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Geometry of Trunks, Branches & Roots

“The great majority of plants express radial symmetry in one form or another.”1

Plants are usually fixed and nonmotile.  They tend to be radial in symmetry.

Animals, on the other hand, tend to have bilateral (dorsiventral) symmetry.

“The trunks and branches of trees usually indicate a radial arrangement in transverse cross-section, and the same is true of roots and vertical stems in general.”2

Credit: Designa – page 247



Geometry of Plant Roots

Plant roots are non-leaf, non-nodes bearing parts of the plant’s body.  It typically grows below the surface of the soil, however there are sometimes aerial roots above ground.


Roots are commonly known for their fractal branching structure.


There is also beautiful geometry involved in the cross-section of roots.


See the image of a cross-section of a barley root.  Notice how similar it looks to Cymatic patterns.

Dr. Hans Jenny & Cymatics Patterns.  Sound = Geometry.



Roots (√)

Recall from Article 8 that supra-rational functions of numbers (such as √2, √3, √5, Φ, π) demonstrate that Number is above all, a relationship.  These “irrational” numbers, or supra-rational as we call them in sacred geometry, cannot be written as a simple fraction.  Instead the numbers go on forever, to infinity, without repeating.


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).  These are the square roots of 2, 3, 5 and Φ (phi).

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

  • √2 – The Generative
  • √3 – The Formative
  • √5 (and its related function of phi, the Golden Mean) – The Regenerative


“The root grows by the constant division of its square shape.”3

Severing the initial square creates a second square two times larger than the first.


The process can continue indefinitely.


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

Square or cubical root cells.  “The root grows by the constant division of its square shape.”


Robert Lawlor teaches us that 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

“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.”4


Some of this transformation includes:

  • plant roots transforming mineral into vegetal
  • leaves transforming sunlight into live tissue
  • rock and stone weathered down into molecular gases and liquid
  • liquid into gas
  • gas into solid
  • light into heat
  • heat into mechanical movement
  • germination of a seed
  • shellfish converting phosphorus and sodium into calcium shells
  • assimilation of food to support mental and spiritual experience

“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 world, 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.”5



Mathematical Roots & Plant Roots

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

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.  Roots have been known to break through concrete, asphalt and other hard surfaces.


The root structure of a single organism can be quite long.  A single tuft of rye grass may have over a billion root hairs, which laid end to end would extend for 350 miles.


Roots aggressively hunt for water, air and minerals.

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.


Mathematical roots are embedded in the square.

The ‘square root’ of 2 equals the diagonal of the square.  Cutting the red square in half creates the length of the side of a second square two times larger (orange square).  This represents the octave principle.  That is, halving = doubling.  In music halving the length of a string creates the same note but one octave higher.  The frequency is then doubled or twice as fast as the same note one octave below.  Halving = doubling.


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.


“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


Robert Lawlor writes, “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.”



The Root and the Germ

R.A. Schwaller de Lubicz teaches the symbolic importance of a seed dividing into root and germ.

The seed immediately divides into root and germ.

This is an alternation of function.


The root principle represents a power botanists call ‘positive geotropism‘.

This is the power to descend, involve and transmute from below.


The germ represents the power of ‘negative geotropism‘.

This is the power to cause growth upwards and outwards.

It provides nourishment until the root begins to function, then it transforms into the first leaves.

It involves the entire ascent culminating in the new seed.

Credit: Bruce Lyon


“These are then polarized, directional opposites of the same power.”7

This alternation of function of root/germ is symbolized geometrically.

The root (diagonal) of one square is equal to the germ (side) of the next and so on…

This illustrates a three-term proportion: a:b::b:c.

The ratio symbolizes:

The geometric root/germ is to the universal root/germ as the universal root/germ is to the botanical expression of root and germ.


“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


Furthermore, the 1st chakra (red ray) is known as the ‘root’ chakra which is located at the coccyx – the lowest point of the spinal cord in a human being.  The journey of the transformation of the soul begins here.  This is the foundation upon which everything evolves.

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


Plato’s words correspond beautifully to the idea that the tree, with its roots and branches, correspond physically to the branching structure of our nervous system, and metaphysically to the layers of our mind.


  • The Deepest Roots: The Cosmic Mind (Infinite or All-Mind)
  • Mid-level Roots: The Archetypical Mind (Galactic Mind)
  • Shallow Roots: Planetary Mind (Akashic)
  • Base of Trunk: Racial Mind (Group or Tribe)
  • Lower Trunk: Individual Unconscious Mind
  • Higher Trunk: Individual Inner Ego
  • Lower Branches: Individual Subconscious
  • Higher Branches: Individual Conscious Mind
  • Leaves: Individual Outer Ego
  • Flower/Fruits: Our thoughts, beliefs, actions, intentions, behaviors, creative products…etc.  This refers to everything we put out in the world on a mental, emotional, spiritual and physical level.




In this article we have seen even more examples of how geometry structures plant life.  We continue to see examples of how the Fibonacci sequence and golden ratio play a role in the growth and function of plants, and we have seen how the mathematical square roots are related to physical roots of plants.


In the next article we will continue our investigation into the geometry of plants moving to the geometry of leaves, flowers and fruits.


“Sweet is the lore which brings;

Our meddling intellect

Mis-shapes the beauteous forms of things;

We murder to dissect.

Enough of science and of art;

Close up these barren leaves;

Come forth, and bring with you a heart

That watches and receives.”  ~ William Wordsworth



  1. Wade, David, Symmetry: The Ordering Principle, Wooden Books, 2006
  2. ibid.
  3. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  4. ibid.
  5. ibid.
  6. ibid.
  7. ibid.
  8. Booth, Mark, The Secret History of the World, The Overlook Press, 2010


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