The Golden Proportion

by Grahame Gardner

During Billy’s dowsing demonstrations at the October weekend in Northampton, he mentioned the detrimental spirals given off by some pendants and jewellery, unless they were constructed to certain geometric principles, in particular the Golden Mean. I noticed this produced a few blank faces in some of the group, followed by a flurry of note taking. To those people who like many others, have heard of the Golden Mean but don’t really know what it is, this article is dedicated.

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 is what is known as an irrational number. These are numbers like Pi (Π), which, as I’m sure you can remember from school geometry, 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. It never falls into a repeating pattern (like 3.141618618618). It’s always different, no matter how long you calculate it. That’s an irrational number, and Pi and Phi are both examples of these.

Figure 1
The pentagram as constructed from the Vesica Piscis

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. 1).

The Pentagram is quite a remarkable symbol and has a very long history. It’s 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 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’.

Figure 2
Golden Proportion in the Pentagram

The Pentagram is interesting because it embodies the Golden Proportion in every single part of it (Fig. 2). 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 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). 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, right down to the spacing of protein molecules in our DNA. So working with the Golden Proportion is very harmonious to the human body, and this is presumably why jewellery embodying this proportion does not emit detrimental energies.

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).

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. 3).

Figure 3
The Golden (Phi) rectangle

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.

Figure 4
The Golden Spiral

Let’s delve a little deeper. If you were to subdivide this second rectangle by making a square within it (Fig. 4), 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 is mollusc shells, in the arrangement of leaves on a plant, and the way flies spiral in towards a light source.

I really cannot do justice to this sacred proportion in this brief article, but before we leave the subject, let me give one more fascinating fact about the Pentagram. 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.

If you would like to learn more about Phi and sacred geometry, there is a longer version of this article on the website of The Geomancy Group.

© 2002  Grahame Gardner & BSD EEG