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In this article we will discuss the principle of alternation in greater detail.

We keep bringing up alternation because it is one of the most important symbolic aspects of the Dyad. It was introduced in Articles 17 and 19.

Alternation is also an important aspect of square roots. We will also explore the meaning and symbolism of square roots as related to the Vesica and the principles of the Dyad.

Much of the following information is sourced from Robert Lawlor’s Sacred Geometry Philosophy & Practice unless otherwise noted.




Alternation refers to alternate succession or repeated rotation.

It is a situation in which one thing repeatedly happens or exists after another, such as oscillation.

For example, electricity is a single fluctuation in the absolute value of an alternating current or voltage from zero to a maximum and back to zero, being equal to one half cycle.

sinewaves lr


Alternation is a major philosophical premise of ancient thought.

It is the foundation of reality – a model of “the pulsation of Cosmic Life” as Robert Lawlor states. All life and the entire universe progress through alternation.

sound wave monopole


As R.A. Schwaller de Lubicz writes, “All in the Universe is made up of alternations and inversion, which are the manifestation of life, and life is continuous creation: the original scission of the ungraspable Unity.

But the alternation and inversion of what?

It is the alternation of “density” in general, and essentially, of one and the same energy. Density means the reduction into volume of space, that is, of energy. The inverse is the inevitable, compensating complementation of each element of alternation.

Thus, the whole Universe is living and all of life’s phenomena contain a concrete element of alternation and an energetic element of the same nature as this concretization. The concrete aspect can be grasped through cerebral conception, the energetic aspect is graspable through the other state of consciousness that is the intelligence of the heart, the source of intuition.”

heartthoughts lr



Reciprocal refers to:

  • (of a course or bearing) differing from a given course or bearing by 180 degrees.
  • (of a quantity or function) related to another so that their product is one.
  • (in mathematics) – the reciprocal of a number is: 1 divided by the number.
    • For example, the reciprocal of 3 is 1/3, and so forth.
    • This is also called the “Multiplicative Inverse”.


In the Science section of Cosmic Core we discuss the Reciprocal Systems Theory of Dewey B Larson. In this system time and space are reciprocal realities of each other. These two states of existence or parallel realities are known as time/space (the unseen, metaphysical) and space/time (the seen/physical).

Alternation relates to the constant interchange between consciousness and matter; between the visible and invisible; between time and space.


The Vesica Piscis and Proportions

The first three shapes to emerge from the Vesica Piscis, the triangle, square and pentagon, form the only relationships or ratios required to generate all the rest (except sevenness – the heptagon).

vescia 3 4 5


Within the Vesica Piscis is contained knowledge about the square roots of 2, 3, & 5.

These “root” relationships are expressed not as whole numbers but as never-ending decimals.

vesica square roots


The Square roots of 2, 3, and 5 are ratios, not actually numbers, but seeds, like pi or phi, that continue to infinity. Like phi, they represent relationships.


√2 = 1.41421356237…

√3 = 1.73205080757…

√5 = 2.23606797749…


These ongoing ratios hold the structural patterns for all numbers and shapes to follow.

Two intersecting circles contain all the proportions necessary to generate the polygons found in nature and the five Platonic volumes (the basis for all volumetric forms).

square roots polygons

Credit: Michael Schneider, A Beginner’s Guide to Constructing the Universe


Also, 2, 3, and 5 are the only numbers required for the division of the octave into musical scales.

As Robert Lawlor states, “Transformation can be seen to occur by means of three general processes:”

  • √2 – The Generative
  • √3 – The Formative
  • √5 – The Regenerative
    • √5 is related to phi, the Golden Ratio – The Transformative

The three roots are a trinity of “generative principles”.

world tree


As we saw earlier, two lines or axes run through the center of the Vesica – horizontal (X axis) and vertical (Y axis).

The ratio of the Y axis to the X axis is the square root of 3 – the deepest nature of the triune. It cannot be expressed by rational language alone.

vesica sq rt 3


Robert Lawlor writes, “The roots of 2 and 5 can also be derived from this cosmogram of the Vesica because there is no synthetic symbolization of Unity which does not evoke all the major principles: as the Koran says, “There is no god which is not all gods.”
vesica piscis poly constructions square color transform highlight

Credit: Bruce Lyons, 2010



Mathematical Alternation

As we said earlier, alternation was a major premise of ancient thought. Alternation was seen as the foundation of reality. All life and the entire universe progresses through alternation.

To the ancients there was no separation between geometry, natural science, cosmology and theology.

To them, the idea of an irrational number (as √2 = 1.4142135…) was a logical absurdity. The decimal system, for them, did not exist.

They viewed the essence of number as a state: tangible, fixed and measurable.

“Ratio, the Latin root of ‘reason’, also means ‘measure’. An irrational number was an unacceptable contradiction.”1

Numerical progressions were set up which can be seen as representations of vibrational systems in that a vibrating string also moves above and below an abstract node or inexpressible still point.

The ratios emerge in such a way that they go alternately above and below the incommensurable root value.

They approach nearer and nearer to the root value with each alternation.


An excellent example: the Fibonacci Sequence Alternating around Phi

The Fibonacci sequence is closely associated with the phi ratio, particularly the multiplicative aspect.

The Fibonacci sequence begins:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…etc.


Next arrange the Fibonacci sequence into ratios of successive pairs:

1/1; 2/1; 3/2; 5/3; 8/5; 13/8; 21/13; 34/21; 55/34…etc.


Then take the decimal value of that ratio. As a limit it approaches the golden section (1.618033988…) but it never gets there.

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.6666

8/5 = 1.6

13/8 = 1.625

21/13 = 1.615384

34/21 = 1.619048… and so on to infinity.


This is called asymptotically reaching a limit.

See the below graph. You can see how the terms approach the golden ratio, bouncing above and below as you move up, yet they never actually reach it.



The graph below shows the same process but out to more terms.

Fib graph


This mathematical alternation also occurs for √2, √3, √5, and π.


Rational and Irrational Numbers

In ancient thought rational and irrational numbers represented two different states of being.

The rational numbers were related to physical reality – the seen world of manifestations.

The irrational numbers were related to the metaphysical reality – the unseen world of consciousness.

flammarion lowres


Irrational Numbers

Irrational numbers symbolize the constant, creative process of acting and reacting energy.

mathematics 1044091 1920

They represent the immeasurable gestating force – the Source reality.

Irrational numbers emanate from an incomprehensible Unity and are symbols for pure archetypal processes such as generation, fusion, regeneration and transformation.


Rational Numbers

Rational numbers are related to manifestation. They are the familiar terms we use in everyday calculations.

numberscolor lr


They are the incomprehensible made comprehensible. The comprehensible is only a momentary limitation of the One indefinable Being into a definable moment.

As Robert Lawlor states, “All that is definable arises out of an Indefinable All.”

“Every aspect of the phenomenal world was seen to be a fixed, instantaneous moment caused by the interaction of complementary components, a moment trapped between light and dark, life and death, day and night, between formation, disintegration and reformation.”2



Root Powers as Whole Number Ratios (Continued Fractions)

Continued fractions reveal the Principles of Alternation.

Just as we saw with the Fibonacci sequence, the ratios emerge in just a way that they go alternately above and below the incommensurable root value.

They approach nearer and nearer to the root value with each alternation.

Expressed in this way they retain their dynamic process quality.

continued fractions copy



Roots (√)

Lawlor tells us, “The root grows by the constant division of its square shape.”

The root cells are a powerful metaphor for the principle of integration and transformation.

root onion

Root cells of an onion


Geometric contemplation is founded on the idea that natural forms are to be understood as symbols revealing metaphysical archetypal principles which guide and control universal evolution.


Mathematical Roots & Transformation

Lawlor writes, “The square root of two is the geometric function which represents the universal metaphor of the root, and the root represents the principle of transformation.”3


  • plant roots transform mineral into vegetal

spring onions 845032 1920


  • leaves transform sunlight into live tissue

hosta 3456852 1920


  • rock and stone weather into molecular gases and liquid

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  • liquid into gas

kettle 653673 1920


  • gas into solid

soap bubble 1959327 1920


  • light into heat

road 3133502 1920


  • heat into mechanical movement
  • germination of a seed

cress 253789 1920


  • shellfish convert phosphorus and sodium into calcium shells

crab 298346 1920


  • assimilation of food supports mental and spiritual experience

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“Everything is in a state of digestion, assimilation, transmutation. This transformation goes on in every passing moment as well as in the long aeons of evolutionary cycles. Transformation is the ubiquitous condition of the worlds, and their evolution from mineral to plant to animal, kingdom emerging out of kingdom, volume forming itself out of the converging vector extension of a preceding volume. There is periodicity, rhythm, oscillation, pattern, frequency, all measurable in time and space units.” 4



Transformation & the Present Moment

Lawlor explains, “This is the genesis of sequential appearances, but the moment itself of transformation, from one state to another, from one quality of being to another, from one form or level of consciousness to another, is always a leap, a jump, an incomprehensible velocity, as it were, outside of time, as when one cell divides into two.

If we approach life or evolution with only the sequential intelligence, only the rational, measuring facility, the reality of genesis will always elude us.

This transformative moment is all that really exists; the phenomenal worlds are a transitory reflection.

soap bubble 3490954 1920

They are the past and future of this one ever-present eternity, the only possible eternity without duration which is the present moment.”5



Mathematical Roots & Plant Roots

An ancient designation exists associating the roots of squares and cubes to the vegetal root.

root cell


The root contains an incredible power of growth. Roots have been known to burrow over 100 feet through desert sand in order to reach water.

africa 1170014 1920


Roots are commonly known to destroy concrete sidewalks, foundations and other structures.



A single tuft of rye grass may have over a billion root hairs, which laid end to end would extend for 350 miles.

root hairs


Roots aggressively hunt for water, air and minerals.

angkor wat 469 1920 forest 2248606 1920


They must constantly secrete acids to dissolve minerals to provide nourishment and protection for the plant.

The root of a plant and mathematical roots are both causative:

Plant roots are embedded in the earth. The Earth is represented by the square or cube.

roots 900653 1920 cartography 1801579


Mathematical roots are embedded in the square.

The √2 is the diagonal of a square. The √3 is the diagonal of a cube. The √5 is the diagonal of the double square.

square root of 2 square root of 3 sq root 5


The plant depends upon the root for stability and nutrition.

The plant root nourishes because it is able to break down (divide) the fixed sense mineral parts of the soil to transform into its own tissue.

The Geometric root is an archetypal expression of the assimilative, generating, transformative function of the root.

The root of 2 contains the power of nature – it destroys in order to progress (it severs the initial square) and contains the power to instantaneously transform 1 into 2. See the image below. The diagonal of the first severed square produces the side of the next larger square. The second square is twice as large as the first. This is another example of how Halving = Doubling, or the law of the Octave.

Lawlor square root 2


Lawlor explains, “The plant grows progressively out of a previous breaking down, but there is no rational theory which can explain how a flower or a squash can spring forth from a tender, narrow stem, like the explosion of one square out of another.”6

pumpkin flower 2705388 1920


“The root is a symbol for the law of sacrifice in nature, for, like a mother, its efforts are not for its own benefit but to uplift the plant it its movement toward light.”7

sprout 1147803 1920



Roots & the Human Body

The root function can be expressed as the intestinal function.

offal 1463369

This involves a transformation of food into energy.

Digestion acts to root us in the physical world.

The root function is also expressed in the brain. On the left we see roots and branches in the brain. On the right we see actual neurons in the brain that closely resemble the pattern of roots and root hairs.

brain 2146817 neurons 440660 1920

The brain transforms crude amorphous mental stuff into reason and understanding.

It is also expressed in procreative power.

Sexual functions, like digestion, act to root us in the physical world.

portrait 3113651 1920


“In the secret history not only is it true to say that what eventually evolved into human life passed through a vegetable stage, but the vegetable element remains an essential part of the human being today. If you removed the sympathetic nervous system from the body and stood it up on its own, it would look like a tree. As one of Britain’s leading homeopathic healers put it to me, in a rather beautiful phrase: The sympathetic nervous system is the gift of the vegetable kingdom to the physical body of man.’”8 – Mark Booth



There is “the kind of soul that dwells in the summit of our body, and it raises us up from the earth towards the heavenly region to which we are naturally akin, since we are not soil-bound plants but, properly speaking, creatures rooted in heaven. For it is from heaven, where our souls originally came into existence, that the gods suspended our heads, which are our roots, and set our bodies upright.

When a man is caught up in his appetites or his ambitions and devotes all his energies to them, the mental processes that go on inside him are bound to be restricted entirely to mortal beliefs, and he himself is bound to be completely and utterly as mortal as a man can be, since that is the part of himself that he has reinforced. But anyone who has devoted himself to learning and has genuinely applied his intelligence cannot fail to attain immortal, divine wisdom, if the truth should come within his grasp.” Timaeus 90 b-c

ufo 2144977 1920



Roots & Lightening

Lightening is the root of the sky.

thunder 953118 1920

It transforms carbon and nitrogen into compounds assimilable for plants.


“The morphic similarity between lightning and the root of a plant is also functionally accurate. Science now speculates that early in earth’s evolution horrendous lightning storms in the atmosphere provided energetic ultra-violet light which transformed methane, hydrogen, nitrogen, and carbonic gases into the proto-molecules for organic compounds. These molecules were deposited by torrential rains into the primal seas out of which life arose. Once again the ‘root’ functions as a transformative principle which supports the uplifting propensity which we call life.”9

bad urach 1674049 1920



  1. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  2. ibid.
  3. ibid.
  4. ibid.
  5. ibid.
  6. ibid.
  7. ibid.
  8. Booth, Mark, The Secret History of the World,Overlook Press, 2010
  9. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982


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