The Golden Mean


The Golden Mean is a name for a proportion which has been important in art and
religion throughout known human history.

It is derived from a series of numbers known as the golden numbers.

It is said that if you divide any line at the most pleasing point, the two parts will be in the
golden proportion to one another.

The ancient Athenians had a precise proportion which they called phi, which may be
defined as the square-root of five plus one over two [(5^.5+1)/2]. This is a precise
expression of the golden mean.

The Athenians considered it important enough to use it in the construction of the
Parthenon.

It is used in the Mona Lisa and the Great Pyramid at Gizeh.

The Greeks and the Renaissance artists considered it to be the most pleasing proportion
to the human eye - unsurprising since it is almost omnipresent in living things, including
our own bodies.

The golden mean has the unique property
phi squared = phi + 1
This means that if you take a golden rectangle (i.e. a rectangle with one side phi times the
other)(construct a golden rectangle) and add a square whose sides measure the same as
the long side of the golden rectangle to the long side of the golden rectangle, then the
resulting rectangle will be a golden rectangle whose sides are phi times the sides of the
original rectangle.



You can also, if you like, take away a square whose sides are the same as the shorter
side of the golden rectangle, from the shorter side of the golden rectangle, and the
resulting rectangle will be a golden rectangle whose sides are 1/phi times the sides of the
original rectangle.

I hope that these properties are illustrated by the colourful but approximate golden
rectangles above.

This is one way in which phi appears as a symbol of generation of the infinite from the
infinitesimal. There are others...

The pentagon and the golden mean

Symbols

The expression of truths beyond the scope of normal language may be achieved by the
use of symbols. Geometrical symbols demonstrate unchanging truth by their nature,
making it possible to speak to people separated by vast gulfs of time or culture, with some
hope of being understood.

In Part Seven of 'The Republic' (the original Greek title of which, means something like; 'to
do with living in groups'), Plato  tells a story known as the 'parable of the cave', to
illustrate the conditions of human knowledge.



Plato's cave  -  ideals and their  material reflections


Imagine a cave running deep into the ground. In this cave, far from the light, are people
who have been fastened, from birth, so that they cannot move or turn their heads but
must always look straight ahead of them.

Behind them is a wall, and behind the wall, a road, and behind and above the road, a fire
which burns high enough to cast shadows of the prisoners on the wall of the cave in front
of them.
Furthermore, there are people walking along the road carrying all sorts of objects,
including models of people and animals. Sometimes the people walking on the road are
speaking, and sometimes not.

All that the prisoners know of the world, of themselves, and of their fellows, is the sound of
their voices and their shadows on the wall opposite, so when they see the shadows of
objects being carried down the road and hear voices, they regard the shadows as real
people who are speaking to them.

Now imagine that one of them is released and made to turn his head to look at the fire. His
movements are painful, the brightness of the fire dazzles him so that he cannot see the
objects that cast the shadows and he still regards the shadows as the reality and the light
and all its confusion as unreal.

When he is told that the objects are nearer reality than the shadows, he does not believe
it. When he is made to look straight at the fire, it hurts his eyes and he tries to return to
his former condition and the world he knows.

Now, he is carried against his will up the cave towards the open air and the sunlight. He
struggles every inch of the way, and when he is finally out he is so dazzled he cannot see
the things he is being told are all around him.

Gradually, however, he starts to make out shadows, then reflections, then the objects that
cast them; after that, he starts to make out stars and the moon, and, finally, the sun.

In time, he comes to think that the sun produces the seasons and the year and gives rise
to all the world and everything that he used to see in the cave. When he thinks of his
former life he is delighted that he has escaped, and when he thinks of his fellows still in
bondage he longs to share his new understanding and freedom.
If he returns to the cave he will be initially blind in the darkness, and unable to distinguish
himself in the wisdom of the cave, which consists of guessing which shadows and sounds
will appear next. The things which he says about the upper world will be baffling and seem
insane, and if he tries to force the cave-dwellers to be free, they will resist, and if they can
use violence on him and put him to death, they will.

In this story, the prisoners represent our normal, mundane level of awareness;
the fire is the sun, giving rise to the shadows which represent the material world, and
the shadow-guessing cave-wisdom is the social, political and material knowledge
that lets us 'get on in the world' and 'get along with people', which is for the most part
(like the refusal of the prisoners to look at the objects which cast the shadows) a
collection of mechanisms to prevent any real contact with the world or other people in it.
To understand the fire represents understanding the laws that govern the material world.
To leave the cave and learn about the outer world represents achieving transcendent
understandings.
Plato held that the highest aim of a thinking being was to attempt to discern the underlying
truth and, having discerned it,  to return to the cave with evidence of the underlying
nature. He held that geometry was an important part of the evidence available to us of the
true light.

On the 'sacred' nature of geometry
construct a golden rectangle

The underlying (or overarching - it's the same) nature of everything is
geometrical. We can talk about it as 'energy grids' or the 'architecture of the
universe', as practised by the 'great architect' in the sky, but what it is, is the way
'big' stuff is when we look at it from a distance, and the way 'small' stuff is when
we look at it from a distance - beyond the levels of the instruments currently
available - and it makes patterns.

So, of course, does every thing on the smaller, physical scale, but we tend not to
notice these, since  they are overlaid with concepts which do not obviously
match up with the underlying structures, in the same way that our surface
utterances seem to differ from the underlying sentence structure of our
language(s).

If we compare all these energy systems, though, we see that they follow the
same splitting, branching pattern (trees, rivers, veins, nerves, lightning,
sentences), which, if we take it back, goes to the One which is the All.
USEFUL LINKS
Twelve around 1