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In this article we will continue our discussion of the geometry of plants.  We will look now at the geometry of plant cells and cellular function.


First we will start by mentioning the four aspects that plants need to survive: light, air, water, and minerals (solids).

These four ‘states’ can be used as metaphors for the human levels of consciousness.

Keith Critchlow writes, “Light is inspiration itself.  Air is the established domain of the Intellect, which becomes the foundation of culture.  Water symbolizes the emotive relationship between people and all living things – the social level.  Finally, the solid state symbolizes our experiences of materiality or the sensorial world of ‘food’, tangibility and physical things, the necessary support for all physical bodies.

Thus photosynthesis performs the primary vital task of bringing the higher level of these four stages right down into the lowest levels of working fuel or physical food.

Not least of the functions of the ‘green’ world is the production of oxygen as a necessary by-product of the action of photosynthesis.  The reciprocity of carbon dioxide and oxygen is again a beautiful metaphor for the symbiosis, or complementary breathing relationship between the vegetal kingdom and the animal kingdom.”



Concentric Circles and Plant Anatomy

First let us recognize the concentric parts of a plant, specifically a tree.  This is the outer bark, the inner bark, the wood and the pith.

Credit: Arborist Now: Urban Forestry


Concentric circles of growth and transformation show up continuously throughout nature and are a key aspect of fractal-holographic geometry representing spheres within spheres or cycles within cycles.

They also represent expanding sound waves (compression waves) ‘frozen in time’.

Compression waves are seen as concentric spheres of pressure expanding away from a sound source at the center resulting in concentric circles of compression and rarefaction.

Note: pressure waves are spherical.  The surface of the wave constantly expands as the pressure wave radiates away from the sound source.

Pressure variations in the atmosphere describe a sinusoidal curve.


Recall from our extensive discussion of physics that all in physical reality is composed of sine waves and sine waves that have folded back upon themselves to form toroids and standing waves.

Note how similar tree ring growth is to Cymatic patterns of sound resonating through a medium (pictured below).  This represents one direction of tree growth.  A tree results from a certain set of wave frequencies and harmonics that oscillate out from the center of the tree and resonate through the fluid-like Aether medium.

These particular wave sets are seen as occurring very, very slowly – much slower than human growth and perception, hence the tree looks frozen in time, yet it is always changing and growing even as we can’t perceive it.

Cymatics Patterns that resemble cross-sections of trees.


In spite of this information we want to emphasize the attitude of Goethe, Rudolf Steiner and Keith Critchlow in regards to plant anatomy.

Critchlow writes, “Botany for quite some centuries has used the unquestioned technique of killing life to analyze its material contents.  This is the natural consequence of believing that everything has a material cause – even including intelligence itself.  This meant that to ‘kill’ in the name of science was permissible.  The stories of Rene Descartes’ animal surgery are enough to raise the hind hairs of one’s head.

So we choose not to encourage any further killing and dissecting of flowers.  Rather, we have chosen to accept each flower as it presents itself to our eye (or camera).  This decision is inspired by Goethe’s ‘Gentle Empiricism’ which seems to us to be the only respectful and responsible approach.”

He continues (on page 249, pictured below) by showing some cross-sections of plant stems and commenting, “These sections through flowers, although beautiful in their symmetry represent an attitude of violence toward the living flower which we cannot approve of.  However, we present this example and our analysis – as an exception.”

Geometry of a cross section of a flower.  1) main axis relative to the flower; 2) cross-section through the flower; 3) bracteole; 4) subtending bract


As with all parts of plants, the cross-sections are built upon geometry.  Often the same geometry that the petals are built on is the same geometry seen in the cross-sections.

The point is that geometry is the form-maker in nature and the geometry works longitudinally, horizontally, and spherically.


However…“No matter what information or data you receive as the result of animal experimentation or dissections for scientific purposes, and no matter how valuable the results appear to be, the consequences of such methods are so distorted that you comprehend less of life than you did before.”1




Geometry of Plant Cells

Plant cells are usually rectangular, cubical or hexagonal.

They tessellate.  This minimizes interstitial space and maximizes support and functionality of the cell.

Plant cell walls are composed of cellulose making them very rigid.

Hexagonal and Cubical Plant Cells



Delaunay Triangulation & Voronoi Tessellations

Delaunay Triangulation and Voronoi Tessellations are a process that nature undergoes to form many structures in nature.  These include hexagonal molecular structures and hexagonal cellular structures of plant seeds, pollen, roots, leaves, flowers and fruits, as well as the cellular structure of algae, fungi, insects and insect wings, and the cellular structure of many animal and human tissues.  This process is also seen in the structure of convection cells.


It is important to understand how this process works geometrically to see why it is so commonly used by nature.

It is all based on triangles.


“The implication is that when a single point relates to two other points in space it becomes a generating angle or vector.”2

In the article Nature by Numbers3 it says, Voronoi Tessellations “are based on a distribution pattern that is easily recognizable in many natural structures, like the wings of some insects and plant leaves.”

The article goes on to show how Voronoi tessellations and Delaunay triangulation can form the structure of a dragonfly wing.

Delaunay triangulation involves a set of discrete points on a plane.  Triangles are formed among the points so that no outside point falls in the circumcircle of the triangles.  Connecting the centers of the circumcircles produces the Voronoi tessellations that are hexagonal and pentagonal in structure.

Visit: to see clearly what is going on here.


Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation.  They tend to avoid sliver triangles (extremely acute triangles that look like slivers).

Reference Construction Lesson #29: Delaunay Triangulation and Voronoi Tessellations.

Cellular structure of a plant stem


Cellular structure of wood cellulose in a block of balsa wood.  Credit: Australian National University



Molecular Structure of Cellulose


It is important to note that life is based on geometry for the efficiency of it.  Geometry equals order and order equals efficiency.  Geometric order just so happens to be beautiful and pleasing to the eye.  But plants grow in this way in order to use the most efficient processes available, getting the maximum amount of inputs with the least expenditure of energy.

Having said that, to say that efficiency is the only raison d’etre for plants would be quite short-sighted.  Throughout Cosmic Core we will continually bring up the idea that Nature is both a scientist, a mathematician and an artist.

There are far too many beautiful variations in species that have no scientific reason for being so.  There is no scientific reason why there should be so many different colorations on beetles, or so many varied structures of diatoms and radiolaria; or why there should be such intricacy and variation of pollen walls.

Nature attempts to create as much variation as it can because Nature (and humanity) is inherently creative.  We are created to create.  Everything created as an inborn need to create in turn.

To ensure these variations will produce a working life form in Nature geometry is used.  Simple geometric principles are used to create complex and myriad forms as we see again and again and again throughout Cosmic Core.

Nature is surely a scientist and an artist!



Geometry of Plant Cell Functions



Keith Critchlow tells us, “Photosynthesis is surely as fine a metaphor for ‘the bringing down of light’ as any within our human comprehension.  That we take this miracle for granted is a reflection of the attitude that pervades the modern world, and that encourages us to take life itself for granted, and in doing so we lose all reverence and respect for it.”



Geometry of Photosynthesis

Photosynthesis is defined as the “process by which green plants and some other organisms use sunlight to synthesize foods from carbon dioxide and water.  Photosynthesis in plants generally involves the green pigment chlorophyll and generates oxygen as a byproduct.”

Or as Keith Critchlow says, “Photosynthesis is the mastery of sustaining life and enlightenment at all levels, symbolically and actually.”


6CO2      +    6H2O →  C6H12O6  +  6O2

Carbon Dioxide             Water                    Sugar                    Oxygen



Hydrogen, nitrogen and magnesium of the chlorophyll molecule are arranged in a complex twelve-fold symmetrical mandala-like pattern, similar to a daisy:

Credit: Robert Lawlor – Sacred Geometry: Philosophy & Practice


Any other arrangement cannot transform the radiant energy of light into life substance, aka, photosynthesis.

“In mythological thought, twelve most often occurs as the number of the universal mother of life, and so this twelve-fold symbol is precise even to the molecular level.

Here we find twelve-fold symmetry as the life-giver or womb which transforms light into the basic spectrum of organic substance.  This is recalled symbolically in the stained-glass window, which transforms light into the color spectrum.”4


Furthermore, there is a strong resemblance of the structure of hemoglobin to chlorophyll.

Hemoglobin in the blood carries oxygen from the respiratory organs (lungs or gills) to the rest of the body.

Chlorophyll is essential in photosynthesis, allowing plants to absorb energy from light.

The layers of a leaf are shown below:

During photosynthesis the sunlight passes through the layers of the leaf (cuticle and epidermis) to the next layer of cells which contains the chloroplasts which perform the essential function of transforming light into food.  “Exactly how they achieve this is still a mystery – a mystery enhanced by the magnificent chlorophyll molecule with its mandala-like form.”5



Geometry of Respiration

Cellular respiration is defined as “a set of metabolic reactions and processes that take place in the cells of organisms to convert biochemical energy from nutrients into adenosine triphosphate (ATP) and then release waste products.”

Pairs of “guard” cells on the underside of a leaf contract and relax to form a vesica piscis-like opening called a “stoma”.  A stoma is a pore of a leaf.  Stomata is the plural form.

This allows the plant to exchange oxygen for carbon dioxide.

The stomata allows for the “breathing” of the plant.

Stomata exist as a feature of all leaves involved in transpiration and breathing (gas exchange), as well as water-shedding.  They are part of the upper and lower surfaces of virtually all leaves.

Note the vesica shaped opening in this electron micrograph of a stoma in a bok choy leaf:


“We still do not know exactly what opens and closes the stomata.  The primary cause is thought to be principally connected with light intensity but we cannot be sure…

This curious organ-like structure poses a mystery as it is associated with gases, liquids and (to a degree) light intensity, and has the quality of proto-nostrils, proto-lips, and proto-eyes.”6

The stomata breathe (lips/nose), weep (eyes) and let in light (eyes).

Recall the information from Article 144 and the studies done that show plants have responses similar to those of an animal nervous system.

“The open-minded scientist will agree that ‘intelligence’ and ‘consciousness’ are still very much on the agenda when it comes to definitions and understandings.”7



Transpiration vs. Respiration

“Respiration is the process by which plants convert carbon dioxide to oxygen as a part of photosynthesis.  [That is, how they acquire energy].  Transpiration is the process of water movement inside of the plant’s system.  [How they lose water].  This means water drawn from roots to leaves, up through the stem and cells of the plant.”8

“While the gases are being exchanged the same facility encourages the evaporation of water as vapor.  This is called transpiration and is a vital aspect of collective leaf behavior – and one of the most mysterious functions of plant and tree life because it is taking water up on a journey quite the opposite to water’s natural tendency.  The transpiration of trees even causes cloud formation above a forest.”9




Auxin is a plant hormone that controls the pattern of leaves on a plant.

Leaves become initiated in localized areas where auxin is absent.

When a leaf is initiated and begins development, auxin begins to flow towards it, depleting auxin from another area on the meristem where a new leaf is to be initiated.

Notice the geometry of an auxin molecule and the merging of the hexagon and pentagon.

Indole-3-acetic acid (IAA) is the most abundant and the basic auxin natively occurring in plants.

It generates the majority of auxin effects in intact plants.

It is the most potent native auxin.

Through plant hormones the plant (as a whole) can react to external conditions and adjust to them, without requiring a nervous system.



Secret of Plant Geometry Revealed:  How Plants Set the Angles of their Branches10

In a study done at the University of Leeds entitled Auxin Controls Gravitropic Setpoint Angle in Higher Plant Lateral Branches, from 2013, scientists have discovered how plants set the angles of their branches relative to gravity.

Control of number of branches and positioning around the main shoot of a plant are well understood (Fibonacci sequence geometry).

Yet scientists have long puzzled over how plants set and maintain the angle of their lateral branches relative to gravity.

The angle of root and shoot branches is not usually set relative to the main root or stem from which they grow but relative to gravity (gravitropic set-point angle).

“In the case of the main root or stem, which grows upright, the mechanism is well understood: gravity sensing cells called statocytes detect that the plant has been tilted, prompting an increase in the movement of a growth-regulating hormone called auxin to the lower side of the shoot or root and driving upward growth in the shoot and downward growth in the root.

When growing vertically again, the statocytes stop sending more auxin to one side than the other and the bending growth stops.”11


Many of the angles in branch and root architectures are at an angle to gravity, rather than being completely upright.

Dr. Stefan Kepinski, of the study, said: “We have found that another growth component — the ‘anti-gravitropic offset’ — counteracts the normal gravitropic growth in these lateral branches. This offset mechanism sustains growth on the other side of a branch from the gravity-sensitive growth and prevents the branch from being moved beyond a set angle to the vertical. It turns out that this countervailing growth is also driven by auxin, the same hormone that causes gravity responsive growth on the lower side of the branch.”12


Branches that are growing close to the vertical have a weak anti-gravitropic offset, while in branches that are growing out at shallow angles away from the vertical the anti-gravitropic offset is relatively strong.

Dr Kepinski said: “The angle of growth of branches is an exceptionally important adaptation because it determines the plant’s capacity to capture resource above and below ground. Depending on what sort of soil a plant is in, it might be beneficial to be foraging for nutrients in the top soil or to be going deeper. Similarly, in the shoot, a plant might gain an advantage from having more steeply pitched branches to avoid shading from neighboring plants. Until now, nobody really knew how non-vertical growth angles, referenced to gravity like this, were set and maintained.”13

Keith Critchlow mentions on this note, “It has been claimed that the patterns of leaf unwindings or spiraling emergence are controlled by the plant hormone auxin.  One can only speculate that the good hormone auxin was educated at a Fibonacci School, as she has a handful of golden strategies to play with.  It seems far more likely that there are a series of interactive collaborative ’causes’ behind such a significant display or a universal law.  To reduce it to a single hormone seems far too reductionist to the author.”14

He goes on to say, “There is a current fashion in the research labs of modern botany to cite ‘auto’ behaviors in plant form.  Phyllotaxis is no exception.  It is as if the Darwinian survival of the fittest was an ‘auto’ pattern of nature, when in fact it is the survival of the fittest to collaborate (or network) that is the truer picture.”




In this article we have seen more examples of how geometry relates to plants.  Geometry is involved not only in the overall structure and growth of plants, but also in the cellular structure and cellular processes such as photosynthesis, respiration and angle-setting.

In the next article we will continue this discussion, looking at the astonishing and beautiful geometry of seeds, pollen and roots.



  1. Roberts, Jane, The Nature of the Psyche: Its Human Expression (A Seth Book)
  2. Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
  4. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  5. Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
  6. ibid.
  7. ibid.
  9. ibid.
  10. University of Leeds, Secret of Plant Geometry Revealed: How plants set the angles of their branches, ScienceDaily, 25 July 2013,
  11. ibid.
  12. ibid.
  13. ibid.
  14. Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011

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