In Part One, we looked at a very simple sequence of numbers that ‘orbited’ or homed-in on a certain value. Now we need to examine that value and look at the sheer magic of what it represents.

This new number was 1.618. It’s derivation is summarised in the diagram above, and described in the previous post. Simply: (red numbers) we add the two previous numbers to get the next. Next: (green numbers) we offset the first line of numbers one place to the right and, using a calculator to three decimal places, we treat the offset numbers of fractions, one number above the other. The third (black) line gives the calculator results, which stabilise at 1.618.

The ‘series’ that generates it – known as the Fibonacci series – came into existence at the time the world was abandoning the old and (by then) clumsy Roman notation (I, II, IV etc) and moving to the Arab-derived numerals that we use today.

The special number 1.618 is known by many names, such a the Golden Ratio and the Golden Mean. It is a number that shows us how we can divide something to protect its ‘wholeness’ in a harmonic way. By doing this, the divided figure will always exhibit pleasing proportions when placed next to (or within) the ‘parent’ figure. For example, Leonardo Da Vinci used it, extensively, in his most famous pictures.

But there are much deeper implications to this than something that looks or feels good, important though that is.

The materialist sees the world as having numbers by virtue of an ‘accident’ that they fit how we see and describe things. The mystic looks for the experience of ‘oneness’ with the processes that created the universe. You can’t find that experience unless you look for it. The universe owes us no debt of making it happen in our minds and hearts – the search must be ours… then the doors of perception will be opened.

Imagine that we have a strip of paper that we are going to divide by cutting with scissors. Let’s say the length of the initial strip is represented by the letter ‘A’. When we cut the strip we will have three values: the initial length (A); and the lengths of the two pieces we produce. We can name the two ‘child’ pieces (a) – the longest, and (b) – the shortest.

Under all circumstances, the original length (A) would be equal to the sum of the two children (a+b) . We can write this A=b+c, the most simple kind of ‘equation’ we could every want to see.

The miraculous Fibonacci number (given the name Phi in the 20th century) gives us the means to divide the original strip of paper such that the longer of the two child pieces bears the same relationship (ratio) to the original strip, as the larger child does to the smaller…

We can keep on doing this – cutting each successive larger portion – with smaller and smaller divisions of the original strip of paper. The whole ‘creation’ will be in harmonic proportions. This generation of smaller and smaller ‘harmonic’ children is called self-similarity.

Nature uses ‘Phi’ all the time. The recent science of Fractals shows how essential self-similar division is for nature to achieve its purposes. A tree is a fractal, for example, as are our lungs. Our blood vessels can carry oxygen to our cells because they follow fractal rules of becoming smaller and smaller within the finite space of our bodies. Only by using such structures can incredibly large processes fit into small spaces. The generation of Phi is not a fractal process, but it perfectly illustrates the marvel of the related fractal structures in nature.

Examples of this in nature include the petals of flowers, such as the sunflower, and the spirals of nautilus sea shells… But there are innumerable examples.

So, how would we actually work out the Phi-derived point of where to cut our twenty-unit strip of paper? We can arrange the self-similar formula so that we have a quadratic equation to solve, but where’s the fun in that!

Instead, we can look at the workings of the older graphical method carried out with the use of compass and straight edge. This brings home the inclusive and ‘connective’ nature of working by hand and is illustrated below:

The horizontal line A-B is the length of paper we wish to divide into the harmonic proportions given by the Fibonacci-derived Phi number 1.618. In this example, the length is 20 units.

To begin, we imagine we have turned the base line (A-B) into a square of four sides and select its right-hand vertical halfway point.

To shorten this, I have simply created point C at the correct half-value (10). The compass is placed on point C and set to the distance of C-B. We begin to draw an upward arc from B to the intersection with the hypotenuse A-C. We then set the compass to a base at the origin – A, and extend its pencil to the previous intersection with the hypotenuse. This time we draw downwards until the curve intersects with the original length A-B. The point of crossing is the length of the largest ‘child’ as above.

The length value, the golden ratio, gives us a new ‘longest child’ length of 12.36 units. We could cut at this point. The relationship of the larger child to the smaller is the same relationship as the original full length to the largest child.

This process could be repeated to infinity using the successive larger pieces. The entire family of larger pieces would inherit the divine proportions of the ‘mother’ length.

In the final post, next week, we will examine how the pentagram combines all the above properties into a single figure of dynamic value to mankind.

Other posts in this series:

One This is Two.

©Copyright Stephen Tanham

Stephen Tanham is a Director of the Silent Eye School of Consciousness, a not-for-profit teaching school of modern mysticism that helps people find a personal path to a deeper place within their internal and external lives.

The Silent Eye provides home-based, practical courses which are low-cost and personally supervised. The course materials and corresponding supervision are provided month by month without further commitment.

Steve’s personal blog, Sun in Gemini, is at stevetanham.wordpress.com.

In the dream the Hermit was speaking. “I am the eye of fiveness,” he said.

I listened… dreams are not always this lucid.

“In the beginning was the division, not the multiplication; and the division contained what divided it, but in another form…”

I was listening, intently. The figure of the Hermit promised great insight…

“No-thing can be a principle. It does not have to be nothing…” He paused, smiling. “But it does have to be a ghost…”

I wondered if you could blink in a dream… apparently not. He placed a dot in the centre of the sky. Then extended his arms into the shape of a draughtsman’s compass and drew, black on azure, a circle.

I smiled, understanding something, at last. “Ah, yes,” I said. “The dot is no-thing, but has existence – if only as a position. The dot is zero.” I paused, aware that my thinking was waking me up… I had to get to a point of…. memory or I would lose it all…

“The circle is everything and everywhere,” I said. “The circle – One – is the arena of existence!”

The way the Hermit faded indicated he was pleased. And then, like the Cheshire cat that left only a smile, he was gone…

That dream was many years ago. I was studying the pentagram and the way it was used in a magical school of the soul. I knew that the geometry – and hence the numeric basis – was closely linked to the organic life we all share.

But I wanted something deeper… and had asked for it.

I consider that my attitudes are roughly half ‘science’ and half mystical. That way, I avoid the worst excesses of both, such as mysticism’s inclination to be fluffy, and to espouse the most complex ‘magical’ theories, even if they are twice as forced as the simplest scientific truths.

Equally, science’s dogmatic adoption of the ‘we are the only truth‘ attitude is to be avoided. Consciousness is not rooted in numbers, but the human mechanism – the body and how it works – is.

So, if you’d like to join in, let’s go in search of what’s at the heart of the pentagram: ‘fiveness’. Stand up and take yourself into a different mental and emotional ‘space’. Tell yourself that you’re not doing something trivial, but something that’s a living key to how you are, or were, before the layers of civilisation, work and family walled us all in numbness.

Stretch out your arms and legs so you look like Leonardo Da Vinci’s famous picture of the ‘Vitruvian Man’. Each hand’s five fingers and each foot’s five toes make up four of the five points of your human pentagram. Your head – the controller, communications receiver and maker of your organic ‘me’ – forms the other. We may reasonably ask what is the fiveness of the head? The question may already have triggered an answer in your mind….

If you’ve never encountered the Fibonacci series, stay with me and I’ll do my best to explain it – very simply. It’s worth the few minutes it will take to understand it.

We begin with zero, then one, because the whole of metaphysics is based on their relationship. Zero is the potential for all numbers to exist. One is symbolically the ‘monad’ – the complete everything from which we come and to which we will return; but One is also the first number, so is doubly useful in this example. We could say that, in Nature, everything is a fraction, yet Nature knows no fractions… Mankind sees only fractions, yet contains the seed of that which caused that division in the first place.

From zero and one, the next term in the series of Fibonacci numbers is generated by adding the two previous numbers. So, (from 0+1) we get another 1. At this point the series starts to take shape, growing quickly as each new number emerges from the sum of the previous two.

The row of green numbers is exactly the same line of numbers as the red ones above. But they have all been shifted one place to the right. What we’re going to do now is to create a fraction (don’t panic – I’ll do the calculations!) from each of the sets of two numbers; one above the other. So the first one would be 1/0 which is an invalid number, since we cannot divide by zero in ordinary mathematics. The next one is 1/1, which is just 1. The next one is 2/1, which is 2. We can see from this that we are swaying from one ‘extreme’ to the other; between the numbers 1 and 2.

Let’s continue to work these numbers to see what it is that we are swaying around… This is a bit like finding you’re a spaceship being pulled into the orbit of an unknown planet… but this planet holds one of the fundamental keys to the Universe…

It takes only ten ‘terms’ of the fractions from the Fibonacci series to produce the hidden planet to which our spaceship is being drawn. If you have a calculator you can check the fractions which lead to it; 5 divided by 3, 8 divided by 5, etc. Each of these divisions gets closer to a number that emerges in the greyed out boxes of term ten, above. From there onwards, the number 1.618 is present in all the results, which continue to ‘sway’ around finer and finer divisions of this mysterious destination.

In fact, we can never get at the final answer, since it is what maths calls an ‘irrational’ number – one that isn’t really a number at all, but is only defined by (in this case) an infinite (never-ending) convergence towards smaller and smaller units.

In practical terms this doesn’t matter. The three decimal places of 1.618 will do us fine. To go beyond this would involve us worrying about one ten-thousandth of a unit, which would be needed in only the most specialised engineering application, such as space travel!

But enough of the maths! We’ve landed on planet Phi… This mysterious number is so important that, like its cousin Pi, it has its own name. Phi is also known as the Golden Mean, the Golden Ratio… and a host of other historic names. It has been with us for a long time.. and very few people know its full significance.

In the next post we’ll examine what it Phi really means; and why it makes the Pentagram and Pentagon such important geometric figures in any world where harmony is important… which is just about everywhere. We’ll also consider why Phi is truly the ghost in the machine…

©Copyright Stephen Tanham

Stephen Tanham is a Director of the Silent Eye School of Consciousness, a not-for-profit teaching school of modern mysticism that helps people find a personal path to a deeper place within their internal and external lives.

The Silent Eye provides home-based, practical courses which are low-cost and personally supervised. The course materials and corresponding supervision are provided month by month without further commitment.

Steve’s personal blog, Sun in Gemini, is at stevetanham.wordpress.com.