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

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

begot

2 Taiji - Yin and Yang in their infinite interplay

begot

3 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

5 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

8 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...