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In this article we will finish our series on the geometry of plants before we move onto the geometry of viruses, bacteria, plankton, sea creatures, insects and animals.


We have mentioned fractals and fractal-holographic patterns many times in Cosmic Core up to this point.  The understanding of the fractal-holographic nature of reality is extremely important and a foundational concept for everything in Cosmic Core.

We will now discuss fractals in a bit more detail, focusing on natural fractals found in nature, Li symmetries and natural spirals.



Fractals & Tree Structure

A fractal is “a natural phenomenon or mathematical set that exhibits a repeating pattern that displays at every scale.”1

Fractals can also be nearly the same at different levels.

Algebraic and geometric fractals are a never-ending pattern that can be zoomed in on infinitely.

The Mandelbrot Set – an algebraic fractal


Fractals have self-similarity on all levels.

They are also called ‘expanding symmetry’ or ‘evolving symmetry’.

They appear to be complex and irregular but possess a recognizable statistical self-similarity.

They are a better description of a physical, observable world – you do not need a straight line, you can use a fractal curve.

Koch Curves, Sierpinski Square & Triangle – Geometric Fractals


Fractals are holographic.  They evolve from the principle of the phi ratio, which comes from the pentagon, which comes from the circle.

Fractals can be seen in all aspects of nature, if you have a sufficiently large perspective to view them.

Fractals led scientists to conclude that things that seem to be chaotic are actually organized.

Highly complex phenomena have a hidden order.


Although in theory, fractals repeat to infinity, in the physical realm they are limited by form.

“Blood-vessels do not reduce indefinitely, any more than the whorls within whorls of the fractal cauliflower extend to infinity.  Nature uses fractal geometry where it is advantageous.”2



Three Kinds of Fractals

There are three kinds of fractals: geometric, algebraic, and natural.3

Geometric fractals include the Sierpinski triangle and Koch curve and algebraic fractals include the Mandelbrot and Julia sets.

We will focus on natural fractals in this article.  In a later article we will discuss all three in detail.



Natural Fractals

Natural fractals are observed in nature.

For instance, the organization of the mineral world can be seen as fractal in structure:

Clay particles – Sand grains – soil Peds – pebbles – rocks – dunes – hills – mountains.


Clouds, lightening, river networks and plants all have natural fractal elements as well.



Lichtenberg “lightning” is a natural fractal.

It is formed by rapidly discharging electrons in lucite (pictured below).

Credit: Paul’s Lab –


Interestingly, the Grand Canyon looks like a giant Lichtenberg lightning scar.

The Grand Canyon from the International Space Station


“The Grand Canyon is approximately 400 kilometers long, 28 kilometers wide, and almost two kilometers deep. Could erosion by the Colorado River be the only factor in its formation?

A few basic facts are necessary to gain a perspective. The Grand Canyon is surrounded by an elevated landscape with the canyon running through it from east to west. The underlying rock strata in the region rises and falls over an area known as the Kaibab Upwarp, while the river descends through an elevation differential of 2100 meters. Water does not flow up over a mountain range nor does it run sideways along sloping terrain, so all theoretical models that insist on water erosion propose that the entire area was slowly uplifted at the same rate as the river eroded the canyon. This process is said to have taken place in a time span of between four million and 400 million years.

The geological models also incorporate natural dams across the river channel that caused reversals in the river flow and were then subsequently breached, allowing the river to resume its previous course. However, a pertinent objection to that theory is that there is no evidence water flowed back into the ends of the giant side channels that join the chasm with the river. Perhaps the most significant challenge to the prevailing theories is the disappearance of almost 1300 cubic kilometers of material that is supposed to have been washed downstream—there is no large delta at the mouth of the Colorado River containing the debris.

Satellite images, as well as pictures taken by astronauts in orbit, seem to indicate that the Grand Canyon is an enormous Lichtenberg figure, in other words, a gigantic lightning scar. As the Electric Universe hypothesis suggests, electric discharge machining (EDM) might account for the Canyon’s appearance: steep walls, thousands of layers, brachiated side canyons at practically every scale, and periodic, hemispherical “nips” cut into each rim.”4



Fractals in Plants

Fractals can be found in many plants as well.  We have discussed many of these in the previous four articles.

Some fractal structures found in plants include:

Mycelium branching patterns

Credit: Kirill Ignatyev



  • broccoli
  • cauliflower





Ferns – Ferns show a preponderance for both fractal branching structures and spirals.


Take a look at Plate 92: Filicinae from Ernst Haeckel’s Art Forms in Nature.

These include:


Cacti – Cacti often show a fractal structure in their growth as well as Fibonacci spiraling processes of growth.



  • Seen in branches and roots.
  • Branches and roots are approximately self-similar (As above, so below.)
  • Formed by repeating a simple process over and over again.


Basic process of tree growth:

  • A sprout comes out of the ground – splits into branches
  • Each branch splits into new branches
  • Each of these branches further splits into branches…and so on…


See page 371 in The Hidden Geometry of Flowers to see the fascinating changes in the shape of trees at different elevations.

This comes from Bill Mollinson’s book Permaculture, from 1988.


Note, natural fractals do NOT keep repeating their patterns forever.  At some point, the fractal repartition breaks down in natural patterns and they cease to be fractals.  This is nature’s way of dealing with infinity in a finite realm.



Branching Patterns

Fairly complex systems are generated from simple rules.

For instance, lightning strikes resemble river systems resemble animal vascular systems.

Below lightning resembles the nervous system, cardiovascular system, leaf veins, plant roots, mud patterns in an estuary, a river system and ice crystals.


Each of these involve efficient distribution of energy in one form or another.

They are the simplest way to connect every part of a given area using the shortest overall distance (or least work).

Any branch of a particular size is always outnumbered by those of the next smaller dimension.

Any branching pattern seen in nature involves natural fractals.



Li Symmetries

Li symmetries are highly common in the natural world.

They are related to fractal branching patterns.

They are self-organizing systems in nature caused by the interaction between processes and materials.


Examples are seen in:

Animal markings

  • Zebra, giraffe, cheetah, snakes, alligators and crocodiles, tropical cone-shell, frogs, lizards, cuttlefish, fish, leopard, ocelot, jaguar, eggshells of common murre and razorbill, seashell patterns, fish scales, section of bones, muscle fibers…



  • goliath beetle, wing casing of grasshopper, wing veins, spider webs



  • Irish Moss seaweed and other seaweeds, vascular cell structure, tissue-forming parenchymatous plant cells, leaf veins, lichens, gill patterns on mushrooms


Stretch patterns

  • tree bark


Landscape patterns

  • cloud patterns, topography of snowlines, drainage patterns of water, mountains & shorelines


  • wind over sand and rock – ribbing of sand dunes


  • water over sand

  • water reflections


  • lava patterns – some look like water reflections, others look like tree bark


Chemical-Mineral processes

  • crack patterns in clay, ceramic, parched earth, old paints and gels


  • ice tracery on windowpanes


  • sand and silt patterns settling in water
  • Fragmented liquid convection rolls


  • Particulate clustering on a liquid medium
  • Kerr magneto-optic effect in thin section of barium ferrite
  • Magnetic maze-domain patterns in silicon-iron polished crystal
  • Magnetic domain patterns

Credit: An inverse transition of magnetic domain patterns in ultrathin films, O. Pormann, A. Vaterlaus & D. Pescia, Nature volume 422, pages 701–704 (17 April 2003),


Mineral patterns

  • agate, malachite, jasper, magnified surface of a diamond, brachiated components in serpentine rock, granite & marble…etc


Li in Neo-Confucian Chinese philosophy refers to “the underlying reason and order of nature as reflected in its organic forms.”5

Alan Watts writes of Li:

“The tao is a certain kind of order, and this kind of order is not quite what we call order when we arrange everything geometrically in boxes, or in rows. That is a very crude kind of order, but when you look at a plant it is perfectly obvious that the plant has order. We recognize at once that it is not a mess, but it is not symmetrical and it is not geometrical looking. The plant looks like a Chinese drawing, because they appreciated this kind of non-symmetrical order so much that it became an integral aspect of their painting.

In the Chinese language this is called li, and the character for li means the markings in jade. It also means the grain in wood and the fiber in muscle. We could say, too, that clouds have li, marble has li, the human body has li. We all recognize it, and the artist copies it whether he is a landscape painter, a portrait painter, an abstract painter, or a non-objective painter. They all are trying to express the essence of li. The interesting thing is that although we all know what it is, there is no way of defining it.       Because tao is the course, we can also call li the watercourse, and the patterns of li are also the patterns of flowing water. We see those patterns of flow memorialized, as it were, as sculpture in the grain in wood, which is the flow of sap, in marble, in bones, in muscles. All these things are patterned according to the basic principles of flow.

So li means then the order of flow, the wonderful dancing pattern of liquid, because Lao-tzu likens tao to water.”


This is an interesting concept, especially since we have been discussing the fluid-like Aether, which can be likened to tao.  All in the universe is motion.  All is in constant flux and flow.  The flow patterns we see, such as wood and marble grain, encapsulate the flow of the universe but on an extremely slow time scale.



Spirals in Nature

Spirals are another common fractal formed by simple repetition, and combining expansion and rotation to generate the spiral.


Goethe notes in The Metamorphosis of Plants that, “Microscopic examination has shown beyond a doubt that the plant’s reproductive organs are brought forth by spiral vessels, as are the other organs…The spiral vessels lie amid the bundles of sap vessels, and are enclosed by them.”


Vessels (trachea) are one of the cell types found in xylem, the water conducting tissue of plants.

“SPIRAL VESSELS are those very delicate air-tubes in the cellular tissue of plants which run unbranched through the different parts of the plant, and whose walls are composed of fibers spirally or circularly twined.

Spiral vessels are either free, when their windings are unconnected with each other, or when the windings are involved with each other in a net-like manner.

Spiral vessels, whose walls are formed of distinct horizontal rings, placed simply one above another, are called annular vessels.

Spiral vessels seldom occur singly, but are generally united by cells into bundles called vascular bundles. These vascular bundles are scattered in the stems of endogenous plants; but in the stems of oxogenous plants they are arranged in one or more concentric circles.

All plants which have spiral vessels are called vascular plants, in contra-distinction to cellular plants, whose substance consists of cells only.”6


The structure of spiral vessels is regarded as having an intimate connection with the tendency of the spiral structure manifested in plants – a tendency which is observed not only in spiral stems, spiral tendrils, spiral flowers, fruit, and the like, but throughout the vegetable kingdom generally in the spiral arrangement of leaves and of the organs which are formed by the metamorphosis of leaves.



Spirals Found in Nature


There are four types of spirals commonly found in Nature:

  • whirlpool eddy
  • wave
  • mushroom vortex rings
  • vortex street



Whirlpool Eddy: a circular movement of water (a vortex)

Wikipedia tells us, “In fluid dynamics an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime.

This phenomenon is naturally observed behind large emergent rocks in swift-flowing rivers.”



Wave:  the passage of undulating motion, heat or sound

A wave is an oscillation accompanied by a transfer of energy.



Mushroom Vortex Rings

Mushroom vortex rings are seen when pouring milk into still coffee or tea.

The poured liquid meets resistance at its head and curls to the side.


Each turn also meets resistance and curls further inward, repeating the cycle of meeting resistance and curling until the force dissipates and resistance disappears, leaving only the spiral path.

A reversal of mushroom vortex rings is called ‘backwater vortices’.

These form behind a flat plate as water or air rushes by it.

These spirals are known as ‘turbulence’.

Mushroom vortexes are often seen in natural structures.  We discuss these in greater detail in our article on the geometry of fungi.


Other examples include:

Bivalve clam shells


Dog noses


Mushroom formation – longitudinal cross-section of many types of mushrooms


The way many leaves unfurl


Mushroom cloud formation


Dust and smoke vortices


Dust spirals trailing a vehicle & vortices behind jets


Vortices in rock


The entry of twins in the womb who whirl into life together

Mushroom vortexes are also seen in the capital of a Greek Ionic column.  This symbolizes the Earth’s subtle energy flowing through the temple, meeting resistance at its head, and rolling back as mushroom vortex rings.



Vortex Street

A vortex street is composed of eddies linked in a chain of alternately opposite spinning whorls.

When a still object interrupts a moving stream or a moving object disturbs still water, a vortex street forms to reestablish balance.

“The central axis of a vortex street is a forward-flowing zigzag from which the separate vortices emerge and balance.  The central rhythm of alternating pulsation gives the whole “street” stability.  Each spiral whirls independently balanced, and all spin in directions that support the direction of the central stream.”7


Flocks of birds use vortex streets in their V formations.

“Only the lead bird must really work at flapping its wings; the others latch onto the undulating spiral wake of turbulence trailing behind it.  They simply relax their wings and let the rolling wave move them up and down and forward.  When the lead bird becomes tired she falls back while another moves ahead to work at splitting the breeze for the others.”8


Sand dunes are carved by one side of the moving vortex street (the wind).

The same occurs with ribbed sand at the shore or in deserts.


We can also see vortex streets in certain cloud patterns and in the patterns of atmospheric flow on Jupiter and Saturn.

Vortex Street cloud patterns on Earth


Vortex Street cloud patterns on Jupiter



Vortex Streets & Plants

“An important part of our own process of expanding vision is to see plants differently, not as static “things” but as dynamic processes, and to develop the ability to recognize plants, in whole and part, as living whirlpools, waves, mushroom rings, and vortex streets.”9

Remember, plants are processes that grow in a fluid-like medium (Aether).  It is only natural that many of their forms resemble spirals and vortex streets.

Each bough, branch, twig, stem and leaf is a complete vortex street and part of greater vortex streets.

Some of these include:

Wheat, barley, rice & millet


The 3-4-5 triangle can be seen in the way many vortex street plants and panicles form.

Credit: Gyorgy Doczi, The Power of Limits, 1981

Racemes & Panicles – found in many wildflowers






Twigs & Alternate Leaf Arrangements








Cut univalve shells


Heloconia (Lobster claws)


Duckweed ferns


Welwitshia mirabilis male cone

Immature cone with stunning geometry.  Credit: Scott Zona


Mature and flowering cone.  Credit: D. Eickhoff


Close-up of cone



Credit: Leafy liverwort by George Shepherd


Credit: Greater Scented Liverwort by James St. John


Here is Plate 82: Hepaticae from Ernst Haeckel’s Art Forms in Nature.  This plate shows liverworts which are herbaceous perennials.  There are many vortex streets to be seen.

These include:

  1. Marchantia nitida, female gametophyte
  2. Marchantia polymorpha, archegonial head from below
  3. Fimbriaria marginata = Asterella marginata, female gametophyte
  4. Fimbriaria venosa = Asterella venosa, female gametophyte
  5. Fimbriaria cubensis = Ceratolejeunea cubensis, female gametophyte
  6. Fimbriaria sanguinea = Schisma sanguineum , female gametophyte
  7. Lunularia cruciata, archegonial head with sporophytes
  8. Jungermannia ventricosa = Lophozia ventricosa, female gametophyte with sporophyte
  9. Jungermannia connivens = Cephalozia connivens, female gametophyte with sporophyte
  10. Lepidozia reptans, female gametophyte with sporophyte
  11. Jubula Hutchinsiae, part of gametophyte
  12. Harpalejeunia ancistrodes, part of gametophyte
  13. Scapania undulata, female gametophyte with mature sporophyte
  14. Scapania subalpina, gametophyte
  15. Scapania umbrosa, female gametophyte with sporophytes
  16. Scapania nemorosa = Scapania nemore, female gametophyte with sporophytes
  17. Scapania aequiloba, female gametophyte with sporophyte cup



So much in reality is formed upon geometry within geometry including spirals within spirals.

“The spiral is as much a mathematically factual phenomenon as it is a metaphor and symbol of paths of energy or botanical consciousness.

Proceeding from center outward is evolution.

Proceeding from periphery to center is involution.

The first develops or dissipates; the second concentrates or integrates.”10



Spiral Geometry of Plants

As we saw earlier, many plants show a preponderance of spiraling.  This principle of spiraling is called by Keith Critchlow ‘turning into being.’

This can be seen in the way a flower bud spirals as it opens.


It can also be seen in the way a leaf or fern unfurls.


The spiral leaf unfolding shown by Critchlow is based upon the √3:1 relationship or half of the equilateral triangle.  “It takes twelve moves to complete one turn back to the base line AB.


On page 153 he shows a spiral based on nineness.  “A spiral based in the nine-pointed star can be shown to take nine stages to get back to the same axis and eighteen stages to make a full turn.”



“Patterns generated by spirals moving in opposite directions are frequent in nature,” says Gyorgy Doczi.  “Here they concern us as special instances of a more general pattern-forming process: the union of complementary opposites.”

Doczi continues: “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 process itself by which the daisy’s harmonious pattern was reconstructed is likewise a jointing of complementary opposites – straight radii and rotating circles.

Credit: Gyorgy Doczi – The Power of Limits, 1981


“If one follows the different curvatures of these spirals through the squares formed by the radiating and rotating sets of lines in diagram a, one can see that spiral A moves from one circle to the next as well as from one radius to the next within a single row of squares.  We will call this a curvature of 1:1, signifying that the rotational and the radiational components of growth are equal.  Spiral B moves through two squares while reaching from one radius to the next, crossing two circles: a curvature of 1:2.  In a similar manner one can say that spiral C has a curvature of 3:1, while spiral D approximates 5:1.

In spite of their differences in curvature, all of these spirals share the qualities of being logarithmic and equiangular, through all stages of growth.”

Credit: Gyorgy Doczi – The Power of Limits, 1981



Spiral Images of Plants

Below is a beautiful visual guide to the spiral geometry of plants.  There are nearly endless examples in nature to be found.  We provide a few examples here.  Many others are shown in the previous series on plants.


Romanesco Broccoli


Crassula ‘Buddha’s Temple’ – succulent


Aloe polyphylla


Eulychnia castanea spiralis (Spiral cactus)

Credit: douneika


Pinecone Ginger (Beehive Ginger)

Zingiber spectabile


Torch Ginger


Costus comosus (Red Tower Ginger) & Costus Barbatus

Credit: Snap Man


Chandelier Plant (top view)






Red Cabbage & Ornamental Cabbage




Magic Dogwood (Cornus florida subspecies urbiniana)


Pine & Fir cones


Cycads – various species

a) Bowenia serrulata seed cone b) Bowenia spectabilis pollen cone c) Ceratozamia decumbens seed cone d) Ceratozamia decumbens pollen cone e) Cycas couttsiana seed cone f) Cycas revoluta pollen cones g) Dioon angustifolium seed cone h) Dioon angustifolium pollen cone i) Encephalartos ferox seed cone j) Encephalartos ferox pollen cone K) Lepidozamia hopei seed cone l) Lepidozamia peroffskyana pollen cone M) Macrozamia lucida seed cone n) Macrozamia lucida pollen cone o) Microcycas calocoma seed cone P) Microcycas calocoma pollen cones Q) Stangeria eriopus seed cone r) Stangeria eriopus pollen cone S) Zamia imperialis seed cone t) Zamia imperialis pollen cones. Photos: all Michael calonje except K: larry Krauss and n: irene terry.


Sago Palm


Rattan Palm trunk w/ spiraling spines & rattan palm male cones


Pelecyphora aselliformis


Ludwigia sedioides (Mosaic plant)


Lobelia deckenii


Amazon Lily Pad

Underside view


Drosophyllum Lusitanicum

Credit: incidencematrix


Camellia & Rose


Lantana flowers




Daisy, Mum, Ranunculus (buttercup) & Rhododendron


Red Hot Poker (Torch Lily)


Bromeliads – Various species


Cacti – various species


Thinking Cactus


Succulents – various species


Moraea Tortillis – curly succulent


Albuca spiralis & Albuca concordiana – succulents




Sulcorebutia rauschii


Oxalis palmifrons

Credit: BGBulbs


Spiranthes longilabris (Giant spiral Orchid)

Credit: NC Orchid


Dragonblood Trees, Yemen




We are now nearly at the end of this series on the geometry of plants – only the geometry of fungi, moss and algae remain.  Throughout this series we have seen that the fundamental geometry that plant life forms upon is the golden ratio and Fibonacci sequence.

Always remember these two things come from the pentagon.


Also recall that everything in nature is built upon consciousness and all in nature is able to telepathically communicate with all else.  If this is hard to understand, just ponder the concept that our thoughts have electromagnetic realities.  This means that our thoughts and emotions have an electromagnetic wave structure – they fall along the electromagnetic spectrum, though outside the spectrum that current scientific instruments can detect.  The entire electromagnetic spectrum surrounds us at all times – both the portions we know, and those we don’t know.  Consciousness is simply a different set of electromagnetic wavelengths and frequencies than those in our currently known spectrum.  We can ‘tune into’ these frequencies – and thus experience telepathy – both between humans and between humans and nature.

This communication is not based on sound, smell, pheromones or chemicals.  It is based upon a fractal-holographic consciousness that is interconnected throughout the universe on all scales.

Review Article 144 for information on plant consciousness.


“We must return, wiser creatures, to the nature that spawned us — not only as loving caretakers but as partners with the other species of the earth. We must discover once again the spirituality of our biological heritage. The majority of accepted beliefs — religious, scientific, and cultural — have tended to stress a sense of powerlessness, impotence, and impending doom — a picture in which man and his world is an accidental production with little meaning, isolated yet seemingly ruled by a capricious God. Life is seen as “a valley of tears” — almost as a low-grade infection from which the soul can be cured only by death.

Religious, scientific, medical, and cultural communications stress the existence of danger, minimize the purpose of the species or of any individual member of it, or see mankind as the one erratic, half-insane member of an otherwise orderly realm of nature. Any or all of the above beliefs are held by various systems of thought. All of these, however, strain the individual’s biological sense of integrity, reinforce ideas of danger, and shrink the area of psychological safety that is necessary to maintain the quality possible in life.”12



  2. Wade, David, Symmetry: The Ordering Principle, Wooden Books, 2006
  4. Armstrong, Michael, The Grand Canyon: Part One, 29 September 2008,
  6. Chamber’s Encyclopedia, Spiral Vessels, 1880,
  7. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, HarperPerennial, 1995
  8. ibid.
  9. ibid.
  10. Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
  12. Roberts, Jane, The Individual and the Nature of Mass Events, Amber-Allen Publishing, 1981

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