# Posts Tagged Phi

## The Golden Mean as the Key to the Universe

From I was a young and hopeful mathematician I have been fascinated by the Golden mean and the Fibonacci Series, with its connection to plants and animals, but I lacked a good explanation for why it was found in all of nature, in all of the universe.

Plato said that the Golden Mean, Phi, is key to the universe. It has puzzled me over time as the interesting and beautiful structures following the Fibonacci numbers in nature aren’t explanation enough, there has to be a spiritual background. If the Golden Mean is the key, the Golden Triangle is the door. So, what is the spiritual explanation behind?

The Creating Trinities

### The Fibonacci Series

The formula for the Fibonacci series is

Fi+1= Fi + Fi-1, where the initial conditions are i= 1, F0= 0, F1= 1

where the subscript describes the position in the sequence.

Notice that the new value depends on the previous two! This is the principle behind all creation. If it only depended on the foregoing, it would be a repetition, like the propagation of single-celled organisms producing copies of itself, but here the the growth factor is phi, the golden mean.

This means we get a sequence that looks like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Growth from point to point in the sequence approaches a fixed value the further we get and this growth factor is phi= Fi / Fi-1 (~1.6180), which we also know from the golden section.

The growth is large, in mathematical language we call it exponential, which can be displayed graphically:

The formula is connected with growth, both with physical growth of plants, animals, planetary systems, galaxies, but also in breeding, sexual reproduction, see Fibonacci Numbers and Nature with many good examples and descriptions.

### The Fibonacci Spiral

One of the most characteristic forms is the Fibonacci curve, occurring everywhere:

And the following two pictures show impressing examples.

A hurricane:

And a galaxy:

### Plant Growth

What’s makes the formula interesting is that the next value depend on the two preceding values, not the last alone, this explains the growth in plants, where each step always builds on the already existing, and the new is not just a repetition of the previous, it’s bigger and better, with a factor of Phi, the Golden Mean.Here an example of plant growth with the number of leaves produced:

Here a collage with many examples on Fibonacci grows:

### The Fibonacci Series and Reincarnation

I read a Theosophic text some years ago where it was stated that man through each incarnation spiritually was growing with the factor Phi. It sounded very interesting, but there was no explanation for why it was that way, but I remembered it, one day I might find out.

One day I found out that the Fibonacci series also described the sexual reproduction, and that was the beginning of the understanding.

The Fibonacci formula tells that one incarnation builds on the two previous incarnations, which following Steiner should be one female and one male incarnation. It sounded reasonable.

Hegel’s dialectic described as thesis plus antithesis, resulting in the synthesis, or with his own concepts: abstract, negative and concrete. You start with the abstract and put it up against its opposite, the negative and from these are the concrete formed:

So we have

Fi+1= Fi + Fi-1        where Fi-1= Thesis, Fi = Antithesis,  Fi+1= Synthesis

If we look at the incarnations of a person, a is the first incarnation as a woman, b is the next incarnation as a man, it will be followed by a second incarnation c as a man, the synthesis of the previous two. Next incarnation d are as a woman and also the next incarnation e and as all incarnations hereafter will be between d and e, they will no longer alternate between the sexes, but all subsequent incarnations will be as a woman:

### The Three Logoi

This is a definition by Rudolf Steiner on the three creative principles in the universe, also called the Logoi’s:

Here we have three definitions of Beings who bring about, who underlie a planetary chain. They are called the three Logoi.

1. The Third Logos produces by means of combining.
2. When out of one substance something else having new life comes into being, this is brought forth by the Second Logos.
3. Everywhere, however, where we have to do with a coming forth out of nothing, we have the First Logos.

This is why the First Logos is also often called the One who is immanent in things, the Second Logos the One who in the quiescent substance in things creates life out of the living, the Third Logos the One who combines everything existing, who puts the world together out of things.

Drawing by Steiner

These three Logoi always manifest in the world in and through one another.

The Third Logos in spiral form, Sam Tageuk or Gankyil:

See much more on this here: The Logos Walks the Earth

Here

Fi+1= Fi + Fi-1        where Fi-1= First Logos, Fi = Second Logos,  Fi+1= Third Logos.

First Logos gives the idea, Second Logos gives life and Third Logos gives the resulting form.

### Video on the Golden Key

This is a beautiful video made on the Golden Mean and the Fibonacci series by Jonathan Quintin:

### References:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html

A non-Euclidean tiling of the disc by regular heptagons.
Image created by David Wright.

Non-Euclidean geometry and Indra’s pearls

The Logoi of the different hierarchies as genders.