published on 27 June 2016 in life
The Fibonacci Sequence has always attracted the attention of people since, as well as having special mathematical properties, other numbers so ubiquitous as those of Fibonacci do not exist anywhere else in mathematics: they appear in geometry, algebra, number theory, in many other fields of mathematics and even in nature! Let’s find out together what it is …
The life of Fibonacci
Leonardo Pisano, called Fibonacci (Fibonacci stands for filius Bonacii) was born in Pisa around 1170.
His father, Guglielmo dei Bonacci, a wealthy Pisan merchant and representative of the merchants of the Republic of Pisa in the area of Bugia in Cabilia (in modern north-eastern Algeria), after 1192 took his son with him, because he wanted Leonardo to become a merchant.
He thus got Leonardo to study, under the guidance of a Muslim teacher, who guided him in learning calculation techniques, especially those concerning Indo-Arabic numbers, which had not yet been introduced in Europe.
Fibonacci’s eduction started in Bejaia and continued also in Egypt, Syria and Greece, places he visited with his father along the trade routes, before returning permanently to Pisa starting from around 1200.
For the next 25 years, Fibonacci dedicated himself to writing mathematical manuscripts: of these, Liber Abaci (1202), thanks to which Europe became aware of Indo-Arabic numbers, Practica Geometriae (1220), Flos (1225) and Liber Quadratorum (1225) are today known to us.
Leonardo’s reputation as a mathematician became so great that Emperor Federico II asked an audience while in Pisa in 1225. After 1228, not much is known of Leonardo’s life, except that he was awarded the title of”Discretus et sapiens magister Leonardo Bigollo” in recognition of the great progress he made to mathematics. Fibonacci died sometime after 1240, presumably in Pisa.
The rabbits of Fibonacci and the famous sequence
Liber Abaci, in addition to referring to Indo-Arabic numbers, which subsequently took the place Roman numerals, also included a large collection of problems addressed to merchants, concerning product prices, calculation of business profit, currency conversion into the various coins in use in the Mediterranean states, as well as other problems of Chinese origin. Alongside these commercial problems were others, much more famous, which also had a great influence on later authors. Among them, the most famous, source of inspiration for many mathematicians of later centuries, is the following: “How many pairs of rabbits will be born in a year, starting from a single pair, if each month each pair gives birth to a new pair which becomes reproductive from the second month?”. The solution to this problem is the famous “Fibonacci sequence”: 0, 1, 1, 2, 3, 5, 8, 13, 21,34,55,89… a sequence of numbers in which each member is the sum of the previous two.
An important characteristic of the sequence is the fact that the ratio between any number and the previous one in the series tends towards a well-defined value: 1.
618… This is the golden ratio or golden section, φ (Phi), that frequently occurs in nature (to know more about: The perfection of the snail).
When Fibonacci illustrated this sequence, as a solution to a “recreational mathematics” problem, he did not give it particular importance.
Only in 1877 the mathematician Édouard Lucas published a number of important studies on this sequence, which he claimed to have found in Liber Abaci and which, in the honour of the author, he called “Fibonacci sequence”.
Studies subsequently multiplied, and numerous and unexpected properties of this sequence were discovered, so much so that since 1963, a journal exclusively dedicated to it, “The Fibonacci quarterly”, has been published.
The Fibonacci sequence in nature
Observing the geometry of plants, flowers or fruit, it is easy to recognize the presence of recurrent structures and forms.
The Fibonacci sequence, for example, plays a vital role in phyllotaxis, which studies the arrangement of leaves, branches, flowers or seeds in plants, with the main aim of highlighting the existence of regular patterns.
The various arrangements of natural elements follow surprising mathematical regularities: D’arcy Thompson observed that the plant kingdom has a curious preference for particular numbers and for certain spiral geometries, and that these numbers and geometries are closely related.
We can easily find the numbers of the Fibonacci sequence in the spirals formed by individual flowers in the composite inflorescences of daisies, sunflowers, cauliflowers and broccoli.
In the sunflower, individual flowers are arranged along curved lines which rotate clockwise and counterclockwise. Credits: The Fibonacci sequence in phyllotaxis – Laura Resta (Degree Thesis in biomathematics)
It was Kepler who noted that on many types of trees the leaves are aligned in a pattern that includes two Fibonacci numbers. Starting from any leaf, after one, two, three or five turns of the spiral there is always a leaf aligned with the first and, depending on the species, this will be the second, the third, the fifth, the eighth or the thirteenth leaf.
Arrangement of leaves on a stem. Credits: The Fibonacci sequence in phyllotaxis – Laura Resta (Degree Thesis in biomathematics)
What Is The Fibonacci Sequence? And How It Applies To Agile Development
- The Fibonacci sequence is a series of numbers where a number is the addition of the last two numbers, starting with 0, and 1.
- The Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
- Written as a rule, the expression is:
- Xn = Xn-1 + Xn-2
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The Fibonacci Spiral And The Golden Ratio
The Fibonacci sequence is often visualized in a graph such as the one in the header of this article. Each of the squares illustrates the area of the next number in the sequence. The Fibonacci spiral is then drawn inside the squares by connecting the corners of the boxes.
The squares fit together perfectly because the ratio between the numbers in the Fibonacci sequence is very close to the golden ratio , which is approximately 1.618034. The larger the numbers in the Fibonacci sequence, the closer the ratio is to the golden ratio.
The spiral and resulting rectangle are also known as the Golden Rectangle .
The Origins Of The Fibonacci Sequence
Fibbonaci (Leanardo Pisano Bogollo , Fibonacci was his nickname) first introduced the series of numbers known as the Fibonacci sequence in his book Liver Abaci  in 1202. Fibonacci was a member of an important Italian trading family in the 12th and 13th century.
Being part of a trading family, mathematics was an integral part of the business. Fibonacci traveled throughout the Middle East and India and was captivated by the mathematical ideas from his travels.
His book, Liver Abaci, was a discourse on the mathematical methods in commerce that Fibonacci observed during his travels.
Fibonacci is remembered for two important contributions to Western mathematics:
- He helped spread the use of Hindu systems of writing numbers in Europe (0,1,2,3,4,5 in place of Roman numerals).
- The seemingly insignificant series of numbers later named the Fibonacci Sequence after him.
Fibonacci discovered the sequence by posing the following question:
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 rabbits begin to bear young two months after their birth?
- Start: At the start no rabbits are born, as the initial pair has not had time to be pregnant and born (0).
- The first month: One pair of rabbits are born (1).
- The second month: Again, one pair of rabbits are born as the new rabbits have not yet matured to bear young (1).
- The third month: Two pairs of rabbits reproduce, and one pair is not ready, so two pairs of rabbits are born (2).
- The fourth month: Three pairs of rabbits reproduce and 2 pairs of rabbits are not ready, so three pairs of rabbits are born (3).
- The fifth month: Five pairs of rabbits produce and three are not ready, so five pairs of rabbits are born (5).
- And so on.
Though Fibonacci’s question concerning the rabbits is an unrealistic scenario, the sequence can be observed in nature, such as in the array of sunflower seeds and other plants, and the shape of galaxies and hurricanes.
Sunflower Seeds are a dramatic demonstration the Fibonacci Sequence in nature.
The Importance of the Fibonacci Sequence
While this series of numbers from this simple brain teaser may seem inconsequential, it has been rediscovered in an astonishing variety of forms, from branches of advanced mathematics  to applications in computer science , statistics , nature , and agile development.
How Τhe Fibonacci Sequence Ιs Used Ιn Agile Development
Now you may be saying to yourself “That’s nice, but what does it have to do with Agile Development?”, and that’s a great question. What does the Fibonacci Sequence have to do with Agile Development? Interestingly, the Fibonacci’s Sequence is a useful tool for estimating the time to complete tasks.
Estimating Tasks In Agile
A big part of managing an Agile team is estimating the time tasks will take to complete. A points system is often used to give a high-level estimate of the scale or size of a specific task. Bigger more complex tasks get more points and smaller tasks get fewer points. Managers can then review and prioritize tasks based upon the assigned scale.
Using The Fibonacci Sequence With Your Team
To use the Fibonacci Sequence, instruct your team to score tasks from the Fibonacci Sequence up to 21.
1, 2, 3, 5, 8, 13, 21
One being the smallest easiest tasks and twenty-one being large projects. As far as why you should use the Fibonacci sequence instead of, say, t-shirt sizes, check back for my next article on 5 Reasons Using the Fibonacci Sequence Will Make You Better at Estimating Tasks in Agile Development.
And for more on leading an Agile eLearning Development team, check out our sweet eBook guide, The Agile Guide to Agile Development.
- 1. 5 Reasons Using The Fibonacci Sequence Makes You Better At Agile Development
- 2. 8 Components And Uses Of Burndown Charts In Agile Development
- 3. Free eBook: The Agile Guide To Agile Development
What is the Fibonacci Sequence (aka Fibonacci Series)? – The Golden Ratio: Phi, 1.618
In the 1202 AD, Leonardo Fibonacci wrote in his book “Liber Abaci” of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi.
This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa.
He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning “son of (the) Bonacci”.
Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .
This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze. Click to enlarge.
The relationship of the Fibonacci sequence to the golden ratio is this: The ratio of each successive pair of numbers in the sequence approximates Phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60.
The table below shows how the ratios of the successive numbers in the Fibonacci sequence quickly converge on Phi. After the 40th number in the sequence, the ratio is accurate to 15 decimal places.
1.618033988749895 . .
Compute any number in the Fibonacci Sequence easily!
- Here are two ways you can use phi to compute the nth number in the Fibonacci sequence (fn).
- If you consider 0 in the Fibonacci sequence to correspond to n = 0, use this formula:
- fn = Phi n / 5½
- Perhaps a better way is to consider 0 in the Fibonacci sequence to correspond to the 1st Fibonacci number where n = 1 for 0. Then you can use this formula, discovered and contributed by Jordan Malachi Dant in April 2005:
- fn = Phi n / (Phi + 2)
- Both approaches represent limits which always round to the correct Fibonacci number and approach the actual Fibonacci number as n increases.
The ratio of successive Fibonacci numbers converges on phi
of the two
|Ratio of each
number to the
one before it
Tawfik Mohammed notes that 13, considered by some to be an unlucky number, is found at position number 7, the lucky number!
The Fibonacci Sequence and Gambling or Lotteries
In the Fibonacci system the bets stay lower then a Martingale Progression, which doubles up every time. The downside is that in the Fibonacci roulette system the bet does not cover all of the losses in a bad streak.
An important caution: Betting systems do not alter the fundamental odds of a game, which are always in favor of the casino or the lottery. They may just be useful in making the playing of bets more methodical, as illustrated in the example below:
|Round||Scenario 1||Scenario 2||Scenario 3|
|Bet 1||Bet 1 and lose||Bet 1 and lose||Bet 1 and win|
|Bet 2||Bet 1 and lose||Bet 1 and lose||Bet 1 and win|
|Bet 3||Bet 2 and win||Bet 2 and lose||Bet 1 and lose|
|Bet 4||–||Bet 3 and win||Bet 1 and lose|
|Bet 5||–||–||Bet 2 and win|
|Net Result||Even at 0||Down by 1||Ahead by 2|
The life and numbers of Fibonacci
R.Knott and the Plus team For a brief introduction to the Fibonacci sequence, see here.
Fibonacci is one of the most famous names in mathematics. This would come as a surprise to Leonardo Pisano, the mathematician we now know by that name.
And he might have been equally surprised that he has been immortalised in the famous sequence – 0, 1, 1, 2, 3, 5, 8, 13, …
– rather than for what is considered his far greater mathematical achievement – helping to popularise our modern number system in the Latin-speaking world.
The Roman Empire left Europe with the Roman numeral system which we
still see, amongst other places, in the copyright notices after films
and TV programmes (2013 is MMXIII).
The Roman numerals were not
displaced until the mid 13th Century AD, and Leonardo Pisano's book, Liber Abaci (which means “The Book of Calculations”), was one of the first Western books to describe their eventual replacement.
Leonardo Fibonacci c1175-1250.
Leonardo Pisano was born late in the twelfth century in Pisa, Italy: Pisano in Italian indicated that he was from Pisa, in the same way Mancunian indicates that I am from Manchester.
His father was a merchant called Guglielmo Bonaccio and it's because of his father's name that Leonardo Pisano became known as Fibonacci.
Centuries later, when
scholars were studying the hand written copies of Liber Abaci
(as it was published before printing was invented), they
misinterpreted part of the title – “filius Bonacci” meaning “son
of Bonaccio” – as his surname, and Fibonacci was born.
Fibonacci (as we'll carry on calling him) spent his childhood in North
Africa where his father was a customs officer.
He was educated by the
Moors and travelled widely in Barbary (Algeria), and was later sent
on business trips to Egypt, Syria, Greece, Sicily and Provence.
In 1200 he returned to Pisa and used the knowledge he had gained on
his travels to write Liber Abaci (published in 1202) in which he introduced the Latin-speaking world to the decimal number system. The first chapter of Part 1 begins:
“These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.”
Italy at the time was made up of small independent towns and regions and this led to use of many kinds of weights and money systems. Merchants had to convert from one to another whenever they traded between these systems.
Fibonacci wrote Liber Abaci for these merchants, filled with practical problems and worked examples demonstrating how simply commercial and
mathematical calculations could be done with this new number system
compared to the unwieldy Roman numerals.
The impact of Fibonacci's book as the
beginning of the spread of decimal numbers was his greatest
mathematical achievement. However, Fibonacci is better remembered for
a certain sequence of numbers that appeared as an example in Liber
The problem with rabbits
One of the mathematical problems Fibonacci investigated in Liber Abaci was about how fast rabbits could breed in ideal circumstances.
Suppose a newly-born pair of rabbits, one male, one female, are put in a field.
Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on.
The puzzle that Fibonacci posed was… How many pairs will there be in one year?
- At the end of the first month, they mate, but there is still only 1 pair.
- At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits.
- At the end of the third month, the original female produces a second pair, making 3 pairs in all.
- At the end of the fourth month, the original female has produced
yet another new pair, the female born two months ago produced her
first pair also, making 5 pairs.
Now imagine that there are pairs of rabbits after months. The number of pairs in month will be (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be new pairs. So we have
which is simply the rule for generating the Fibonacci numbers: add the last two to get the next. Following this through you'll find that after 12 months (or 1 year), there will be 233 pairs of rabbits.
Bees are better
What Is The Fibonacci Sequence? Why Is It So Famous? » GetsetflySCIENCE
Well I have a question for all nature lovers, admirers that while observing some flowers, leaves you must have been awestruck by the pattern of it and I am sure no one would have ever known (of course if you already knew about Fibonacci sequence then that’s different) that such pattern actually has a name and even some possible explanation. Such beautiful patterns are what we are going to discuss at length and I am sure you will just be amazed to find about it more and more.
It is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written as a rule, the expression is xn = xn-1 + xn-2.
For example, 0+1=1, 1+1=2, 2+1=3, 3+2=5 and so on………
So, simply it means it is a series of number that follows a unique integer sequence. These numbers generate mathematical patterns that can be found in all aspects of life. The pattern can be seen from the human body to the physiology of plants and animals.
Leonardo Pisano Bigollo(Fibonacci)
It is named after Fibonacci, also known as Leonardo of Pisa or Leonardo Pisano, Fibonacci numbers were first introduced in his Liber abaci in 1202. He was a son of a Pisan merchant, Fibonacci travelled widely. Math was very important to those in the trading business, and his passion for numbers was cultivated in his youth.Knowledge of numbers is said to have first originated in the Hindu-Arabic arithmetic system, which Fibonacci studied while growing up. He wrote many books about geometry, commercial arithmetic and irrational numbers. He also helped develop the concept of zero.
How does Fibonacci sequence work?
The Fibonacci sequence is derived from Fibonacci numbers. The numbers are as follows 0,1,1,2,3,5,8….and so on.
These numbers are obtained by adding two previous numbers in the sequence to obtain the next higher number. The formula is Fn =Fn-1 +Fn-2.
Every third number is even and the difference between each number is 0.618 with reciprocal of 1. 618.These numbers are known as golden ratio and golden means.
What is the Fibonacci Sequence – and why is it the secret to musical greatness?
22 November 2018, 16:21 | Updated: 28 January 2019, 09:23
The Fibonacci Sequence appears frequently in music and art. Picture: Getty Images / Classic FM
Geniuses from Mozart to Leonardo da Vinci have used the Fibonacci Sequence. But what is it and why does it make great music?
The Fibonacci Sequence has been nicknamed ‘nature’s code’, ‘the divine proportion’, ‘the golden ratio’, ‘Fibonacci’s Spiral’ amongst others.
What exactly is the Fibonacci Sequence?
Simply put, it’s a series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…
The next number in the sequence is found by adding up the two numbers before it. The ratio for this sequence is 1.618. This is what some people call ‘The Divine Proportion’ or ‘The Golden Ratio’.
Fibonacci Numbers and Lines Definition and Uses
Fibonacci numbers are used to create technical indicators using a mathematical sequence developed by the Italian mathematician, commonly referred to as “Fibonacci,” in the 13th century.
The sequence of numbers, starting with zero and one, is created by adding the previous two numbers.
For example, the early part of the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, 377, and so on.
This sequence can then be broken down into ratios which some believe provide clues as to where a given financial market will move to.
The Fibonacci sequence is significant because of the so-called golden ratio of 1.618, or its inverse 0.618. In the Fibonacci sequence, any given number is approximately 1.618 times the preceding number, ignoring the first few numbers. Each number is also 0.
618 of the number to the right of it, again ignoring the first few numbers in the sequence. The golden ratio is ubiquitous in nature where it describes everything from the number of veins in a leaf to the magnetic resonance of spins in cobalt niobate crystals.
- Fibonacci numbers and lines are created by ratios found in Fibonacci's sequence.
- Common Fibonacci numbers in financial markets are 0.236, 0.382, 0.618, 1.618, 2.618, 4.236. These ratios or percentages can be found by dividing certain numbers in the sequence by other numbers.
- While not officially Fibonacci numbers, may traders also use 0.5, 1.0, and 2.0.
- The numbers reflect how far the price could go following another price move. For example, if a stock moves from $1 to $2, Fibonacci numbers can be applied to that. A drop to $1.76 is a 23.6% retracement of the $1 price move (rounded).
- Two common Fibonacci tools are retracements and extensions. Fibonacci retracements measure how far a pullback could go. Fibonacci extensions measure how far an impulse wave could go.
Fibonacci numbers don't have a specific formula, rather it is a number sequence where the numbers tend to have certain relationships with each other.
The Fibonacci number sequence can be used in different ways to get Fibonacci retracement levels or Fibonacci extension levels. Here's how to find them. How to use them is discussed in the next section.
Fibonacci retracements require two price points to be chosen on a chart, usually a swing high and a swing low. Once those two points are chosen, the Fibonacci numbers/lines are drawn at percentages of that move.
If a stock rises from $15 to $20, then the 23.6% level is $18.82 ($20 – ($5 x 0.236) = $18.82). The 50% level is $17.50 ($15 – ($5 x 0.5) = $17.50).
Fibonacci extension levels are also derived from the number sequence. As the sequence gets going, divide one number by the prior number to get a ratio of 1.618. Divide a number by two places to the left and the ratio is 2.618. Divide a number by three to the left and the ratio is 4.236.
A Fibonacci extension requires three price points. The start of a move, the end of a move, and then a point somewhere in between (the pullback).
If the price rises from $30 to $40, and these two price levels are points one and two, then the 161.8% level will be $16.18 (1.618 x $10) above the price chosen for point three. If point three is $35, the 161.8% extension level is $51.18 ($35 + $16.18).
The 100% and 200% levels are not official Fibonacci numbers, but they are useful since they project a similar move (or a multiple of it) to what just happened on the price chart.
Some traders believe that the Fibonacci numbers play an important role in finance. As discussed above, the Fibonacci number sequence can be used to create ratios or percentages that traders use.
These include: 23.6%, 38.2%, 50% 61.8%, 78.6%, 100%, 161.8%, 261.8%, 423.6%.
These percentages are applied using many different techniques:
- Fibonacci Retracements – These are horizontal lines on a chart that indicate areas of support and resistance.
- Fibonacci Extensions – These are horizontal lines on a chart that indicate where a strong price wave may reach.