Yes, Phi is one of the most fascinating numbers in Mathematics. Geometrically it is related to the proportions of the sides of simple figures (e.g. the rectangle).

Because of the beauty of such proportions it is variously referred to as "Divine Proportion, Golden Mean, Golden Proportion or Golden Ratio.

In terms of the rectangle if one side (the vertical side) has length of one unit, the other horizontal side will have length of x units (where x = (x + 1)/x.

This value of x (which is an irrational number) = 1.618….. and is what is referred to as Phi.

(I describe later how to geometrically construct the rectangle that demonstrated the Golden Proportion).

It has surprising connections with some simple numerical sequences, one of the most famous of which is the series of Fibonnaci numbers.
 
 

Perhaps I should clarify the famous Fibonnaci sequence. This originated in the 13^th century in an attempt to understand the rate at which rabbits reproduce.

"A man puts a pair of rabbits in a certain place surrounded by a high wall. How many pairs of rabbits can be reproduced from that pair in a year, if the nature of these rabbits is such that each month each pair bears a new pair which from the second month on becomes productive?"
 
 

This sequence is very simple.

It starts with 1, 1. These two numbers are added to get the next term in the sequence i.e. 2

So we now have 1, 1, 2 . Subsequent terms are obtained by adding the last number (in the sequence) to the number immediately preceding it.

So the Fibonnaci sequence develops as follows (up to 1000)
 
 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …..

Now one of fascinating features of this series is that when we divide any term by its precedent, its value approximates the value of Phi (1.618…). This approximation greatly improves with higher sequence terms.

Thus for example 987/610 = 1.61803 (is the value for Phi correct to five decimal places).

So what is amazing is how a relationship as between a number in the sequence and its precedent continually generates a value ever more closely approximating the true value for phi.

Even more remarkable connections can be made using the bi-directional approach in what is now a quantitative numerical context.

Now we can take ratio of numbers in two ways. Thus instead of taking a number in a sequence and dividing by its precedent, we can simply reverse this process and divide the preceding number by its successor.

Thus whereas we initially obtained the value of 987/610, we can reverse this and obtain the value of 610/987 = 1.61803. Incredibly this value approximates to Phi - 1.

Again this relationship will hold for all such reverse number sequences.

When we take any number in the sequence and divide by its immediate predecessor we approximate the value of Phi.

When we reverse this procedure and divide the predecessor by its successor we approximate the value of Phi - 1.

Now if we subtract the latter from the former value we approximate (extremely closely) the number 1.

We make this the first term in a new series.

We next obtain the ratio of any number in the sequence to the number that precedes it by two..

Thus in relation to 987 this number that precedes it by two is 377.

The ratio of 987/377 is 2.61803 (i.e. Phi + 1)

Again, this number will be approximated for any ratio involving any number in the sequence and that preceding it by two.

We then get the reverse ratio (in our example 377/987 = .38197 (correct to five decimals).

This time when we add both results we get the integer 3.

This now becomes the next integer in our second series of values.

So we now have a sequence starting with 1,3

Using the same approach as the Finonacci sequence (by adding the last number to its predecessor) we come up with a new series (The Lucas Numbers)

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843,…..

Now if we keep applying our bi-directional approach to the Fibonnaci series, we in fact generate this alternative sequence.

For example if we now pair each number (in the Fibonnacci sequence with the number that precedes it by 3) we can generate the next term in the Lucas sequence (i.e. 4).
 
 

Thus again using 987 as our starting point, the number preceding it by 3 is 233.

The ratio of 987/233 is 4.236 (correct to 3 decimal place).

Next we reverse the ratio and get 233/987 = .236

Thus subtracting the two values we get 4.

(Once again this result will obtain regardless of our starting number.

So there is a definite pattern. We keep alternating the sign when combining the two ratios.

In the first case to generate 1, it is minus (i.e. we subtracted the two ratios).

In the second case to generate 3, it is plus (i.e. we added the two ratios).

In the present case it is again minus (i.e. we subtracted the two ratios).

So to generate the next integer in the Lucas sequence we must again add the two ratios.

In relation to any number in the Fibonacci sequence, we are now looking at its pairing number (which precedes it by 4).

Thus again starting with 987, the number preceding it by 4 is 144.

The ratio 987/144 = 6.854 (correct to 3 decimal places)

The reverse ratio 144/987 is .146 (correct to 3 decimal places).

Adding these two results we get 7 (which is the next value in our Lucas sequence).

All other terms in the Lucas Numbers Sequence can be generated by continuing this approach.

Perhaps Forum participants would like to use their calculators to obtain the next term in the Lucas series (11).

Here starting with any term, one will pair it with the number that precedes it by 5. (When combining ratios one will now subtract results).

I must see I find the consistency of these patterns truly amazing.
 
 

What I find even more astounding is that using the same bi-directional procedures, the new Lucas Sequence can generate exactly the same results.

For example let us take the last term in the Lucas Sequence 843. The term preceding it is 521.

The ratio of these two terms is 1.618 (to three decimal places) which of course is the value of Phi.

Now when we reverse the ratios we get 521/843 = .618 (to 3 decimal places) = 1 - Phi.

Again subtracting the results we get 1 (the first terms in the same sequence).

When we pair any number (in this sequence) with the one preceding it by 2, we can generate the next number (3).

Thus using 843 as starting point, 322 precedes it by 2.

Now 843/322 = 2.618 (to 3 decimal places) = 1 + phi.

The reverse ratio 322/843 = .382 (to 3 decimal places).

Adding these two results we obtain 3 (the next term in the Lucas Sequence).

So we have here a fascinating example of a self-recursive sequence (when subjected to a mathematical bi-directional approach).
 
 

Indeed we can demonstrate another simple connection as between the original Fibonacci and Lucas Numbers.

If we take the Fibonacci and combine the last number in the sequence with the number preceding it by 2, we obtain the Lucas sequence.

So we have 1, 1, 2, 3, 5, 8… (in the Fibonnaci) and start with 1 in the Lucas.

Thus the next term in the Lucas is (1 + 2 = 3), then 3 + 1 =4, 5 + 2 = 7 etc).
 
 

I could demonstrate other fascinating mathematical connections, but I think this should be enough for the moment.

Now let us turn to the (complementary) qualitative interpretations.

First of all Phi, is an (algebraic) irrational number.

It can be expressed as the solution to the simple equation x = (x + 1)/x. (as we know its value = 1.618 (correct to three decimal places).

Now why is it irrational?

On the face of it this seems a bit puzzling.

The most common representation of the ratios (that generate the value of Phi) is in the geometrical form of a rectangle.

If in this triangle the length of the vertical sides = 1, the length of the horizontal sides

= 1 + x, where x = (x + 1)/x.

However the very construction of this rectangular figure requires the use of the (linear) ruler and (circular) compass.

I know it is very difficult to visualise this in words, but I will explain how it is done.

Using a ruler, start by drawing a square of unit length. Extend the bottom line of the square rightwards (say by another unit).

Bisect the square with a line drawn from the midpoint of the (bottom) horizontal side to the (top) horizontal side. From this midpoint (on the bottom horizontal) draw a diagonal line to the top right hand corner of the square.

Using this diagonal line as radius, use a compass trace out an arc of a circle (in a rightwards direction) which will cut the extended (bottom) horizontal line.

The length extended horizontal line up to this cut point (by the circle) now serves as the horizontal (bottom) side of a new rectangle. The vertical side remains unchanged as 1 unit.

Complete the rectangle.

The horizontal and vertical sides are now in the ratio of the golden mean (i.e. the horizontal length divided by vertical length of rectangle = Phi (1.618… .).

Now the irrational notion in mathematics always involves the relationship as between the line (1) and circle (0).

With (algebraic) irrational numbers the connection is indirect (as in this case).

Again the construction of our rectangle (which geometrically illustrates the golden mean) indirectly requires the use of the circle (i.e. compass) in its construction.

With (transcendental) irrational numbers (e.g. Pi), the connection is more explicit. Thus to geometrically represent pi, we must have both the line (diameter) and circle (circumference).

Now there is an exact complementary (qualitative) psychological correspondence to both notions.

In direct terms the structures of HL1 (the subtle realm) are intuitive (the circle); however indirectly they can be represented in rational terms (the line).

Thus, in the structures of the subtle realm, we have the perfect psychological complement to (algebraic) irrational numbers.

The structures of HL2 (the causal realm) - as I have already stated in discussing Pi, are seen in more balanced terms as the relationship as between both line and circle.

Thus in the structures of the causal realm we have the perfect psychological complement to (transcendental) irrational numbers.

Once again Phi (the golden ratio) corresponds to the (algebraic) irrational numbers and has its complementary psychological explanation in the understanding of the HL1 (the subtle realm).

In more direct terms I can give wonderful Jungian interpretation to Phi.

Indeed it provides a dramatic confirmation of the bi-directional approach.

In my earlier demonstration of number ratios, I explicitly used a bi-directional method.

Firstly, I took the ratio of whole to part (i.e. "higher" to "lower" number), Then I took the ratio of part to whole (i.e. "lower" to "higher" number).

By combining both numbers I was able to consistently generate (quantitative) integral numbers (which had a very meaningful interpretation).

In qualitative terms we can give an exactly complementary interpretation.

I have consistently argued on this Forum that - in dynamic terms - movement is bi-directional (not one-directional). Therefore in relation to psychical development both directions must be recognised.

Indeed I have argued that bi-directional approach is the very basis of a truly integrated (as opposed to a differentiated) method of understanding.

Now I think it is truly remarkable how in the context of a seemingly unrelated numerical sequence (involving irrational numbers) that the bi-directional method literally produce integral numbers.

Thus the quantitative results - using the bi-directional approach - that leads to the generation of integer numbers serve as a powerful archetype for qualitative integration in the psychological realm (which again depends on the use of a bi-directional approach).

Your final point Dave is very interesting.

Yes, Phi is irrational and the Greeks made great use of it geometrically. However you have to remember that they had no real concept then of an irrational number.

Now the square root of 2 is the most obvious of all irrational numbers (and arises in the most simple example of a right angled triangle where opposite and adjacent sides are of length 1 unit. Phi is far more complicated as its value depends on the solution to a quadratic equation. So I think it is understandable that the problem of the irrational should first surface in the context of the square root of 2 (and not Phi).

Dave this has proven to be a fascinating exploration. I was myself amazed at how easily important numerical relationships could flow from the explicit use of a bi-directional approach. I had studies Fibonnacci numbers before and missed this simple connection.

Thank you for giving me the opportunity to "vindicate" the value of the bi-directional approach (in the quantitative as well as qualitative spheres).