Return to Free Library

Return to Geometry Menu

Previous Article                                                                         Next Article

 

We will delve deeper into phi spirals in this article.

 

Spirals & Helices

As we have previously discussed, spirals and helices are the most widespread shape in nature.  They are found at every scale of reality.

Spirals are the purest expression of moving energy.

They express the geometry of self-similarity.

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

 

Three principles of Nature’s spiral:

  • Spirals grow by self-accumulation

 

  • Every spiral has a ‘calm eye’
    • Spirals move faster at its eye than farther out.

 

  • Clashing opposites resolve into spiral balance

 

One important aspect is that where there is one spiral, there is always another, going in the opposite direction.

This is due to the principle of the Dyad (2), that is – polarity.

Let alone, clashing opposites will always resolve into spiral balance.

“A spiral is balance-in-motion made visible, a graph of forces displaying change without change, playing out before our eyes.”1

“Change without change” can refer to an object swirling down a whirlpool.  It remains pointed in the same direction as it spirals around.

“Change without change” can also refer to the principles of standing waves such as is seen in Cymatics.  The overall geometry stays the same as its components (colloids in the case of Cymatics) cycle around in a continuous flow.

Cymatics patterns.  The colloids continually move within the bounds of the geometry.

 

 

Helices

Helices are a symmetrical spiraling about an axis.

Helices always have a particular handed-ness.  This means they spiral either clockwise or counter-clockwise.

 

Things that are mirror images of each other, as in one spirals clockwise and other counterclockwise, are called enantiomorphs.

They are chiral if it has both right and left handed-ness.

A chiral molecule is non-superimposable on its mirror image.

The aromatic molecular structure of oranges and lemons are enantiomorphs.  They are identical except for their handed-ness.

 

 

Logarithmic Spirals

In a logarithmic spiral 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.

The logarithmic spiral was called the “equiangular” spiral by Rene Descartes in 1638.

This was due to the fact that any line drawn from the pole to the curve cuts it at the exact same or “equal” angle all around.  Later Jacob Bernoulli (1654-1705) called it the spira mirabilis – the ‘marvelous spiral’.

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

 

These are found in leaves and shells, cacti and seed-heads of plants, whirlpools and galaxies.

They self-replicate or grow by self-accumulation.

It continues this way inwards and outwards, theoretically, to infinity.

 

 

Examples of logarithmic spirals found in nature include:

The golden spiral/Fibonacci spiral

 

The Root 2 spiral

 

The Root 3 spiral

 

The Square Root Spiral – a.k.a. The Spiral of Theodorus

 

 

The Holographic Nature of Reality

The self-replication of spirals is very important because it creates a fractal, holographic pattern, where the smaller portions mirror the whole (cosmos) itself.

Physicist David Bohm says, “The essential feature of quantum interconnectedness is that the whole universe is enfolded in everything, and that each thing is enfolded in the whole.”

 

The marriage of the whole & its parts is illustrated by:

  • proportional symmetry
  • the golden section

 

“This simple cut [the golden section] appears to be the driving impulse of nature itself, fractalizing with self-similarity into all the parts, and driving the growth process through spiraling golden angles and Fibonacci numbers.”2

 

 

The Golden Rectangle & Golden Spiral

Reference Construction Lesson #65: The Golden Spiral & Finding the Calm Eye

 

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

The golden spiral is an exponential, logarithmic spiral.

“It grows from within itself and increases according to the Fibonacci process of accumulation.”3

The golden spiral can be formed from a golden rectangle:

 

Remove successive squares from each golden rectangle and fill each square with a quarter arc.

This produces the golden spiral.

Every segment has the same curvature, although the spiral size differs.

 

The golden section in one of the daisy’s spirals.  Each stage of growth shares the same proportions (see shaded triangles at right.) Credit: Gyorgy Doczi – The Power of Limits, 1981

 

 

Furthermore, the familiar heart symbol originates from two golden spirals meeting one another and becoming one.

 

 

The 2nd Golden Spiral

There is another golden spiral that often gets overlooked.  “In the classic golden spiral (shown above), the width of each section expands by 1.618 with every quarter (90-degree) turn, and its proportions bear little resemblance to those of the nautilus spiral.  However, another spiral exists that is just as golden.  This spiral expands by a factor of 1.618 with every 180-degree rotation.  Note how it expands much more gradually.  Clearly, a golden spiral based on a 180-degree rotation is much more similar to the nautilus spiral than a golden spiral based on a 90-degree rotation.”4

Credit: Page 158 in Gary Meisner’s The Golden Ratio: The Divine Beauty of Mathematics

 

Approximations of both of these 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

 

 

Jakob Bernoulli (1654-1705) gave the spiral a motto – “Eadem mutato resurgo” – “Although changed, I arise again the same.”

“This process of squares chasing a single receding golden rectangle is mathematically endless.  Theoretically, there is no end to the trail of squares cut out of a continually resurrecting golden rectangle, seducing the squares forever onward.”5

 

“The spiral results from the play of the rational square with the transcendental phi ratio.  It is the play of life itself, clothed by the four states of matter that make it visible.”6

 

 

Nodes

Nodes occur where the resulting spiral intersects the corner of each new square as it expands.

Each node is a phi-ratio farther out along the spiral from the center than the previous one.

Nodes are useful for determining the fractal portion of the phi spiral that is being used by nature in some instances.

Nodes are numbered with red dots on the right-hand image.  Credit: http://cosmometry.net/

 

 

Calm Eye of the Golden Spiral

The calm eye represents the zero at the beginning of the Fibonacci sequence.

To find it, draw one diagonal from corner to corner within a large golden rectangle and then another diagonal in the next smaller golden rectangle.

The eye is located at the point where the diagonals of every golden rectangle cross.

 

The Golden Spiral maintains the same Center of Gravity though it expands in Size.

 

This is seen in the:

  • nautilus – does not need to relearn how to balance as it grows and expands in size

 

  • ram – does not need to relearn balance as its horns grow in size

 

  • trees – stays balanced as the branches spiral around the trunk, regardless of branch size

 

 

Fibonacci and Golden Mean Spirals

Phi goes to infinity and Fibonacci has a finite center.

Thus, Fibonacci is the physical manifestation of phi found in nature.

It is the finite reflection of the infinite principle embedded within it.

The Fibonacci sequence and the Fibonacci spirals are contained within all of life.

 

When comparing a Fibonacci and Golden Mean spiral they are off the closer you get to the center, but by the time you get to steps 55 and 89 in the Fibonacci spiral the two spirals are practically identical.

“The difference is in the areas where they originate.”

Credit: page 210 in The Ancient Secret of the Flower of Life by Drunvalo Melchizadek

 

Fibonacci originates in finity.  It is the physical, visible side of reality.

Phi originates in infinity.  It is the metaphysical, invisible side of reality.

 

 

 

Phi Vortex

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

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

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

 

 

Phi Double Spiral

The Phi double spiral is composed of 2 phi vortexes.

1 is clockwise; the other counter-clockwise.

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

Credit: Marshall Lefferts – cosmometry.net

 

A toroidal energy field is at the center of each fundamental Aether unit at every point in space and time.  It is at the center of each photon, each 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.

 

 

Phi Boundary Condition

At exactly 3 nodes out from the center point of a phi double spiral, there appears a distinct circle.

This is the result of the phi spiral first expanding outward from the center, then as it completes its first three fractal arcs (nodes) it momentarily contracts back upon itself before expanding outward again.

This creates a boundary condition, a circle that defines a potential surface boundary and a complementary inside and outside relationship to the energy field.

 

Credit: Marshall Lefferts – cosmometry.net

 

This boundary can be seen as the necessary limit of a given entity required for the entity to come into form.

Any node on the spiral can be used – the boundary condition will always be 3 nodes out from the node used as center.

See: http://www.cosmometry.net/fractal-holographic-scaling-of-field-patterns for more info.

 

This boundary condition represents the boundary from the seen to the unseen, physical to metaphysica, lspace/time to time/space.

 

 

  1. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  2. Olsen, Scott, The Golden Section: Nature’s Greatest Secret, Walker Books, 2006
  3. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  4. Meisner, Gary B., The Golden Proportion: The Divine Beauty of Mathematics, Race Point Publishing, 2018
  5. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  6. ibid.

Return to Free Library

Return to Geometry Menu

Previous Article                                                                         Next Article