Two Leonardos, Two Golden Spirals

The place is Pisa, Italy, the date is c.12th Century, and Fibonacci is our first player…
Pisa – 1 January 1923 – New York State Archives

Calculus – Approximation or Limit Value

Any two successive numbers in the Fibonacci Sequence result in a ratio close to the Golden Ratio. This is called the ‘Approximation’ or ‘Limit Value’ in the branch of Math known as Calculus.

Leonardo Pisano Bigollo – a.k.a. Fibonacci

A Riddle of Rabbits

‘If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there, if we assume that every month a pair of rabbits produces another pair, and that those rabbits begin to bear young two months after their birth?’

Leonardo Pisano Bigollo posed ‘The Rabbits Riddle’ not long after he returned from his travels in India and the Middle East. He was a young man in his early twenties and had returned to Italy where he had grown up. It is unlikely that he would have known then, that he was to become probably the most well recognised Mathematician of the Middle Ages.

Seemingly picked up during his travels, Leonardo was attributed with introducing the West to the ‘Arabic Numerical System’. It was based on his research of Math in Commerce, in relation to his family’s significant trading business and is all recorded in his book ‘Liver Arabic’.

The Arabic Numerical System replaced the Roman Numerical System which was still being used by many Europeans and the Arabic Numerical System was an important change. Many believe that it allowed a much broader scope of the application of Math and that in Europe, developments would otherwise have been far behind the rest of the world – or would they?

Roman Numerals 1,2 and 3, are represented by I, II, and III

Similarly Chinese numerals 1,2, and 3, are represented by , , and

This is significant. And the subsequent numbers represented by each system are also significant. We expand on this in the next blog on Sacred Geometry.

In the meantime, continuing with The Rabbits Riddle, it was set by Leonardo as a way to illustrate a particular and very important sequence of numbers. Incorporating his nickname Fibonacci, the answer to the riddle is made up of the numbers that have become known as ‘The Fibonacci Sequence’.

The Fibonacci Sequence

A sequence of numbers that are made up of the number that is the sum of the previous two numbers in the sequence, and which starts with 0.

Using Fibonacci’s Rabbits Riddle, it goes something like this:

  • End of month one – The First Pair of rabbits are too young to breed 0
  • End month two – 1 Pair of rabbits are born 1
  • End of month three – 1 Pair of rabbits are born 1 {the other rabbits are too young}
  • End of month four – 2 Pairs of rabbits are born 2 {the other rabbits are too young}
  • End of month five – 3 Pairs of rabbits are born 3 {the other rabbits are too young}
  • End of month six – You get the idea…

Numerically, The Fibonacci Sequence is expressed as:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89

But there’s more to it…

Phi φ

The term “Phi” was coined by American mathematician Mark Barr in the 1900s. Phi has continued to appear in mathematics and physics, including the 1970s Penrose Tiles, which allowed surfaces to be tiled in five-fold symmetry. In the 1980s, Phi appeared in quasi crystals, a then-newly discovered form of matter.

Phi is also used as a symbol for the Golden Ratio and on other occasions in math and science.

This use is separately encoded as the Unicode glyph ϕ. And the Fibonacci sequence provides yet another way to derive Phi mathematically.

It is expressed numerically as:


SPIRALS – Look closely

The Fibonacci Based Spiral

Expressed visually by the shapes and path the numbers create, the Fibonacci based spiral can be seen in this now familiar image:

Each of the larger coloured squares created represents a number in The Fibonacci Sequence and is based on the sum of the two previous numbers. So, the Length of a Line in a Square of 5, plus the Length of a Line in a Square of 8, equals the Length of the Line of their sum, 13, and so on…

The Fibonacci based Spiral gets closer and closer to a Golden Spiral as it increases in size. This is because of the ratio of each number in the Fibonacci series in relation to the one before it, and as it converges on Phi, 1.618… Beginning with Zero, then 1, it then moves on to the next number: 0+1=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8, and so on…

The Golden Ratio

The Golden Ratio is also sometimes referred to as ‘The Golden Mean’ or ‘The Golden Section’.  

Take a Line of Length, any line, any length. Divide the line into two unequal lengths. When the length of the shorter line is subtracted from the length of the longer line, it equals the sum of the original Line of Length.

So a Line of Length of 5cm, divided by two unequal lengths, say 3cm and 2cm, when added together equal 5cm, which is the Length of the original line. This is expressed mathematically as:

The Golden Ratio is found in geometry, appearing in basic constructions of an equilateral triangle, a square, and a pentagon, all of which fit inside a circle, as well as in more complex three-dimensional solids such as dodecahedrons, icosahedrons, and Bucky-balls.

Illusion and Confusion

However, the Golden Ratio based spirals, Fibonacci based Spirals, and the true Golden Spirals, are not the same, not at all…

Examples of the Golden Ratio, Fibonacci Based Spirals, and The Golden Spiral, can be found all around us in Nature, Science, Art, and Architecture.

  • Flower Petals – The number of petals on some flowers follows the Fibonacci sequence.
  • Seed Heads – The seeds of a flower can often be found at the centre of the flower head and seen to migrate outward filling in the remaining space. Sunflowers follow this pattern.
  • Pinecones – The spiral pattern of the pinecone spiral upwards in opposite directions. The number of steps the spirals take usually match the numbers in the Fibonacci sequence.
  • Tree branches – The way some tree branches form or split is another example of the Fibonacci sequence. Many root systems also exhibit this formation pattern.
  • Shells – Shells like snail shells and nautilus shells, are perfect examples of the Golden Spiral.
  • Spiral Galaxies – The spiral galaxies like the Milky Way have a number of spiral arms, each of which has a logarithmic spiral of roughly 12 degrees. The shape of these spirals are identical to The Golden Spiral. The Golden Rectangle can always be drawn over any spiral galaxy image.
  • Hurricanes – Most hurricanes reflect The Golden Spiral.
  • Human Body – The measurement of the human belly button to the floor, against the top of the head to the bellybutton follows the Golden Ratio.
  • Animals – We are not the only examples of the Golden Ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants, and honeybees, also exhibit this proportion.
  • DNA Molecules – A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. Although in the Fibonacci series, 34 and 21 are also successive numbers.

Leonardo Da Vinci

In 1509 Luca Pacioli wrote a book that refers to the number as the “Divine Proportion”. It was illustrated by Leonardo da Vinci and Da Vinci later called this ‘Sectio Aurea’ or the ‘Golden Section’.

The Vitruvian Man drawing by Leonardo da Vinci around 1490, is sometimes called the ‘Canon of Proportions’ or less often, ‘Proportions of Man’. Accompanied by notes from the work of the Roman architect Vitruvius in book 111 of his treatise ‘De Architecture’, the drawing is based on the correlations of ‘ideal human proportions’ using the geometry described by Vitruvius.

We think there are more hidden messages behind the Vitruvian Man drawing, but we’ll expand on this under a different subject heading.

Da Vinci also used the Golden Ratio to define all of the proportions in his work ‘The Last Supper’, including the dimensions of the table and the proportions of the walls and backgrounds. However much of the detail is lost to damage, and in particular the area around the feet of the depiction of the man known as Jesus where it is said a doorway was cut through the painting by accident {?} No doubt the original would have revealed further information to us. It was completed on 9 February 1498, alas less than six years later and according to Giorgio Vasari, it had deteriorated to an almost undistinguishable “muddle of blots”…

The Golden Ratio also appears in the ‘Mona Lisa’ painting. Indeed, the Golden Ratio was used in many other Renaissance paintings and sculptures including those by Michelangelo, Raphael, Rembrandt, Seurat, and Salvador Dali.

A True Golden Spiral

In geometry, a True Golden Spiral is a logarithmic spiral whose growth factor is φ, the Golden Ratio. So a True Golden Spiral gets wider (or further from its origin) by a factor of φ 1.618… for every quarter turn it makes.

Imitations and Approximations

There are several similar spirals that approximate, but do not exactly equal, a golden spiral. These are often confused with the True Golden Spiral.

For example, a Golden Spiral can be approximated by first starting with a rectangle where the ratio between its length and width is the Golden Ratio. This rectangle can then be split into a square and a similar rectangle and that rectangle can then be split in the same way. After continuing this in a sequential number of steps, the result will be an almost complete splitting of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, approximates a Golden Spiral.

The Fibonacci Based Spirals

Another approximation is a Fibonacci Spiral. It is constructed similarly to the above except that you start with a rectangle split into two squares and then with each following step, adding it to the rectangle’s longest side – a square of the same length. Because the ratio between consecutive Fibonacci numbers approaches the Golden Ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added.

When the Fibonacci mathematical formula is applied to quantify or measure the movement of Energy or Consciousness within time or space, the Fibonacci Spiral loses its connection of going back to the Zero Point or Source. Instead, the sequence uses the previous number to add into itself to get to the next higher number of the sequence. It has no centre connection, so it attaches to another and consumes what is in its path to expand. 

Consumptive Modelling and Consciousness Suppression

To be clear, the Fibonacci Based Spirals are a deviation from the true intention and organic purpose of a Golden Spiral. It is used in the propagation of artificial intelligence machinery, and these spirals are acquired through forming bonded attachments to the original and divinely created Krystal Spiral.

The Fibonacci based spiral structures are based on Consumptive Modelling. They create Metratronic Bodies – Reversed Merkaba Ratios and this in turn has has major implications for generating Fallen Consciousness. It leads to the disconnection from higher dimensional connections, creates consciousness traps, and cuts off the being from exchange with the true eternal and organic divine source flow. In short, they create false realities.

Fibonacci based spirals illustrate the math used to perpetuate the war over energy and perpetuating the consciousness suppression on planet earth. This is realized in math because when the sequence grows in number or size, it is due to the consumption of the previous values in order to grow itself larger. The larger it gets, the more it consumes, progressively moving out and away from the organic Krystal spiral of the original core manifestation body or divine template body, as it expands.

The Fibonacci based spiral is a mathematical model of Metatronic Reversals recorded in the consciousness fields that essentially attach like a parasite to a living host, collecting waste products, such as accumulating toxic Miasma fields, {Karmic and associated blocked Energy} which form into conscious energy distortions in the Phantom Matrix that most of humanity exists in and which was designed to infect the Universal Tree of Life… yeah, it’s serious.

Look again… it’s very subtle, but it’s there for those who can see.

The Krystal Spiral

The main difference between the true organic Krystal Spiral, and the Fibonacci based spirals and sequences, can be understood in terms of their respective relationships to their creation point. The Krystal Spiral and the mathematical sequence from which it emerges comes directly from the Zero Point or central point of all union. The Krystal Spiral perpetually retains a living breathing connection through the preservation of what came before, as it expands through true Multiplication, not false Consumption

… Nikola Tesla did stellar work on this for us. He knew all about it. This is why he was ridiculed and persecuted and why his work was used elsewhere but kept hidden from us for so long. So, before we go deeper into the Krystal Spiral, and The Universal Time Matrix, in the next blog we look at Sacred Geometry and more of the sacred knowledge pulled by Math Magician Nikola Tesla.