Leonardo Pisano Fibonacci was an Italian mathematician from the 13th century known for introducing the decimal numeral system and the Fibonacci sequence to Western Europe. The Fibonacci sequence is a series of numbers where each subsequent number is the sum of the previous two, starting with 0 and 1. This sequence appears frequently in nature, such as in the spiral pattern of flower petals, seed heads, pinecones, and branching in trees and galaxies. It is also used in computer programming, poetry meter, and as a technical analysis tool in finance to determine support and resistance levels in stock prices.
What patterns can we find in nature? Plants, flowers and fruits have all kinds of patterns, from petal numbers that are in the Fibonacci sequence, to symmetry, fractals and tessellation.
The fibonacci sequence and the golden ratio #Scichallenge2017Miléna Szabó
#SciChallenge2017
In this presentation I would like to show how important mathematics is. It is shows up in everyday life through nature.
"In order to understand the Universe you must know the language which it is written and that is Mathematics." /Galileo Galilei/
In this presentation, we see that how golden ratio and Fibonacci series are calculated or working? and What is the problem faced by Fibonacci in Fibonacci series.
Maths in Art and Architecture Why Maths? Comenius projectGosia Garkowska
THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
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by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
What patterns can we find in nature? Plants, flowers and fruits have all kinds of patterns, from petal numbers that are in the Fibonacci sequence, to symmetry, fractals and tessellation.
The fibonacci sequence and the golden ratio #Scichallenge2017Miléna Szabó
#SciChallenge2017
In this presentation I would like to show how important mathematics is. It is shows up in everyday life through nature.
"In order to understand the Universe you must know the language which it is written and that is Mathematics." /Galileo Galilei/
In this presentation, we see that how golden ratio and Fibonacci series are calculated or working? and What is the problem faced by Fibonacci in Fibonacci series.
Maths in Art and Architecture Why Maths? Comenius projectGosia Garkowska
THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WHY MATHS?
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
"Harmonic and Other Sequences" presentation includes a brief historical background, problems and solutions to the simplest problems which you may face in your Mathematics.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
2. Who was Fibonacci?
Also referred to as Leonard of Pisa, Fibonacci was
an Itallian number theorist. It is believed that
Leonardo Pisano Fibonacci was born in the 13th
century, in 1170 (approximately) and that he died in
1250. Fibonacci was born in Italy but obtained his
education in North Africa. Fibonacci is considered
to be one of the most talented mathematicians for
the Middle Ages. Few people realize that it was
Fibonacci that gave us our decimal number system
(Hindu-Arabic numbering system) which replaced
the Roman Numeral system. When he was studying
mathematics, he used the Hindu-Arabic (0-9)
symbols instead of Roman symbols which didn't
have 0's and lacked place value. In fact, when using
the Roman Numeral system, an abacus was usually
required. There is no doubt that Fibonacci saw the
superiority of using Hindu-Arabic system over the
Roman Numerals. He shows how to use our current
numbering system in his book Liber abaci.
3. Introduction
In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer
sequence:
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the
previous two.
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation
with seed values
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
Fn= Fn-1 + Fn-2
F0=0, F1=1
5. The problem
A certain man put a pair of rabbits in a place surrounded on
all sides by a wall. How many pairs of rabbits can be
produced from that pair in a year if it is supposed that
every month each pair begets a new pair, which from the
second month on becomes productive?
6. How he solved it:
Fibonacci's experiment shows that over a period of time, a pair of rabbits will
reproduce at a rate expressed in his identified sequence of numbers. One pair
of rabbits will create a pair of offspring. When those rabbits mature, they will
create another pair of offspring, and during that time, the older, original pair of
rabbits will have created an additional pair of offspring, etc.
8. Fibonacci and Poetry
In English, we tend to think of poetry as lines of text that rhyme, that is, lines that end with similar sounds as in this children's song:
Twinkle twinkle little star
How I wonder what you are.
Also we have the rhythm of the separate sounds (called syllables). Words like twinkle have two syllables: twin- and -kle whereas words such as star have
just one. Some syllables are stressed more than others so that they sound louder (such as TWIN- in twinkle), whereas others are unstressed and quieter
(such as -kle in twinkle).
If we let S stand for a stressed syllable and s an unstressed one, then the stress-pattern of each line of the song or poem has the rhythm SsSsSsS.
In Sanskrit poetry syllables are are either long or short. All the syllables in the song above take about the same length of time to say whether they are
stressed or not, so all the lines take the same amount of time to say. However cloudy sky has two words and three syllables CLOW-dee SKY, but the first
and third syllables are stressed and take a longer to say then the other syllable.
Let's assume that long syllables take just twice as long to say as short ones.
So we can ask the question: In Sanskrit poetry, if all lines take the same amount of time to say, what combinations of short (S) and long (L) syllables can
we have?
For one time unit, we have only one short syllable to say: S = 1 way
For two time units, we can have two short or one long syllable: SS and L = 2 ways
For three units, we can have: SSS, SL or LS = 3 ways
Any guesses for lines of 4 time units? Four would seem reasonable - but wrong! It's five! (SSSS, SSL, SLS, LSS and LL)
The general answer is that lines that take n time units to say can be formed in Fib(n) ways. This was noticed by Acarya Hemacandra about 1150 AD or
70 years before Fibonacci published his first edition of Liber Abaci in 1202.
Prof George Eckel Duckworth's book“Structural patterns and proportions in Virgil's Aeneid: a study in mathematical composition”argues that Virgil
consciously used Fibonacci numbers to structure his poetry and so did other Roman poets of the time.
9. Fibonacci and Plants
The leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we will see how
it is that this plant exhibits Fibonacci qualities.
In the case of tapered pinecones or pineapples, we see a double set of spirals – one going in a clockwise direction and one in the opposite direction.
When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.
Similarly, sunflowers have a Golden Spiral seed arrangement. This provides a biological advantage because it maximizes the number of seeds that can
be packed into a seed head.
As well, many flowers have a Fibonacci number of petals. Some, like this rose, also have Fibonacci, or Golden Spiral, petal arrangements.
10. Fibonacci and Animals
The shell of the chambered Nautilus has Golden proportions. It is a logarithmic spiral.
The eyes, fins and tail of the dolphin fall at Golden Sections along the body.
Humans exhibit Fibonacci characteristics, too. The Golden Ratio is seen in the proportions in the
sections of a finger. It is also worthwhile to mention that we have 8 fingers in total, 5 digits on each
hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.
The cochlea of the inner ear forms a Golden Spiral.
11. Fibonacci and Computer Science
Computer Scientists like the Fibonacci sequence because it is a good example of something
that can be programmed easily using what is known as recursion. Recursion just means you
define something using a simpler version of itself: If we write the 5th Fibonacci number (which
is 8) as fib(5), the 4th (which is 5) as fib(4) and so on then we can calculate it as:
That tells a computer to calculate fib(5) by calculating fib(3) and fib(4) first, both simpler
Fibonacci calculations, and then add them together. fib(4) and fib(3) are worked out in the
same way using simpler calculations again. We can write this to work for any number (let's
call it n) as:
That just says that for any number n that is bigger than 1, work out the nth Fibonacci number
by first working out the previous two, fib(n-2) and fib(n-1), and adding them. We then just
have to say how to do the simple cases you eventually end up at, when n is either 1 or 0:
Define fib(n) = fib(n-2) + fib(n-1) if n > 1
Define fib(5) = fib(3) + fib(4)
Define fib(n) = fib(n-2) + fib(n-1) if n > 1
| fib(1) = 1
| fib(0) = 1
12. Fibonacci and Finance
In finance, Fibonacci retracements is a method of technical
analysis for determining support and resistance levels. They are
named after their use of the Fibonacci sequence. Fibonacci
retracement is based on the idea that markets will retrace a
predictable portion of a move, after which they will continue to
move in the original direction.
The appearance of retracement can be ascribed to ordinary price
volatility as described by Burton Malkiel, a Princeton economist in
his book A Random Walk Down Wall Street, who found no reliable
predictions in technical analysis methods taken as a whole. Malkiel
argues that asset prices typically exhibit signs of random walk and
that one cannot consistently outperform market averages.
Fibonacci retracement is created by taking two extreme points on
a chart and dividing the vertical distance by the key Fibonacci
ratios. 0.0% is considered to be the start of the retracement, while
100.0% is a complete reversal to the original part of the move.
Once these levels are identified, horizontal lines are drawn and
used to identify possible support and resistance levels.