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In this article we will discuss a side topic of the Pentad – that of spirals and helices.

There are many forms of spirals, yet all spirals are processes of transformation and growth.

The growth may happen at different rates and in different proportions, yet they all speak to the idea of pressing forward – upwards and onwards – and the associated process of transformation that occurs with this growth.



Spirals & Helices

Spirals and helices are the most widespread shape in nature.

They are found at every scale:

Sub-atomic particle tracks


Centripetal flow of gravity


DNA double helix


Wind & water flow


Cloud patterns


Smoke vortices


Vortices in the Northern Lights

Credit: David Cartier Sr./NASA


Hurricanes & tornadoes


Bacteria (Spirochetes) & Virus (helical) morphology

Spirochete.  Credit: Mathieu Picardeau – Institut Pasteur

Helical Virus


Plant structures


Phytoplankton blooms in the ocean

Credit: SeaWiFS Project/NASA/Goddard Space Flight Center/ORBIMAGE


Seashells & sea creatures

Spider webs, insect anatomy and insect movement through water

Credit:Brito, Rute M., Schaerf, Timothy M., Oldrovd, Benjamin P., Brood comb construction by the stingless bees Tetragonula hockingsi and Tetragonula carbonaria, Swarm Intelligence, 2010, DOI:10.1007/s11721-012-0068-1, Link



Animal horns, trunks, tails and other anatomy


Human anatomy & physiology


Planetary vortices


The movement of planets and Suns through space




Spirals are the purest expression of moving energy.

They also express the geometry of self-similarity.

As Michael Schneider writes, “The spiral’s role in nature is transformation.”


There are three principles of Nature’s spiral:

  • spirals grow by self-accumulation
  • every spiral has a ‘calm eye’
  • clashing opposites resolve into spiral balance


“In symbolic geometry the essential principle of the Pentad is life and regeneration expressed by the property of self-similarity.  Inwardly, this characteristic indicates the possibility of spiritual regeneration, rebirth from the human to the divine as sought by every one of history’s religions.  Methods for attaining it differ, but curiously they share the spiral as a universal symbol of spiritual transformation, just as spirals indicated the transformation of forces in nature.”1



Four Types of Spirals Found in Nature

When discussing spirals in nature and fluid dynamics, it is important to remember that the pervasive Aether medium [the space/time ‘Vacuum’ of space] is fluid-like.  Therefore fluid dynamics play a major role in the formation, structure and function of all in reality.



There are four types of spirals commonly found in Nature:

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




“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.”2




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

It is also a disturbance that transfers energy through matter or space.


There are two main types of waves:

  • mechanical waves
  • electromagnetic (EM) waves


Mainstream physics tells us that mechanical waves propagate through a medium, and the substance of the medium is deformed, but electromagnetic waves do not require a medium.

Here we disagree and say that all waves require a medium and the medium that EM waves require is the Aether.  We spend many, many articles detailing the whys and hows of this discussion.  Read Articles 98-129 for more information.

Never forget the importance of waves.  All waves consist of wavelengths and amplitudes and all wavelengths contain harmonics or sets of mathematically proportional wavelengths encoded within them.

In two dimensions waves look like sine waves.  In three dimensions they look like corkscrews or helices.


But remember, most waves actually move spherically outwards from a point source.

When these spherical waves oscillate between two realms in three dimensions the waves are pulled into the center of their source from the circumference and propelled back out again from the center.  This motion forms a torus flow process.

As wave packets oscillate back and forth from time/space to space/time, interference patterns are created.  These interference patterns are geometric.  (See Cymatics – Article 121).  The geometry formed is a result of the specific wavelength, amplitude and harmonics of the wave.  The geometry is not a ‘thing’, it is an Aetheric flow field upon which matter precipitates.

Just as harmonics are embedded within every wavelength, geometry is also embedded into every wave frequency.  This is a fact of nature that cannot be denied.

Cymatic patterns: Geometry is embedded within wave frequencies.  Sound (vibration) creates geometry.


Again, this topic is covered extensively throughout the Science section of Cosmic Core.



Mushroom Vortex Rings

These 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 vortex rings are seen in:

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 daisy 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.”3


Flocks of birds use vortex streets in their V formations.

Michael Schneider tells us, “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.”4

Henri Weimerskirch, in 2001, fitted pelicans with heart rate monitors and found that birds at the back of the V had slower heart rates than those in front, and flapped less often.

Johannes Fritz fitted northern bald ibis with equipment to record the bird’s position, speed and heading, several times a second.

“The recordings revealed that the birds fly exactly where the theoretical simulations predicted: around a meter behind the bird in front, and another meter off to the side. Some ibises preferred to fly on the right of the V, or on the left. Some preferred the center, and others the edges. But on the whole, the birds swapped around a lot and the flock had no constant leader.”5

It was also found that they were tracking the good air throughout the flap cycle, when it was thought to be only 20% of the time.  They are able to instantly respond to the wake that hits them.


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

This is the same with ribbed sand at the shore or in deserts.



Vortex Streets & Weather

The Coriolis Effect is a form of a vortex street.

The dictionary defines the Coriolis effect as “an effect whereby a mass moving in a rotating system experiences a force acting perpendicular to the direction of motion and to the axis of rotation.  On the Earth, the effect tends to deflect moving objects to the right in the northern hemisphere and to the left in the southern and is important in the formation of cyclonic weather systems.”

Due to the spin of the Earth and the Coriolis effect, low-pressure storms north of the equator spin counterclockwise, meshed with high pressure spirals (fair weather) spinning clockwise.


High & low pressure spirals are linked as a daisy-chain vortex street in bands around the planet.

The differences in temperature, motion, direction and moisture are resolved as vortex streets.

Vortexes can also be seen with ocean currents.

Tornadoes result from a clashing of contrasting temperatures and winds.


Certain types of clouds such as marine stratocumulus clouds can form “cloud streets” along the direction of the wind flow.  “When the flow is interrupted by an obstacle such as an island, a series of organized eddies can appear within the cloud layer downwind of the obstacle.  These turbulence patterns are known as von Karman vortex streets.”6


Vortex streets can also be seen in the patterns of atmospheric flow on Jupiter and Saturn.

Jupiter’s Great Red Spot


The vortex at the center of Saturn’s polar hexagon



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.”7

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.


Some of these include:

Wheat, barley, rice & millet


Racemes & panicles


Tillandsia – Bromeliad


Twigs & Alternate leaf arrangements

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




Cut univalve shells


Heloconia (Lobster claws)


Duckweed ferns


Welwitshia mirabilis male cone…




Types of Geometric Spirals:

  • The Archimedean Spiral
  • The Fermat Spiral
  • The Greek Ionic Volute Spiral
  • Logarithmic Spirals:
    • Square Root 2 Spiral
    • Square Root 3 Spiral
    • Square Root 5 Spiral
    • Spiral of Theodorus – The Spiral of Square Roots
    • The Fibonacci Spiral
    • The Golden (Phi) Spiral
      • Phi Vortex
      • Phi Double Vortex



The Archimedean Spiral

The Archimedean spiral is one whose distance from the center point grows at a fixed rate.

The distance between successive coils is always the same.

This is seen in a coil of rope, clock springs, record grooves, and a roll of paper towels.


The curvature (or angle) does not stay the same – it gets broader as it widens.  This is in direct opposition to logarithmic spirals whose angle does not change as it grows.

Below is an image of a double Archimedean spiral.  It has two coils: one going clockwise and the other anti-clockwise.

The helix is a 3D version of the Archimedean spiral.


Archimedan spirals and helixes such as this are seen in:


Bolts, screws & coil springs


Prehistoric Spirals in Art

Gyorgy Doczi writes that, “Intertwined spiral mazes from Neolithic times, identical with the Cretan Labyrinth, the Maori tattoo, and the American Indian Tapu’at, are carved into the rocks of barrow tombs in New Grange, Ireland.  These double spirals have been interpreted as symbols of death and rebirth, because as one follows the line coiling inward, on finds another line coming out in the opposite direction, suggesting both burial in the tomb and emergence from the womb: the dinergy of life and death.



Axonemes in Microtubules

Some of the most minute and important elements within living cell structures (such as red and white blood corpuscles) group themselves in double spiral patterns.  The cores of these microtubules, referred to as axonemes and seen here enlarge in the electron micrograph are a faithful match to the double spirals of prehistoric tombs, the tattoos of the Maoris, and the Mother Earth patterns of the American Indians.”

Axoneme, the core of a single axopod of a microtubule, shown in cross-section.  Enlarged 90,000 diameters.  Credit: Gyorgy Doczi – The Power of Limits



Markings on some shells

Credit: Samuel Colman – Nature’s Harmonic Unity



Certain plant structures


Uncoiling ferns




Coiling earthworms, centipedes and millipedes


The hive of the tetragonula carbonaria species of sting-less bee from Australia

Credit:Brito, Rute M., Schaerf, Timothy M., Oldrovd, Benjamin P., Brood comb construction by the stingless bees Tetragonula hockingsi and Tetragonula carbonaria, Swarm Intelligence, 2010, DOI:10.1007/s11721-012-0068-1, Link



Some spider webs


Human fingerprints


Double helix of our DNA


Electromagnetic waves & the movement of photons through space


The movement of planets through space


Reference Construction Lesson #59: Constructing the Archimedean Spiral.



Fermat Spiral

The Fermat Spiral, named after mathematician Pierre de Fermat, is also called a Parabolic Spiral.

It looks like two Archimedean spirals put together, one going clockwise and the other anticlockwise.

Successive whorls enclose equal increments of area.

This accounts for its appearance in phyllotaxis – the arrangements of leaves on a stem and florets on a flower face.  Below are common phyllotaxis patterns.



Greek Ionic Volute Spiral

The Greek Ionic Volute Spiral is a unique and beautiful spiral.

“This spiral is the characteristic feature of the Ionic column.  The most perfect specimen of these volutes is to be found on the capitals of the portico of the Erechtheum.  No record of how the Greek architects set out thee curves has come to light, and the only satisfactory method of setting out the curve has been arrived at by examining and measuring the curves on the marbles.  It has been found by Mr. John Robinson that the convolutions of the curves can be circumscribed by similar rectangles, and that the ratio of the sides of these rectangles varies from 6:7 to 5:6.”8

Samuel Colman writes, “We know that the Greek artists created from lines of the shell most beautiful things many of which can never be improved upon for they were constructed under the influence of fundamental laws of Nature.  Of these the Ionic volute is perhaps the most perfect example.”


It is a complex spiral to draw.  Beautiful directions can be found here:



Logarithmic Spirals

In logarithmic spirals the curve appears the same at every scale.

Any line drawn from the center meets any part of the spiral at exactly the same angle for that spiral.

Zoom in on the spiral and there is another identical spiral waiting for you.

These are found in spider webs, leaves and shells, cacti and seed-heads of plants, flowers, the nerves of the cornea, whirlpools, tropical cyclones and galaxies.


It self-replicates – growing by self-accumulation.

Theoretically it continues this way inwards and outwards to infinity.

There are many types of spirals found in nature.  However, “those inexpressible in whole numbers are the ones which seem to occur with the greatest regularity and were particularly revered in all traditional civilizations in their temple architecture or sanctuaries.”9



Square Root Spirals

The most commonly seen logarithmic spirals found in nature are based upon the square root of 2, 3, 5 and the golden ratio.  These four spirals comprise a family of spirals that Nature uses in many ways.

All these square root ratios emerge from the Vesica Piscis – the ‘womb of creation’.

With side lengths = 1: The square root of 2 = the diagonal of a square; the square root of 5 = the diagonal of a double square; the square root of 3 = the length of the Vesica Piscis.


The Square Root 2 Spiral

The Square Root 3 Spiral

The Square Root 5 Spiral – approximated below


Square Root Spiral – Spiral of Theodorus



Introduction to the Golden Ratio

We will now move onto the most important aspect of the Pentad – that of the golden ratio.  We will introduce it here and discuss it in greater detail over the next five articles.


The Regenerating Pentad

The Pentad displays the property of fractal self-similarity by its very nature.

This is seen in the geometric property of the pentagram replicating smaller and larger versions of itself that spiral inwards and outwards as it grows.


Pentagons and pentagrams both exhibit self-similarity.

“The Pentad holds the principles of the geometry of regeneration.  It is nature’s way of producing endless variety using a single self-reproducing scheme.”10



The Regenerating Pentad Mirrored in Nature:

This regeneration of the Pentad is seen in the ability of vegetation and some animal creatures to regenerate another whole like themselves from a part, or from a seed.

These include nearly every plant, starfish, earthworms, and flatworms.


This is also seen in the veins in a leaf that mirror the branching pattern of the whole leafless tree.


Each ridge of a fern leaf models the whole leaf and the whole plant.


Broccoli and cauliflower resemble miniature trees.


A tree’s leaves swirl around their twigs in the same pattern that twigs swirl around branches, as branches around boughs, as boughs around the trunk.



The Pentagon & the Golden Ratio

The pentagon is the pure geometric embodiment of the golden section, or phi (Φ).


Recall that the golden section is the only way to divide a line or volume so that the small part is to the large part as the large part is to the whole.  It represents how an entity can be a part of the whole and the whole simultaneously – how one can be finite and infinite at the very same time.


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

Φ = (√5 +1)

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 is to φ, the Golden Section.



The golden ratio is a relationship that represents the accumulative process that manifests the pentagon.

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.”11


Golden Spirals

In golden spirals the distance between the golden spiral coils keep increasing, growing wider as it moves away from the source or narrower as it moves toward it.

They are based upon the phi proportion of 1.618…

The golden spiral is an exponential, logarithmic spiral.

The golden spiral “grows from within itself and increases according to the Fibonacci process of accumulation.”12

The golden spiral can be formed from a golden rectangle.  The short side =1.  The long side equals 1.618 (phi):

The image above is a Fibonacci rectangle.  It is nearly identical to a golden rectangle.  The difference is the two smallest equal squares in the center.  In a golden rectangle these would not be equal (they would be 1:1.618) and the ratio would continue inwards indefinitely.


Removing successive squares from each golden rectangle and filling each square with a quarter arc produces the golden spiral.

We will discuss the golden spiral in greater detail over the next few articles.



Fibonacci Spirals

Fibonacci Spirals are intricately related to the golden spiral.  The Fibonacci spiral is how Nature expresses an infinity in the physical realm.   It is almost identical to the golden spiral.

The only real difference is where they originate.

Credit: Drunvalo Melchizadek


As we have seen, Fibonacci spirals are common throughout the natural world:

  • seashells
  • broccoli
  • cauliflower
  • sunflowers
  • many different plant species
  • ram’s horns
  • our ears and fists
  • water whirlpools
  • galaxies



The Lucas Spiral vs. the Fibonacci Spiral

The Lucas spiral is pictured below.  It is based upon the Lucas series.  See Article 178.  The Lucas series is found in 1.5% of observed phyllotactic plant patterns.

Wikipedia states, “The Lucas spiral approximates the golden spiral when its terms are large but not when they are small. 10 terms, from 2 to 76, are included.”

Compare the Lucas spiral to the Fibonacci spiral below:



Phi Vortex

Duplicating and repeating the phi spiral in a circular array with a common center point creates a phi vortex.

These vortices can be clockwise or counterclockwise.

This is commonly seen in water and air flow, as well as galaxies.



Phi Double Vortex Spirals

The phi double vortex spirals involve two phi vortexes.

One moves clockwise; the other counter-clockwise.


This is a fundamental field pattern.  It is a cross-section of a spherical/toroidal energy field.



A toroidal energy field is the center of each fundamental Aether unit at every point in space and time.  It is at the center of each photon, atom; each cell; each life form’s energy field; each planet, star, galaxy and galactic cluster.

The torus shape and function remains the same.  It only differs in size and strength of flow.


We discuss these concepts in great detail throughout Cosmic Core.


“Our own inner spiral process whirls us into the mystery of the infinite within us.  The spiral shows us that we are comprised of a continuum, not fragments, as we ordinarily appear to ourselves.  A clue waits coiled in the spiral’s principles of self-replication, self-accumulation, self-recurrence, and self-similarity.  The recurring key word is ‘self’.  The message of the spiral is growth and the transformation of our Self.”  Michael Schneider


  1. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  3. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  4. ibid.
  5. Yong, Ed, Birds that Fly in a V Formation Use An Amazing Tick, 15 January 2014,
  7. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  8. Spooner, John Henry, Elements of Geometrical Drawing, 1901
  9. Critchlow, Keith, The Hidden Geometry of Flowers: Living Rhythms, Form and Number, Floris Books, 2011
  10. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  11. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  12. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994


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