre of the base of one side to the top.}
We will find that is is line (EZ) that will help us to square the circle, but the apothem has an interesting characteristic as well.
Now let's look at this pyrmid in 2D, at an oblique angle from above.
Squaring the Circle with the Great Pyramid
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(ABCD) is the base of the Great Pyramid. This is lettered similarly to the wire frame version on the previous page. We also find the centre of this square is at (E), and we divide the square into four equal smaller squares (AiEh), iEfD), (fEgC) and (hEgB) For the purpose of this exercise, the side (CD) of the base equals 2. |
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Construct square (i JKD), thus creating double square (JKfE). |
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Create diagonal (EK) which intersects (i D) at (l). iD = 1, therefore the diameter of the circle is also 1. Remember, (EK), the diagonal of a double square, = (√5) = .618 + 1 + .618. |
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Put the point of your compass at (E) and extend it along the diagonal (EK) to point (m) where the circle intersects (EK), and draw the arc downward to intersect line (K D f C) at (n). If (EK) = (5), and lines (lm) and (lD) and (li) all = .5, the diameter of this circle is 1. This makes line (Em) = .618 + 1, or 1.618. Line (Em) is called the apothem. |
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Draw line (En) which intersects line (AilD ) at (o). Put compass point at (f) and extend it to (n). Again put your point at (E) and draw the circle which happens to have the radius (E o). (f n) is the height of the Great Pyramid. This circle comes remarkably close to having the same circumference as the perimeter of the base (ABCD). |
Let's go back to the original right triangle (EYZ)
if (EY) = 1, then (YZ) = phi
and (EZ) = (√phi)
EY = 1, The apothem is phi/1.618. This makes the famous 51+° degree angle.
This makes the height of the Great Pyramid the square root of phi (ÿ).
Using a2 + b2 = c2
or (EY)2 + (EZ)2 = (YZ)2
or 12 + √ø2 = ø2
Is this possibly true? - Yes!
12 + 1.272019652 = 1.618033992
or
1 + ø = ø2
or phi plus one = phi square!!!
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Ø - Phi — 1.618:1 — the Phi Rectangle
 The Golden Section, phi, 1.618: |
In the Beginning was the One. In order to observe itself, it cut part of itself away to make 'Other'. This Golden Section is in beautiful proportion. As the subdividing continued away from the One, they continued in this phi ratio. This can be used to go back to the One as well. It is in this sense that three is farther away from the One than two is. Have you ever noticed that it is easier mathematically to go away from One than to go towards it? In other words, it is easier to add and multiply than it is to subtract and divide. |
3:5 : : 5:8. This ratio indicates that it is part of this series: 1 • 2 • 3 • 5 • 8 • 13 • 21 • 34 • 55 • 89, and so on. This is called the Fibonacci Series. Start anywhere in the series, add the number below, and you get the next number (for example, 21 + 13 = 34).
As one ascends up the series, any number in the series, when divided into the next one up, gets closer and closer to (but never hits exactly) 1.618, phi, the Golden Section.
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On a line create square (ABCD) where AB = 1. |
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Divide lines (AD) and (BC) in half at (F) and (E). (BC) = 1, (EC) = .5. Double square (ECDF) is thus created with a diagonal of (ED). |
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Using (ED) as a radius swing arc from (D) downwards to intersect the initial base line at (G). Extend line (AFD), and create a perpendicular to line (BECG) at (G) so that it intersects line (AFD) at (H), thus creating phi rectangle (ABGH). |
The formula that shows this is:
(BE) = 1/2. (ED) = 5/2 .5 + 1.118 = 1.618
Phi and the Square Root of Five:
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Extend arc (DG) through (A) to (I). Note the clear relationship between phi and the square root of five. |
Solomon's Temple also contains phi. The Vestibule (DCBA) measures twelve cubits by twenty cubits. 12 to 20 can be reduced to 6 to 10 and further to 3 to 5. Three and five are two numbers in the fibonacci series. 3/5 = 1.6, a close approximation to 1.618, or phi (Ø).
There is also a solid astronomical alignment to the Equinox Sunrise for this chamber as well.
Designed by Phidias, the Parthenon is the Queen of Greek Temples, and personifies their interest in Sacred Geometry. If the height of the Parthenon is 1, its width is phi (Ø) 1.618, and its length is 2.236 (√5). You can see this relationship more clearly at the bottom of the preceding page.
1.618 (phi or ø) + .618 = 2.236 (√5). So there is a clear relationship between phi (ø), and the square root of five (5).
These are the 5 sacred geometrical ratios - pi (π), (√2),(√3),(√5) and phi (ø). They are found in sacred spaces all over the world.
Remember, sacred geometry is basically simple, AND you must do it with your hands, if you want to really gnow sacred geometry. (Western Man's left brain got the best of him, and so, unfortunately, the closer we get to the present, the more complicated it gets - but it should be basically simple!)
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The Double Square and √5
 The Double Square: Short Side = 1 Longer Side = 2 Diagonal = Square Root of Five, 2.236 |
The Double Square is found in some of the best known sacred spaces in the world, from the King's Chamber in the Great Pyramid and Solomon's Temple in the Bible to the interior of Calendar II, an important underground stone chamber in Vermont, USA. The diagonal of a double square is to the shorter side as the square root of five is to one. The square root of five = 2.236. |
| Solomon's Temple provides numerous examples of sacred geometry. The holy place (EFCD) is the place where good Jews who had been properly cleansed could go. This space measures twenty cubits by forty cubits. Another place where a double square is found is the Calendar II underground stone chamber site in central Vermont in the USA. It measures ten feet by twenty feet. |
(CDEF) Double Square in Solomon's Temple |
Calendar II, a drystone walled underground stone chamber in central Vermont, USA. Archaeologists say this is a colonial root cellar, but it exhibits all of the characteristics of sacred space, and was probably built by Native Americans.
The interior of the chamber is 20 feet long by 10 feet wide , or 2 to 1. It has seven massive ceiling slabs, and a rectangular flue hole in the ceiling at the back of the chamber. The mouth of the chamber is oriented towards the Winter Solstice Sunrise.
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Square Root of Three — 1.732 : 1 — Vesica Pisces
The Vesica Pisces is created by two identical intersecting circles, the circumference of one intersecting the center of the other. The vulva-shaped space thus created is called the Vesica Pisces.
This is the lid of the Chalice Well designed by Bligh Bond in the early part of the 20th century. It covers one of the most powerful Holy Wells in Britain. The Chalice Well has numerous examples of vesicas. Even Nature demonstrates this.
Vesica Yew
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| the Gothic Arch is the top half of a Vesica Pisces - see Chartres Cathedral. It This vescia is the sacred geometric shape of the Piscean Age. | Gothic arch on the tower on the Glastonbury Tor. This site was a hermitage and retreat for early Christian monks | Gothic arch in the Gallilee of Glastonbury Abbey. Note circular Romanesque arches behind in the Mary Chapel. |
The Vesica Pisces Two Circles share a common (AB). Radius AB = 1 The intersecting circles create a Vesica Pisces. The minor axis of this Vesica Pisces (AB) = 1, The major axis (CD) = the square root of three, 1.732 Proof: in right triangle (EBC), EB = 1/2 AB, or .5 CB is also a radius of the circle whose center is B, so CB = AB = 1. Therefore using the Pythagorean Theorem CD is perpendicular to AB; therefore angle CEB = 90° |
SYMBOLS USED:
∴ Therefore
* Times
∞ Infinity
CD:AB :: √3:1 CD is to AB as the square root of 3 is to 1
a2 +b2 = c2
EB2 + CE2 = CB2
EB2=1/2=.52
52=0.25 + CE2 = (CB)=1
∴ CE2 = 0.75
CE = √0.75
CE = 0.8660254038
CE is 1/2 of the major axis CD,
∴ CD = 2 * 0.8660254038
CD = 1.7320508076∞
√3 = 1.7320508076∞
Therefore CD = 1.7320508076∞
and CD:AB :: √3:1 Write comment (0 Comments)
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Square Root of Two — 1.414 : 1 — the Square
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In sacred geometry, the square represents the physical world. It can be defined totally. If its side is one, its perimeter is exactly four, and its area is one square - exactly. The Square is yang. The diagonal of a square whose side is 1 is √2, or 1.4142135624∞. This is what is called an "irrational number." While there are several ways to describe an irrational number, one definition is, "a decimal number that never ends, never repeats." Never ends? That takes it to infinity (∞) - into the spiritual. So while the square represents the physical, it has within it, a connection to the spiritual. All sacred geometrical figures have this connection. |
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The square was found was in the Holy of Holies (the back room) of Solomon's Temple (G,H,F,E). This was where the Hebrews kept the Ark of the Covenant and other most sacred treasures. (The dimensions here are taken from the first part of the Ezekiel Chapter 41.) |
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On top of Glastonbury Tor sits an impressive stone tower. The Tor and its tower dominate the Somerset Levels. Its visual impact shows that this is one of the major power centres in a place repleat with Earth Energies. |
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This is a view taken from inside the tower looking upward. It is a pefect square. The Tor is dedicated to Archangel Michael. One of his jobs is to ground spiritual energies. This is often symbolically depicted as him spearing a dragon in to the Earth. |
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pi — 3.1416 : 1 — the Circle
 The Circle: Radius (CD) |
Pi (3.1416 : 1) is found in any circle. In sacred geometry, the circle represents the spiritual realms. A circle, because of that transcendental number pi, cannot be described with the same degree of accuracy as the physical square. The circle is yin. It is a good shape to do all kinds of spiritual activities in. It is good for groups to work in circles. There are many examples of sacred spaces that are circular. |
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Ring of Brodgar, Mainland Orkney is a true circle. Most stone rings in the British isles are not actually circular. Dr Alexander Thom proved this with his pioneering work in the sixties. Some of the true circles are Merry Maidens in Cornwall, Stonehenge and the Ring of Brodgar. |
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Howard Issac, Seneca Native American, stands in front of a Medicine wheel at the home of Wolf Clan Mother Grandmother Twylah Nitsch on the Seneca Cattaraugus Reservation in western New York. Note the clear demarcation of the Four Directions. The circle encloses it all. |
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This circular dance pattern/labyrinth is found in the floor of the nave of Chartres Cathedral. The labyrinth is the same distance inside the front door as the Rose Window is above it. This circular stain-glass masterpiece and the labyrinth are exactly the same size. |
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Dear Reader - Mid-Atlantic Geomancy intends to be graphic intensive. This page is all text. I felt it necessary to get some basic sacred geometrical information out in this manner. It's only one page, so please chew through it.
Many, like myself, have suffered from math abuse at school (I have dyslexia). The math you will use here is basic and different. If you come to a section that is too difficult/threatening, skip just as little as you can, but please, continue! : )
If our Maker wants to talk with us, S/He will get us anywhere and anytime. There was a Christian-baiter named Saul who was travelling the road to Damascus on his way to bait some early Christians. God grabbed him and struck him blind. He became that misogynist Paul of New Testament fame. God/dess can grab your attention in the middle of Times Square or Piccadilly Circus. Although, as Great Mystery (my word of God/Goddess/the One) is everywhere, I don't know about you, but I wouldn't choose to attempt to contact Him/Her in such a place.
If we want to talk with God/dess, experience has shown that it helps to be in the right environment. Spiritual seekers from Mayans through Christians, Native Americans, Egyptians and Hindus to the Neolithic builders of the stone rings in Britain and Ireland (and many more) found that by constructing their sacred places using certain geometrical ratios - just a small handful of them - they could more easily connect with their Maker.
Yes, it is possible to speak with our Creator anytime (even in Times Square); however, sacred geometry makes this easier, and different ratios make different connections easier. The ratios have to do with different spiritual activities like healing, foretelling the future, long-distance communication, levitation and, most important, heightened ability to communicate with our Maker. These ratios help us to vibrate at the appropriate frequency to aid us in accomplishing the particular spiritual activity we have in mind.
The Five Basic Sacred Geometrical Ratios
When one looks at sacred enclosures globally, there is a group of five mathematical ratios that are found all over the world from Japan's pagodas to Mayan temples in the Yucatan, and from Stonehenge to the Great Pyramid. These ratios are:
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pi - (π) - 3.1416... : 1 - Pi is found in any circle. If the diameter is 1, the circumference is 3.1416 (C = D).
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Square Root of Two - (√2) - 1.414
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Square Root of Three - (√3) - 1.732
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Square Root of Five - (√5) - 2.236
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phi - (Ø) - 1.618 : 1 - Phi is the Golden Section of the Greeks. It was said to be the first section in which the One became many.
These are all irrational numbers. I have seen pi taken to 1500 decimal places with no discernable pattern to it (is that Chaos?). Let's take a closer look at each of these special numbers, and see how we can find them in the sacred geometry used by geomancers around the world. All five of these numbers gain their meaning only when beaten against the One. They are all ratios of x : (to) 1. The One is where it begins. "In the beginnng was the Word" (a vibration).
While sacred geometry is primarily understood through your hands and eyes, by doing it, Numbers also play an important part in that they provide "proof" to our analytical side, and in this king of geometry, the Sacred geometry of Nature, some amazing "co-incidences and realizations occur.
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