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We will now start a five-part series on the Golden Ratio (phi), one of the most important aspects of the Pentad.


Phi = 1.618033…etc



The Golden Ratio

Bruce Rawles writes, “The Golden Mean is an intra-dimensional doorway though which matter emerges into manifest 3-D reality.  Thus the Golden Mean is the “fingerprint” of creation.  When we re-create this moving and always expanding sequence, we have in effect – the exact movement of creation in the expansion process.”1

Before we move into the Golden Ratio itself, let us review what a ratio and proportion is.  This was first covered in Article 8.


Ratio (logos)

A ratio is the relation of one number to another.  For example:

4 to 8        or       4:8

As John Michell writes, “A number on its own has no inherent meaning; its symbolism arises purely from its relationships to other numbers.”

It is a comparison of two different sizes, quantities, qualities or ideas.

Ratios are expressed as a:b and constitute a measure of difference.

“Not only then is a ratio a:b the fundamental notion for all activities of perception, but it signals one of the most basic processes of intelligence in that it symbolizes a comparison between two things, and is thus the elementary basis for conceptual judgment.”2

That is, ratios represent the compare/contrast function of our rational thinking.

As we will see, proportions involve the compare/contrast function on a new level, representing the correlating/synthesizing aspect of our intuition as we learn to harmonize seeming differences into greater wholes.



Proportion (analogia) – Four-termed

A proportion is a repeating ratio that typically involves four terms.

For example:

4 is to 8 as 5 is to 10        or      4:8 :: 5:10

It is a relationship of equivalency between two ratios and is expressed as a:b::c:d.


In proportions there are no fixed quantities, only fixed relationships.

As we said above, proportions represent differences coming together to form greater wholes.

As Robert Lawlor writes, “It represents a level of intelligence more subtle and profound than the direct response to a simple difference which is the ratio, and it was known in Greek thought as analogy.”3

Gyorgy Doczi reminds us, “In all realms of our experience, we are finding the need to rediscover proper proportions.  The proportions of nature, arts, and architecture can help us in their effort, for these proportions are shared limitations that create harmonious relationships out of differences.  Thus they teach us that limitations are not just restrictive, but they also are creative.”


A ratio that involves four terms is called a four-termed discontinuous proportion.

“When we think with four elements, that is with two different ratios, we locate our thought within the range of manifestation of the natural world since four is the number-symbol indicative of the finite, rational, measurable world of procreated form.”4



Geometric Mean – Three-termed Proportion

A geometric mean involves three terms and is equal to the square root of the product of these three terms.

For example, the geometric mean of 1 and 9 is 3.

√(1×9) = 3

Thus this example is written as:

1 : 3 : 9          or         1:3 :: 3:9


A ratio that involves three terms is called a three-termed continuous geometric proportion.

It is called the ‘perfect proportion’ because it is a direct proportional relationship, an equality of proportion bound by one mean term.



“Knowledge by Identity”

“Knowledge by Identity is a mode of perceptual awareness comparable to a three-term continuous proportion.

We acknowledge an external source of experience.

We recognize it is in a continual flow of relationship without our inner perception and cognition.

Thus, it is the relationship, not the external object that we are experiencing.

As Robert Lawlor writes, “The objective world then is interdependent with the entire physical, mental and psychological condition of the perceiving individual, and consequently will be altered by changes in his inward condition.”



Two-Termed Proportion

The only two-termed proportion is the Golden Proportion or Phi (φ).

The Golden Proportion is a three-term proportion constructed from two terms.

It relates to the ‘first mystery of the Holy Trinity: The Three that are Two.’

Gyorgy Doczi writes, “The two parts of the golden section’s proportions are unequal:  one is smaller, the other larger.  They are often referred to as minor and major.  Minor and major here are opposites united in a harmonious proportion.

The two chief modes of Western scales, the minor (considered to be the sad one) and the major (associated with brightness) differ from each other only in the length of steps between certain of their intervals, just as the minor and major parts of the golden section differ from each other only in their lengths.  And just as the joining of minor and major parts delights us in the visual harmonies of the golden section, so the joining of minor and major scales, called modulations, enchant us when we hear them in the movement of chords and melodies.”


φ is the only division that fulfills this characteristic:

1/φ + 1/φ2 = 1

The Golden Proportion completes the mathematical metaphor for the Trinity: “Three that are Two that are One.”


The smaller term is to the larger term in the same way as the larger term is to the whole (smaller plus the larger).

This can be expressed as:



Phi is a proportion, not a number – “a proportion upon which the experience of knowledge (logos) is founded.”5

“On any given line there is only one point that will bisect it into two unequal parts in this uniquely reciprocal fashion, and this one point is called the point of golden section.  The complete reciprocity of this proportion strikes us as particularly harmonious and pleasing, a fact that has been proven by many scientific experiments since the end of the last century.” ~ Gyorgy Doczi



“If we once again use proportion as model for the perceptual activity based on the recognition of differences, we have in this unique Golden Proportion ‘within’ Unity a case where the perceived difference (that which we experience as an object) plus the perceiver of that object are symbolized as contained within a sustained awareness of an all-encompassing Unity, a:b::b:1.  This perceptual state corresponds to the goal of dynamic mediation.”6


Recall that mediation is the binding of two extremes by a single mean term, or the binding of two ‘opposite’ tones by harmoniously placing a third tone between them.

This mediating term can be likened to our own individual consciousness, or the state of our minds and emotions.

As Keith Critchlow writes, “There was an analogical connection in the ancient Greek mind between this harmonic mean and human relationships; the term Golden Mean was applied to human affairs in the maxim: ‘Do unto others as you would have them do unto you.’”

As is repeatedly mentioned throughout Cosmic Core ratios and proportions are about relationships (relationships between types and levels of consciousness) and numbers represent qualities of consciousness.

The Golden Ratio specifically symbolizes our relationship with our Source – the metaphysical reality, our Divine Infinite natures, Cosmic Consciousness…whatever you prefer to call it.  This means that the Golden Ratio symbolizes the potential of our relationship with Unity and Infinity (that of the whole universe, seen and unseen).

Our consciousness is the harmonizing and binding element between our individualized nature and our Unified Infinite nature (our Higher Self).  The more we use our consciousness to harmonize with the universe, the world, other people, nature and ourselves, the more we become like the universe.  That is we symbolically grow transparent and act as vessels of light from which unity, harmony and infinity can shine through.  That is, our partial or individualized self (the small portion of the golden ratio) will begin to resemble our Higher Self (the large portion of the golden ratio) in the way that our Higher Self resembles the Whole of the Infinite Creation, or Cosmic Consciousness.

That means our individualized selves (the selves that we are now with our consciousness that feels small and separate from all others) will begin to resemble the whole of the Infinite Universe.  We will thus become more like the Infinite Universe.  As we exercise the compassionate, harmonizing and creative aspects of our consciousness, we will become exponentially more compassionate, more powerful, more peaceful, more harmonious in every way, wiser, more empathetic, more creative, more honest and open and more able to travel the universe and communicate with consciousness throughout the universe.



Logos, Creation & the Golden Ratio

Scott Olsen writes concerning “The creative moment of original scission” of Unity:

“Concerning the Logos (ratio or word), at the start of St. John’s gospel, 1:1, we read: ‘In the beginning was the Logos, and the Logos was with God and the Logos was God.’  The only ratio that is simultaneously one and with one is the golden section.”


Word = Logos = a three-termed proportion.

“With God” can be read “In God”.


Read it again:

In the Beginning was the Logos

And the Logos was within God

And the Logos was God.

These three lines intuitively describe the geometric implications of the Golden Proportion and the knowledge that Unity is within everything and IS everything.  This Unity, or Source, can be called God if you so choose.

Robert Lawlor writes, “In a sense, the Golden Proportion can be considered as supra-rational or transcendent.  It is actually the first issue of Oneness, the only possible creative duality with Unity.  It is the most intimate relationship, one might say, that proportional existence – the universe – can have with Unity, the primal or first division of One.  For this reason the ancients called it ‘golden’, the perfect division, and the Christians have related this proportional symbol to the Son of God.”



Golden Ratio

The Golden Ratio brings us back to the question of the Dyad: How does the One Become the Many?

The Golden Ratio helps answer this question.

It is the only possible creative duality within Unity.

It represents duality, because of the splitting in two parts, as exemplified by the Vesica Piscis.

It represents creativity because the two parts are asymmetric – not equal – this allows for continual growth and change.

This is represented by the triangle – the first form to emerge from the vescia.  The triangle represents harmony, and relates to a harmonization among consciousnesses as described above.

A circle is drawn around the emergent triangle.  This automatically creates the golden ratio as seen in the black and green line below.  Phi = 1.618…

AB = 1;  BC = .618…;  AC = 1.618…

The golden ratio comes from an equilateral triangle in a sphere.  AB : BC :: AC : AB


Another geometric example follows:

As was seen in Construction Lesson #38 taken from Robert Lawlor’s Sacred Geometry: Philosophy & Practice.  

“We being by drawing a circle, acknowledging it as the geometric metaphor for a homogeneous, non-differentiated space.  As in our other diagrams, this unity-space must divide into duality in order to create.  So we begin by dividing the unity-circle into two halves, a division which occurs within the initial unity…A curious paradox arises when we divide a unitary circle in two in this manner, forming the basis for the traditional yin-yang symbol.  The two circumferences of the inner circles are together equal to that of the larger circle, but the area contained within the two is only half that of the original circle.  One has become Two…

When we form geometrically the container of the two circle by drawing an arc from eahc end of the vertical diamter tangential to the two circles, terminating both the upper and lower arcs at the horizontal diamter, we see that these two arcs have cut the vertical radius  (center point to A) into the Golden Section…

Thus, given with the original scission or contraction into two is the plan of the return [to unity]: the pentagon, the symbol of life, with its fivefold symmetry.”

This means that simultaneous with the first division forward from 1 to 2 (the instant creation began), came the path of the return back to unity: the 5, or golden section.

Symbolically, this means the doorway or path back towards Unity is always open no matter how polarized a person’s consciousness is and how separate everything in reality appears to be – unity is always within if they so choose to focus in that direction.

As we will continually encounter in the study of Sacred Geometry, symbolism is the name of the game, and all symbolism revolves around consciousness, levels of consciousness and how it is perceived and used.



History of the Golden Ratio

The Golden Ratio is called by many names:

  • Phi ratio, or Phi (from Phidias, Greek sculptor who used it in the Parthenon)
  • Sacred cut
  • Golden mean
  • Divine proportion (divina proportione)
  • The section
  • Extreme and mean ratio
  • aurea sectio (golden section)
  • Godlike proportion
  • Divine section
  • Continued division
  • The medial section
  • The golden cut

It was used in ancient Egypt & the Pythagorean tradition, but was first officially defined by Euclid (325-265 BC) in his Elements.

The earliest known treatise on the Golden Ratio was by Fra Luca Pacioli, illustrated by Da Vinci (who coined the term aurea sectio – golden section).  This treastice was called De Divina Proportione and was written between 1496 and 1498 and published in 1509.


“By discussing mathematical proportion – especially the mathematics of the golden ratio – and its application in art and architecture, Pacioli hoped to enlighten the general public about the secret of harmonic forms.”7

Fra Luca Pacioli and Pupil by J. de Barbari – 1440/50-1516


The English term ‘golden section’ was first published in Martin Ohm’s 1835 Pure Elementary Mathematics.



The Pentagon & the Golden Ratio

The pentagon is the pure geometric embodiment of the golden section.

Φ is inseparably related to the √5 function and the pentagon.

With edge lengths 1, a pentagon naturally produces Φ.



The pentagon-pentagram relationships of the golden section can clearly be seen.

The side of a pentagon is in relation to its diagonal as 1: (√5 +1)/2 or 1:φ , the Golden Section.


The Golden Ratio is a relationship that represents the accumulative process of manifestation.

It is a fundamental measure that seems to crop up almost everywhere in nature.

However, it is not found everywhere in nature.

“But it can be said that wherever there is an intensification of function or a particular beauty and harmony of form, there the Golden Mean will be found.”8



The Golden Ratio & Symmetry

Perfect symmetry is static – one part nullifies the other.

Asymmetry is dynamic.

The Golden Section is asymmetric – it sets things in motion – it creates the tension that occurs to create movement and it keeps the movement going.

“It is the asymmetric push, the dynamic energy of the golden ratio manifesting as life, form and consciousness that provides the impetus to the rhythmic swing, the initial push of the pendulum.”9

“Therefore the φ proportion is the perfect division of unity: it is creative, yet the entire proportional universe that results from it relates back to it and is literally contained within it, since no terms of the original division steps as it were, outside of a direct rapport with the initial division of Unity.”10



Observation of Phi

There are four primary ways phi is observed in physical reality:

As a spiral (found in shells, plants, water and air vortices, galaxies…etc.)


As a ratio of quantity (found in plant seed heads)


As a ratio of scale (found in human body, plants and animals)

Credit: Gyorgy Doczi – The Power of Limits, 1981


As angles of both rotation and scaling (found in plants and many other phenomena)

The golden angle = approximately 137.5°



Qualities of the Golden Ratio

As we saw above, the golden ratio is the unique ratio such that the whole is to the larger portion as the larger portion to the smaller portion.


This results in a continuous geometric proportion:

 Whole     =   Large Part  = Φ

Large part    small part


The proportion a:b::b:(a+b) or approximately 1.618 in a numerical value.


Reference Construction Lesson #61: Construction of the Golden Mean.


No other proportion behaves so elegantly around Unity.

Unity can act as the Greater (whole), mean (longer segment) or the Lesser (short segment).

Credit: page 302 in Designa.


The Golden Proportion is an irrational (supra-rational) function of number.

This means it is inexpressible as a simple fraction.

It is seen as a key opening a door to a higher reality of Number and demonstrates that Number is above all a relationship.

Φ= (1 + √5)/2

or 1.618033988749894848204586834365638117720309180…


φ = 0.61803398875… (pronounced ‘fee’)  φ = 1/2(√5-1)

Φ = 1.61803398875… (pronounced ‘fi’)   Φ = 1/2(√5+1)


The Golden Ratio “symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage.”11

Phi as a continuing fraction:  From unity to infinity.



The golden ratio (phi) has some unique properties and makes some interesting appearances:


Φ = Φ2 – 1    therefore    1 + Φ = Φ2

Φ2 = Φ + 1 = 2.618……. (Φ + 1 = Φ * Φ)

1/Φ = Φ – 1 = 0.618…


1 + Φ1  = Φ2

Φ + Φ2 = Φ3

Φ2 + Φ3= Φ4

Φ34 = Φ5…ad infinitum…


Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the golden ratio raised to successively higher (or lower) powers is automatically generated:

Φ, Φ2, Φ3, Φ4, Φ5, etc…


The cube of φ is a volume arrived at by simultaneously adding and multiplying:

1/φ  =  1  = φ + 1 * φ


1 + φ  = φ  * φ  = φ + 1


φ + φ2 = φ3 = φ * φ * φ = φ * φ2

A geometric representation of the continuing fraction seen above.  From unity to infinity.



Phi is Additive & Multiplicative

Addition is the most common process of growth.

This is seen in: cells; wealth; knowledge, experience, population numbers…


Multiplication is an accelerated form of addition:

4 x 4 = 4 + 4 + 4+ 4

What was linear, now becomes a square, surface or plane.

This is seen in: stem to flower to fruit; a seed absorbing moisture suddenly germinating…etc.


“If the transformative power of redemption is fixed to the material cross, the cross of addition +, then the moment of resurrection comes when this principle allows the cross to fall + to x, and an exponential growth occurs, an incomprehensible, non-sequential leap to another level of being.”12

Crossing of + and x: the weaving of warp and weft; spirit and matter; light and sound.



Phi & √2 – Models for Evolution

The golden spiral is made up of ‘whirling squares’ or a series of √2 ratios.


The Square Root of 2: √2

We have extensively discussed the √2 in previous articles.  Review Articles 24-26 & 37 if necessary.

√2 is the way the Golden Ratio expresses itself in Nature.

Remember, the √2 expresses the diagonal of a square.  You can see how the golden arc of the diagonal relates to the straight line diagonal of the square.

The straight line diagonal = √2.

The arc of the diagonal = Φ.


Φ & √2 are both geometrical proportions.

Through the creation of √2 we are propelled outside the original square.

“This marks the beginning of an endless, ever expanding progression and proliferation, leading further and further away from the original Unity.”13


√2 cannot have a geometric internal division of Unity.

It is strictly a procreative power that functions generatively on the external plane.

It represents a quantitative, statistical evolution – to which evolutionary adaptation conforms and is represented by the straight-line version of the golden spiral – the version that represents the rational thinking aspect of our consciousnesses.



Phi – φ

Phi, on the other hand, “has its goal the image of the perfection of the original Unity.”14

It is the only continuous proportion that yields a progression in which the terms representing the external universe (φ2, φ3) are an exact continuous proportional reflection of the internal progression (1/φ2, 1/φ3).


This represents an evolution guided from within; where the manifest world is an image of the Divine.

This curved version of the golden spiral represents the intuitive aspect of our consciousness.

“The Golden Proportion represents indisputable proportional evidence of the possibility of a conscious evolution as well as of an evolution of consciousness.”15

This means physical evolution and spiritual evolution are mirrored and are meant to be brought into harmony in order to progress.

Harry H. Gillespie writes, “The path isn’t a straight line; it’s a spiral.  You continually come back to things you thought you understood and see deeper truths.”

We will finish this article now with words of wisdom by Gary B. Meisner: “In its own unique way, phi touches upon some of the most fundamental questions of philosophy and the meaning of life.  When we discover common threads in the mathematical design of things in our world, especially where it seems unexpected or unexplained, it can beg the question of whether there could be something more than chance at work – a grander plan of design with some guiding purpose, or even a designer.  Others may seek to explain these same observations as coincidences arising from natural processes in adaptations and optimizations.  Everyone has an underlying belief system that influences their interpretation of everything they see and hear, no matter how much evidence is presented to the contrary.  These fundamental questions of where we came from, why we are here, and where we are going, are mysteries that we all must ponder with an open mind and an open heart.”16


  2. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  3. ibid.
  4. ibid.
  5. ibid.
  6. ibid.
  7. Meisner, Gary B., The Golden Proportion: The Divine Beauty of Mathematics, Race Point Publishing, 2018
  8. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  9. Olsen, Scott, The Golden Section: Nature’s Greatest Secret, Walker Books, 2006
  10. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  12. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  13. ibid.
  14. ibid.
  15. ibid.
  16. Meisner, Gary B., The Golden Proportion: The Divine Beauty of Mathematics, Race Point Publishing, 2018


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