Showing posts with label Fibonacci. Show all posts
Showing posts with label Fibonacci. Show all posts

Sunday, 27 May 2018

Square and Compasses

"For in God we live and move and have our being..." - Acts 17:28

When I was 9 years old, I got in trouble at school for drawing a "rude" picture on a piece of paper and passing it around the classroom. This is the "rude" picture :)



For those who have problem figuring out what the picture means: it is a stylised depiction of a couple having a sexual intercourse in a "missionary" position (a man on top). 

This was a time when I was just finding out that there is such a thing called "sex" and was finding it all very funny.

Anyway, time passed, the sex became less funny and more fun and I completely forgot about this incident...

Then recently, while I was looking for something on the internet, I came across a picture of the Masonic Square and Compasses symbol:


The Square and Compasses is the single most identifiable symbol of Freemasonry. This symbol is a mystery. No one knows where it came from or what it means.

Most Freemasons, when asked the meaning of their Square & Compasses logo, state:

"Both…are architect’s tools…to teach symbolic lessons…"

I had of course seen this symbol before and I generally accepted that the symbol is probably in some way related to the idea of the God being the Great architect of the world.

If you look at this picture you will see why:




You can see that the ancient Hebrews saw the universe as a stone built structure with the stone flat Earth as the base and the stone Heavenly dome as the roof. Inside of this structure was all there was, the life itself. 

Now this was very interesting. The origin of the Masons is deeply rooted in the religion of the First Temple. So they probably shared the same world view described in the above drawing. And if we look at the above picture the idea of God being the Great Architect starts making a lot of sense. Only a great architect could have built the stone universe...

That architect would have used the Compasses to draw the top of the universe, the great stone semi circular dome of Heavens. He would also have used the Square to draw the bottom of the universe, the great stone flat earth. 

And this is exactly how the Compasses and Square are positioned in the Masonic symbol: the Compasses at the top, representing the Heavens, the Square at the bottom, representing the Earth. 

It seams that at least some Masons shared this view. 

Albert Pike, in his "Morals & Dogma" says: "The Compass, as the Symbol of the Heavens…and the Square, as the Symbol of the Earth...". 

Albert Pike actually goes further to say that the Compass (Heaven) represents the spiritual portion of the double nature of Humanity…and the Square (Earth) represents the material, sensual, and baser portion of the double nature of Humanity." 

J. S. Ward states in his "Interpretation of Our Masonic Symbols" that "…the compasses stand for…the spiritual side of man, while the square appertains to the material world…"

This is linked to the Euclid’s 47th problem of “Squaring The Circle,” said to be the primary goal of the Masonic craft. 



Squaring the circle, however, does not in this case refer to a mathematical problem. It is a description of the the fact that man has both physical and spiritual nature.

Now the reason why man has both physical and spiritual natures is because we are all children of the father sky and mother earth.


Sky, because we can't touch it, but because it has direct influence on life, is perceived as spiritual. The word spirit comes from breath, air...
Earth, because we can touch it, and because we and everything else that is alive, lives and grows out of earth, is seen as material. The word matter comes from mother...

In Genesis, God forms "Adam", out of "the dust of the ground" and "breathed into his nostrils the breath of life" (Genesis 2:7). This is basically a description of the creation of life by Father Sky and Mother Earth...

This reminded me of a Serbian riddle

Question: Tall (high) father, wide mother
Answer: sky and earth

This riddle contains the ancient belief in Father Sky and Mother Earth which was preserved in Serbian tradition.

Father Sky and Mother Earth are at the core of the old European belief system.

In Serbian tradition the Father Sky and Mother Earth are known as:

Father Sky:

Dajbog - giving god
Djed - grandfather, male ancestor

Mother Earth:

Dajbaba - giving goddess
Baba - grandmother, female ancestor

It is the dynamic interplay between the Sky and the Earth, the intercourse between the Father Sky and Mother Earth, which creates all life.

Which brings me back to my primary school doodle:


A father (on top) and mother (on the bottom) creating life...Compass and the Square. Heaven (Sky) father and Earth mother creating life.

In English speaking countries, the Compass and the Square symbol has a capital letter G in the space between the Compass and the Square.



Among the most widely accepted interpretations are that: [G] stands for God, and is to remind Masons that God is at the center of Freemasonry. In this context it can also stand for Great Architect of the Universe (a reference to God). 

So there is one GOD who creates all. The first thing he created was chaos. He had to. He created everything :) Then he "separated" the chaos into the sky and earth. Sky and earth then created living nature. Living nature then created all living things including man.

This is interesting. God Begets One. One begets Two. Two beget Three. Three beget all living things.

Now where did I see this before?

I know. In Taoism.

The Tao begot one (wuji, chaos).
One begot two (taiji, Heaven and Earth, Yang and Yin).
Two begot three (man heaven earth, life).
And three begot the ten thousand things (All things).

And in Fibonacci sequence. 1, 1, 2, 3... I wrote about the Fibonacci sequence and it being at the core of Taoism in the post entitled "Fibonacci".

Fibonacci sequence describes Genesis, the creation of life, the materialisation of the divine, the squaring of the circle...

In case of Taoism behind everything is unknowable Tao. It is unknowable because it is beyond material universe (chaos) which is all we can ever really know. 
In case of Judeo Christianity behind everything is unknowable GOD. It is unknowable because it is beyond material universe (chaos) which is all we can ever really know.
In case of Science, behind everything is unknowable "something" that kicked off the Big Bang. It is unknowable because it is beyond material universe (chaos) which is all we can ever really know.

There is basically no difference between these three definitions of the beginning of the creation of life. They are all beliefs based on logical deduction that there must have been something or someone that created our universe (chaos) from which (or in which) sky and earth were formed (created) and which in turn created life and us, humans. But we can never have a proof that this belief is true or any way to see this something or someone. 

So from our point of view the only thing that really matters are the two things that directly create, maintain and destroy life: sky and earth. It is because of this that sky and earth were recognised as Father Sky and Mother Earth and were as such deified and venerated as God Father and Mother Goddess.

Sky father was often identified with the Sun. This is why Sky gods are often interchangeable with Sun gods. For instance in Slavic religion Dabog, the Sun god is also the giver of rain hence he is actually the Sky god. Dabog is imagined as a radiant youth who rides across the sky every day from east to west in golden chariots pulled by four white horses. 

There are very strong indications, based on archaeological and historical data, that the First Temple, the Temple of Solomon, the Temple Masons are trying to rebuild, was a solar temple, built on top of an ancient solar observatory, and built by the followers of a solar cult. More precisely the cult of Helios (El). Helios who was the old sun god imagined as a radiant youth driving across the sky from east to west in golden chariot pulled by four white horses...I wrote about this in my posts "Boaz and Jachin" and "Sun god from the first temple". The solar cult of the First Temple was only replaced by the lunar cult of the Second Temple after the return from Babylon. Not all jews agreed with this religious change. Some, like the Essenes, regarded the cult of the Second Temple as a heresy and dedicated themselves to the "rebuilding of the First Temple solar cult"... Now the God of the Jews, El, Helios, Sky Father, was the God of the Early Christians...

There are also very strong indications, based on ethnographic and archaeological evidence, that Virgin Mary is the Christianised Mother Earth. The old holy places related to the Mother Earth worship were converted to holy places dedicated to Virgin Mary (or the main Female Saint of the area). The old rituals performed to appease Mother Earth were transferred to Virgin Mary (or the main Female Saint of the area). I wrote about this in many posts, such as "Shepherd's chapels from Velebit", "Babje leto - Grandmother's summer"...

What does the intercourse between the Father Sky and Mother Earth produce? Life. Living nature including us humans...
What did the intercourse between God and Mary produce? Christ who is God who became Man. God materialised. 

So based on the above analogy, Christ is just a symbol. Symbol of life itself, symbol of nature. Life is materialised divine, god who became matter.  

Squaring (materialising) the circle (Devine, sun, sky, God) = creating life...

If Christ is life, living nature and we are part of life, living nature, then we are Christ. We truly live and move and have our being in Christ...
If Christ is life, living nature, god who became matter, then we are all god materialised. Just like the mystics the world over have been teaching for millennia.

I wonder how many masons and non masons are aware of all this?

What do you think? Does this make sense?

PS:

Chinese creator god Fuxi 伏羲 and mother goddess Nüwa 女娲 in the process of creating life. They are holding square and compasses. Astana (Xinjiang), Tang époque (618-907). 


What an incredible coincidence... :) 

Saturday, 24 February 2018

Fibonacci

Leonardo of Pisa, known as Fibonacci, was the 13th century Italian mathematician. In his 1202 book Liber Abaci he popularized the Hindu–Arabic numeral system in the Western World.

He also stumbled across a very interesting sequence of numbers while contemplating a curious problem involving rabbits. Fibonacci started with a pair of fictional and slightly unbelievable baby rabbits, a baby boy rabbit and a baby girl rabbit. They were fully grown after one month. and did what rabbits do best, so that the next month two more baby rabbits (again a boy and a girl) were born. The next month these babies were fully grown and the first pair had two more baby rabbits (again, handily a boy and a girl). Ignoring problems of in-breeding, the next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature. Fibonacci asked how many rabbits a single pair can produce after a year with this highly unbelievable breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month).


He realised that the number of adult pairs in a given month is the total number of rabbits (both adults and babies) in the previous month. He carried the calculation up to the 13th place and ended up with this sequence of numbers, which was after him named Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

Dividing each Fibonacci number by the previous Fibonacci number produces an interesting sequence of ratios:

1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
5/3 = 1.666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...

The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number usually represented by the symbol ϕ (Phi) which is an irrational (endless random decimal number) with the value 1.6180339887…

As a matter of fact, the golden ration is the most irrational number of all.

If we plot Fibonacci numbers using squares, and draw connected arches with radius equal to the size of the consecutive squares we get Fibonacci spiral.

  
This spiral is closely related to another spiral, called the golden spiral. The golden spiral is created by plotting the equation r=e90⋅θ⋅ln(φ). This spiral grows by a factor of φ every quarter rotation. That is, if the distance from the center of the spiral to the edge of the spiral is 1 unit at some angle, then 90 degrees farther along the spiral, the distance from the center to the edge will be about 1.618 units.


If we look at these two spirals we will notice that they look very similar. The Fibonacci spiral is made up of quarter circles which grow in relation to the Fibonacci sequence, while the golden spiral grows at a constantly increasing rate. It seems odd that they should look so similar. Yet it turns out the the Fibonacci spiral is a very good approximation of the golden spiral. Which you can see on this graph showing both spirals overlayed on top of each other:



The golden ratio value ϕ = 1.6180339887... and its reciprocal value φ = 1/ϕ = 1/1.6180339887…= 0.6180339887... and the Fibonacci number sequence which converges to the Golden ratio number appear very often in nature where they are linked to the growth processes in living organisms. Golden ratio has also been linked to beauty, elegance and perfection and is found embedded in dimensions of many ancient monuments.

Now in the history of mathematics we can read that Fibonacci was not the first person to describe the number sequence which is today named after him. Apparently part of the same number sequence was described much earlier by Indian mathematicians in connection with Sanskrit prosody. It seems that the writers of Vedas used some of the Fibonacci numbers as a bases for the meter of the Vedas.

The clearest exposition of the sequence is found in the work of the Indian prosodist and mathematician Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135 AD):

Variations of two earlier meters [is the variation]... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].

This means that Indian scholars described the Fibonacci sequence 500 years before Fibonacci. Some Indian scholars, like Susantha Goonatilake and Parmanand Singh even claim that the first mention of the Fibonacci sequence should be attributed to the Iron Age prosodist Pingala, the author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody, who lived in the 2nd - 3rd century BC. This would push the discovery of the Fibonacci numbers by the Indian scholars to 1500 before Fibonacci. However this claim that Pingala was the first to describe Fibonacci number sequence is based on the cryptic formula "misrau cha" (the two are mixed) found in Pingala sutras and the claim that it is somehow related to the much later Virahanka's description of the Fibonacci meters. But this is a great stretch and I am not sure if this is indeed the case...

The thing is there is another description of the Fibonacci sequence which is even older than the purported Pingala's one. And this description of the Fibonacci sequence is not cryptic or obscure. In fact it is at the foundation of Taoism. And yet it seems to be invisible to people researching the history of mathematics.

The description of the Fibonacci sequence is found in the chapter 42 of the Tao Te Ching.

The Tao begot one.
One begot two.
Two begot three.
And three begot the ten thousand things (All things).

The ten thousand things carry yin and embrace yang.
They achieve harmony by combining these forces.

The Tao Te Ching or Dao De Jing, is a Chinese classic text traditionally credited to the 6th-century BCE sage Lao Tzu (Old Master). The oldest excavated portion dates back to the late 4th century BCE. The Tao Te Ching, along with the Zhuangzi, is a fundamental text for both philosophical and religious Taoism.

Generations of scholars have debated the historicity of Laozi and the dating of the Tao Te Ching. But the linguistic studies of the text's vocabulary and rhyme scheme point to a date of composition after the Shi Jing (The Classic of Poetry, dating from the 11th to 7th centuries BC) yet before the Zhuangzi (an ancient Chinese text from the late Warring States period (476–221 BC) which contains stories and anecdotes that exemplify the carefree nature of the ideal Daoist sage).

So here we have the beginning of the Fibonacci sequence:

1. Tao - The unconditional and unknowable source and guiding principle of all reality

begot

Wuji - Ultimate, boundless, infinite, undivided, primordial universe

begot

Taiji - Yin and Yang in their infinite interplay

begot

Tian-Ren-Di - Heaven, Man (Life, Chi (Life force)), Earth ( Chapter 25 Tao Te Ching)

begot all things which carry Yin and embrace Yang and achieve harmony by combining these forces. This combining of Yin and Yang in living things born by Heaven and Earth goes through

Wu Xing The Five Phases, also known as the Five Movements, the Five Processes, the Five Steps/Stages each dominated by a different one of the Five Elements, the Five Agents and the Five Planets. During each of these five phases "a different one of the five types of chi is dominat".

and this endless recombination, change of Yin and Yang is described by the

Bagua - eight divinatory symbols, which were first mentioned in the Taoist classic I Ching dated to the Western Zhou period (1000–750 BC). Bagua is used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each symbol consists of three lines, each line either "broken" or "unbroken," respectively representing yin or yang. Due to their tripartite structure, they are often referred to as "trigrams" in English.


1, 1, 2, 3, 5, 8...All things

The Fibonacci sequence which is in biology linked with the growth of the living things, in Taoism describes the creation and the endless change and evolution of life...

Considering that the classics which talk about this "sequence" date to "some time between the 11th century BC and the 5th century BC", this is by far the earliest description of the Fibonacci sequence. I  think it's time to change the History of Mathematics books...