INTRODUCTION TO SACRED GEOMETRY
Sacred Geometry works with 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? Let’s begin with the rational.
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.
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. All five Sacred Geometrical ratios we are working with, the square roots of two (1.414), three (1.732), and five (2.238), phi (1.618), and pi (3.1416) are irrational numbers.
There are certain kinds of irrational numbers that are called transcendental numbers. Like irrational numbers which are defined by what they are not (they are not rational numbers), transcendental numbers are so identified because they are not algebraic numbers. 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 is 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. One infinite equation which relates to the value of pi is this:
π/4 = 1 – (1/3) + (1/5) – (1/7) + (1/9) – (1/11) + (1/13) – (1/15) +….
and so on into infinity.
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:
Pi – (p) – 3.1416… : 1 – Pi is found in any circle. If the diameter is 1, the circumference is 3.1416…(C=pD).
Square Root of Two – (√2) – 1.414…:1
Square Root of Three – (√3) – 1.732…:1
Square Root of Five – (√5) – 2.236…:1
Phi – (φ) – 1.618…:1 – Phi is the Golden Section, said to be the first section in which the One becomes the Many.
All five of these numbers gain their meaning only in relation to the One. They are all ratios of x :1. The One is where it begins.
These five numbers are all irrational numbers. These are numbers like Pi p , which is a number that defines the ratio between a circle’s radius and circumference. Everyone knows that the decimal part of Pi (3.141618…) goes on forever. This decimal never falls into a repeating pattern (like 3.141618618618), but is always different, no matter how long you calculate it. That’s an irrational number. Irrational numbers are the keystone of sacred geometry because they manifest the infinite in normal space.
Simplest shape of all, the circle is manifestation of the One. The discovery of the circle arrives as the child discovers self and begins to distinguish his/herself from their surroundings.
This is really the power of the circle. It’s a microcosm of Universe, a horizon, a boundary between self and the rest of Universe. With no beginning and no end, the circle’s circumference is a profound statement about the transcendental nature of Universe. Expanding from the nothingness of its dimensionless centre to the infinitely many points of its circumference, the circle implies the divine generation of shape and form from nothing to everything.
The transcendental number that the circle generates is Pi p (3.1415926….). The circumference is calculated from the radius by the formula 2pr, where r is the radius of the circle. Now 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.
Pi (p) 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, and thus relates to Heaven. It is a good shape to do all kinds of spiritual activities in. It is good for groups to work in circles. Examples of sacred spaces based upon the circle are Stonehenge, the Ring of Brodgar in Orkney, and the Merry Maidens circle in Cornwall.
The Vesica Piscis
The circle divides into the Vesica Piscis, the One becomes Two. Self generates Other. The circle replicates by contemplating itself, reflecting its light, and casting an identical shadow. The vesica piscis embodies all concepts of duality, the two circles both attracting and repelling each other, giving polarity and tension.
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 (Latin for ‘fish bladder’. It represents the birth portal, the cosmic Yoni of the Goddess. All subsequent numbers and shapes can be produced through the portal of the vesica using the sacred geometer’s tools. The irrational number produced by the vesica piscis is the square root of 3. If the radius of each of the circles is 1, then a vertical line drawn down the centre of the vesica has a length of root 3 (1.7320 …. ).
The vesica piscis occurs often in sacred architecture, and a common manifestation is the Gothic arch. A Gothic arch is simply the vesica part (ABC) with vertical extensions from the centre of the sides. Moslem arches also use the vesica, but turned on its side through 90 degrees. 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 a perfect example of this. The vesica is also representative of the shape of the human aura, and is used as such in many religious paintings.
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 (quarters, winds, elements…). We are perhaps more familiar with this shape than any other, since it permeates our lives in practically everything we build or make.
For a geomancer working with sacred space, the important part of the square is the diagonal. If the square has sides of one unit in length, then the length of the diagonal is the square root of 2 (1.41421 … ). Again, this is an irrational number. You can easily prove this for yourself using Pythagoras’ theorem. Stand in the centre of the square and you stand at the centre of two infinite lines. A classical example of this was the Holy of Holies in Solomon’s Temple, where the Ark of the Covenant was kept.
The Double Square
As it sounds, this is two squares side by side, a rectangle with a short side of 1 unit and long sides of 2. This represents the 2:1 ratio of the octave in musical terms. Again, Pythagoras’ Theorem shows the diagonal is the square root of 5.
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 the sacred geometrical 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. 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 – in sacred geometry we’re 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 = 5, therefore hypoteneuse = √5).
The square root of five (2.237…) is another one of these 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 lines of theoretically infinite length. What better space to commune with the One? So in the three-dimensional double-cube of the Kings’ Chamber, every diagonal of the space is root 5, and we can see how the possibilities are magnified when we move into three dimensions.
The Double square is also the shape of the Kaaba in Mecca, and The ‘Holy Place’ in Solomon’s Temple. This was the main part of the temple, not quite as sacred as the Holy of Holies.
The Phi φ Rectangle and the Golden Proportion
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 φ 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.
Perhaps the most difficult concept to grasp, this is also the most cosmic. The Phi proportion, Golden Section or Golden Mean is one of Nature’s universal constants that is all around us, from the growth patterns of plants to the proportions of our own bodies, and yet goes largely unknown. It governs the processes of life and growth. 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 concept of beauty is determined by how closely the facial features of others approach Golden Mean proportions. By definition, the Golden Mean is a way of dividing something into two unequal parts, such that:
the whole = the large part = Phi (φ).
the large part the small part
Numerically, the ratio is 1:1.618033… ,another irrational number.
To try and clarify this, 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.
The Pentagram is quite a remarkable symbol and has a very long history. It is very difficult to draw accurately geometrically, and this is partly why it has developed the occult associations that it has today. It was worn as a hidden sign of recognition by advanced initiates of the Pythagorean mystery school around 500 BC and, one thousand years later, the secrets of its construction were only passed on orally, never written about, 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, published the secret in his book ‘De Divina Porportione’.
The Pentagram is interesting because it embodies the Golden Proportion in every single part of it (Fig. 7). Take 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 φ units. 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.
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. It is possible to construct a set of Golden Mean dividers and go around measuring everything in this way. They look very similar to the drawing instruments called ‘pantographs’, used to enlarge or reduce a drawing mechanically.
The Golden Proportion also manifests in Nature as the spiral of the nautilus shell, the orbits 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, 89…. and so on. Start anywhere in the series, add the number below, and you get the next number (for example, 21 + 13 = 34). The 3:5 : : 5:8. ratio indicates that it is part of the Fibonacci Series.
As one ascends up the series, any number in the series, when divided into the next one up, gets closer and closer to Phi (φ).. but never hits it exactly as it can’t, being an irrational number 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 (φ), the Golden Section.
So how can we use Phi in our sacred spaces? To make a Phi Rectangle, we begin with a square (ABCD). Now divide it into two equal parts by drawing a vertical line exactly down the middle of the square (EF). Place your compasses at the bottom point of this line (E), and set the radius to one of the top corners of the square (D). Draw an arc down to where the base line of the square would be if it was extended, and then extend the base line until it cuts the arc at (G). Do the same thing from the top point of the vertical line (F), and extend the top side of the square outwards until it cuts that arc at (H). Connect those two new points with a vertical line (GH), and there you have your Phi rectangle (ABGH).
Now if you were to subdivide this second rectangle (DCGH) by making a square within it (Fig. 9), 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. You see this pattern is nautilus shells, in the way plants grow, and the way flies spiral in towards a light source.
Solomons Temple also contains phi φ. 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 φ.
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 φ 1.618, and its length is √5, 2.236. And 1.618 + .618 = 2.236.
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, which are both adjacent terms in the Fibonacci Sequence, another Phi relationship.
References & Sources
Sig Lonegren, articles on Mid-Atlantic Geomancy
Grahame Gardner, article in The Dragon’s Egg, Imbolc 2002
John Michell, The Dimensions of Paradise, Thames & Hudson 1988, ISBN 0500013861
John Michell, The New View over Atlantis, Harper & Row 1983
David Furlong, The Keys to the Temple, Piatkus 1997, ISBN 0749917458
Robert Lawlor, Sacred Geometry – Philosophy and Practice, Thames & Hudson 1982, ISBN 0500810303
Michael S. Schneider, A Beginner’s Guide to Understanding the Universe ‑ The Mathematical Archetypes of Nature, Art and Science. A Voyage from 1 to 10.
Gyorgy Doczi, The Power of Limits: Proportional Harmonies in Nature, Art and Architecture.