Saturday 24 February 2018


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


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


Taiji - Yin and Yang in their infinite interplay


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...

Monday 5 February 2018

Yin and Yang

In Chinese philosophy, yin and yang, literally dark and bright, describe how seemingly opposite or contrary forces may actually be complementary, interconnected, and interdependent in the natural world, and how they may give rise to each other as they interrelate to one another.

Yin is characterized as slow, soft, yielding, diffuse, cold, wet, dark, and passive; and is associated with water, winter, north, earth, the moon, femininity, and nighttime.

Yang, by contrast, is fast, hard, solid, focused, hot, dry, bright, and active; and is associated with fire, sky, summer, the sun, masculinity and daytime.

Duality is found in many belief systems, but yin and yang are parts of a Oneness that is also equated with the Tao. The term "dualistic-monism" or dialectical monism has been coined in an attempt to express this fruitful paradox of simultaneous unity/duality. Yin and yang can be thought of as complementary (rather than opposing) forces that interact to form a dynamic system in which the whole is greater than the assembled parts. Everything has both yin and yang aspects. Either of the two major aspects may manifest more strongly in a particular object, depending on the criterion of the observation. This constant balanced interplay of yin and yang is known as "Taiji". The concept of Taiji is first found in written sources in works of Zou Yan (305 – 240 BC) who was an ancient Chinese philosopher best known as the representative thinker of the Yin and Yang School (or School of Naturalists) during the Hundred Schools of Thought era in Chinese philosophy..

Taiji is graphically represented by the symbol known as "Taijitu" or "Taiji tu" meaning Taiji symbol, which is more commonly known simply as Yin - Yang symbol:

Yin is the black side with the white dot in it, and yang is the white side with the black dot in it.

The oldest preserved drawing of this symbol in China appears in the works of the Song era philosopher Zhou Dunyi (1017–1073 AD), author of the "Taijitu shuo" (Explanation of the Diagram of the Taiji), which became the cornerstone of Neo-Confucianist cosmology. His brief text synthesized aspects of Chinese Buddhism and Taoism with metaphysical discussions in the Yijing. 

What is the origin of this symbol?

In a work entitled "A Geomedical Approach to Chinese Medicine: The Origin of the Yin-Yang Symbol" which was published in 2012 by Stefan Jaeger, we can read that:

The Yin-Yang symbol is tightly connected with the annual cycle of the earth around the sun,and the four seasons resulting from it. Using the gnomon , a pole stuck in the ground the ancient Chinese were able to record precisely the position and the length of the sun’s shadow throughout the solar year.

This enabled them to precisely determine the length of the year, which they found to be about 365.25 days. Furthermore, they divided the circle of the year into segments, including the vernal equinox,autumnal equinox, summer solstice, and winter solstice. In addition, they used concentric circles around the pole, helping them to record the length of the sun’s shadow every day. As a result, they measured the shortest shadow during the summer solstice, and measured the longest shadow during the winter solstice. After connecting the measured points and dimming the part that reaches from summer solstice to winter solstice (Yin), they arrived at a chart like this one:

The resemblance between this chart and the modern Taichi-tu or Yin-Yang symbol is striking. This diagram provides visual evidence that the original Yin-Yang symbol describes the change of a pole’s shadow length during a year. This corresponds well with the meaning of "Taiji" in Chinese which is "Great pole"...

Now this is an amazing explanation for the origin of Taiji. Except that there is one problem with it. The above Taiji like shadow diagram shows the shadow change at the 70° latitude north. Latitude represents the angle from the equator. Now this is the globe with some of the latitudes marked on it.

The Arctic Circle lies at the latitude 66° 33´.

So the only way Ancient Chinese astronomers could have gotten the Taiji like shadow diagram would have been if they had their gnomon stuck into the ground somewhere in northern Siberia. Which is highly unlikely to have happened. If they had their gnomon stuck into the ground anywhere in China, the resulting shadow diagram would have looked something like this:

I mean you can see the resemblance to the Taiji symbol but it is definitely far away from this, shadow diagram at the 70° latitude:

Anyway the thing is, the article is right in one thing:  

The Yin-Yang symbol is tightly connected with the annual cycle of the earth around the sun, and constantly changing daily amount of light and darkness and the effect this change has on Earth's temperature.

In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. Earth currently has an axial tilt of about 23.4°.

Earth's axis remains tilted in the same direction with reference to the background stars throughout a year (regardless of where it is in its orbit). This means that one pole (and the associated hemisphere of Earth) will be directed away from the Sun at one side of the orbit, and half an orbit later (half a year later) this pole will be directed towards the Sun. This is the cause of the lengthening of the day from winter solstice to summer solstice and the shortening of the day from summer solstice to winter solstice. 

Now you would think that the earth's temperature would also increase from winter solstice to summer solstice and decrease from summer solstice to winter solstice. But this is not the case. 

This is Sun's sunlight cross. It marks the transitional points on the sunlight cycle in the northern hemisphere:

1. Winter solstice - the shortest day and the longest night
2. Spring equinox - the equal day and night
3. Summer solstice - the longest day and shortest night
4. Autumn equinox - the equal day and night

This is Earth's, climate, vegetation cross as recorded in Celtic and Serbian calendar. It marks the transitional points of the climatic, vegetative cycle in the northern hemisphere:

Celtic calendar:

1. Imbolc- the beginning of the spring
2. Bealtaine - the beginning of the summer
3. Lughnasa - the beginning of the autumn
4. Samhain - the beginning of the winter

Serbian calendar:

1. St Sava - the beginning of the spring
2. St George - the beginning of the summer
3. St Ilija - the beginning of the autumn
4. St Mitar - the beginning of the winter

As you can see the sun cross and earth cross are out of sync. The earth cross is rotated forward by 45 degrees and the earth circle cardinal points fall right in between the sun circle cardinal points. This is because the earth climatic, vegetative cycle lags behind the solar cycle. 

I talked about this in detail in my post "Two crosses". At the end of that post I drew this diagram showing transformation of Sun's light cycle into Earth's temperature, climatic, vegetative cycle. 

The Sun cross transitions into the Earth cross. This transition is governed by the slow accumulation and release of the heat which is transferred from the Sun to the Earth through sunlight...

This is the never ending wheel of life...

And then two days ago while I was having a discussion about Celtic calendar, I had people say to me:

"How can 1st of February, Imbolc, be the first day of spring? I have foot and a half of snow on the ground all around me and its freezing. This makes no sense..."

I replied that the first day of spring is the first day of spring, not because it is warm, but because it can't get any colder. The 1st of February, actually the 4th of February, the real Imbolc, the mid point between the Winter Solstice and the Spring Equinox, is the coldest part of the year on the northern hemisphere.  This is due to the way the Sun - Earth system works. So even though on the day of the Winter Solstice the days start getting longer, they continue to get colder. Until the 4th of February, the mid point between the Winter Solstice and the Spring Equinox. From that day on, the Earth starts getting warmer.

It is exactly the same on the opposite side of the year. The first day of autumn is the first day of autumn, not because it is cool, but because it can't get any hotter. The 1st of August, actually the 2nd of August, the real Lughnasadh, the mid point between the Summer Solstice and the Autumn Equinox, is the hottest part of the year on the northern hemisphere. This is again due to the way the Sun - Earth system works. So even though on the day of the Sumer Solstice the days start getting shorter, they continue to get wormer. Until the 2nd of August, the mid point between the Summer Solstice and the Autumn Equinox. From that day on, the Earth starts getting cooler.

As I was writing this, something clicked. What I just wrote sounded very familiar. Very very familiar.     

And then something clicked. 

The Yin - Yang diagram.

The winter solstice, the beginning of the lengthening of the day, is the white dot in the middle of the black part of the diagram. It is the seed of light in the middle of the darkest part of the year. But even though the days start lengthening, the weather gets colder until the mid point between the winter solstice and spring equinox. Imbolc, the beginning of the spring and the time of maximum cold...

The summer solstice, the beginning of the shortening of the day, is the black dot in the middle of the white part of the diagram. It is the seed of darkness in the middle of the whitest (lightest, brightest) part of the year. But even though the days start shortening, the weather gets wormer until the mid point between the summer solstice and autumn equinox. Lughnasadh, the beginning of the autumn and the time of maximum heat...

This endless cyclical change of the duration of the day and night is what is seen as the endless cyclical interplay between the light (Yang) and darkness (Yin) which heats (Yang) and cools (Yin) the Earth and creates seasons and makes life possible.

This endless cyclical change is possible because of endless cyclical interplay between Father Sky (Sun) and Mother Earth which is 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
Triglav - three headed male god. He consisted of Jarilo (sun as a young man), Vid (sun as an adult, husband), Perun (sun as an old man, grandfather).

His characteristics were light, heat, dryness. He dominated the white (light, bright) part of the year and was most powerful in the middle of this period, at the end of the summer and the beginning of autumn which is the hottest and driest period of the year.

Mother Earth:

Dajbaba - giving goddess
Baba - grandmother, female ancestor
Troglava - three headed female goddess. She consisted of Vesna (earth as young girl), Mokoš (earth as an adult woman, wife), Morana (earth as an old woman, witch).

Her characteristics were darkness, cold, wetness. She dominated the dark part of the year and was most powerful in the middle of this period, at the end of the winter and the beginning of spring which is the coldest and wettest part of the year.

At their maximum power, both Father Sky (Sun) and Mother Earth are destructive. The Father Sky (Sun) on his own would turn the world int a dead hot dry desert (pure Yang). The Mother Earth on her own would turn the world into a dead cold wet desert (pure Yin). 

It is the eternal dynamic interplay (Taiji) between the Sky (Sun) - (light, heat, dryness) Yang and the Earth - (darkness, cold, wetness) Yin, the intercourse between the Father Sky (Sun) and Mother Earth, which makes life possible. And this interplay is perfectly depicted by the Taijitu or Yin - Yang diagram.

But is this all just a coincidence? I don't know, but I know that the Yin - Yang (darkness - light) symbol perfectly depicts only one process that I know of: This endless cyclical change of the duration of the day and night which causes endless cyclical change of the heating (Yang) and cooling (Yin) of the Earth. I don't really know any other system which behaves in the way described by the Yin - Yang symbol. Do you? If you do please let me know what that process is.

Very interesting right? 

Well here is something even more interesting. These are two granary "shrines" from Old European Cucuteni–Trypillia culture which thrived between 5200 BC and 3500 BC in Eastern Europe covering the region from the Carpathian Mountains to the Dniester and Dnieper regions, centred on modern-day Moldova and covering substantial parts of western Ukraine and northeastern Romania.

And this is the decoration engraved on the wall of one of these granary "shrines":

Is this the Yin - Yang symbol? And if it is, do you think the ancient Tripillian's tried to depict the same process that I just described? After all, granaries are only full because of the intercourse of the  Mother Earth and Father Sky (Sun), Yin and Yang...And if the Tripillians already invented the Yin - Yang symbol during European Neolithic, is it possible that it was somehow transferred to the East over the following millenniums?

Or maybe this is just a similar looking symbol which has nothing really to do with Yin - Yang symbol?