Golden Rectangle Fibonacci Squares It's not magic, it's math! Ms. Moshtagh's Classroom Liber Abaci & what follows... Golden Ratio Bringing it to Life: Two Problems Where Fibonacci all began... Fibonacci was born in Pisa, Italy 1175 AD - 1250 AD Greatest European mathematician of the Middle Ages Full name Leonardo of Pisa (Leonardo Pisano in Italian) His name was short for Filius Bonacci (meaning son of Bonacci) which was combined to create the nickname Fibonacci As a young boy he daydreamed of numbers day and night In 1192 he brought his son, Fibonacci to Bugia with dreams for his son to become a merchant Fibonacci was taught in North Africa in calculation techniques in an Arabian school of accounting As the son of a merchant Fibonacci traveled freely around the Byzantine Empire visiting many centers of trade His travels allowed him to recognize the advantages to the mathematical systems and calculating schemes in popular use at the time His father, Guilielmo Bonacci, was a secretary of the republic of Pisa and a customs officer and merchant in the North African city of Bugia He discovered the Hindu–Arabic numeral system to be particularly simple and efficient (this system gave a competitive edge to the merchants who used it) Ended travels in 1200 and returned to Pisa Then he wrote several important mathematical texts compiling the math skills/techniques he obtained from his travels and making his own contributions 1202 right after Fibonacci’s travels he wrote the Liber abaci which introduced Arabic numerals including digits from 0-9 and place value (fractions, multiplication/divisions/subtraction/addition, etc.) He also added a compilation of the techniques of Arabic arithmetic and algebra involving practical calculation The Liber abaci also gave techniques for solving day to day problems found in commercial activities (i.e. money changing and conversions) Holy Roman emperor Frederick II, a man of influence at the time, became aware of Fibonacci’s work & was impressed when he was able to provide solutions for problems he posed This caused widespread interest in Fibonacci’s works The book popularized Hindu–Arabic numerals in Europe replacing the complicated and time consuming use of Roman Numerals at the time In 1240, Fibonacci was honored with a salary for advising on matters of accounting and teaching the citizens, and this is the last record of him after 1228 The Fibonacci Sequence In the Liber Abaci, Fibonacci presented this problem:

How many pairs of rabbits will be produced in a year beginning with a single pair if in every month each pair bears a new pair which becomes productive exactly two months after birth? His problem considers the circumstances that the rabbits do not die or escape, the female always produces only ONE pair of baby rabbits and the pair of kits are not of the same sex Explanation of the Table:

Month 1: The original pair of rabbits

Month 2: The original pair of rabbits

Month 3: The original pair of rabbits gives birth, totaling 2 pairs of rabbits

Month 4: The original pair gives birth again making two pairs and the pair produced in month 3 remains, this totals three pairs

Month 5: The original pair gives birth again, the pair produced in month three gives birth as well and the pair in month four remains, this totals 5 pairs

And so on….... If you list the number of pairs in order you form a sequence, the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 The rule?

Each successive number is a sum of the previous terms The sequence is neither arithmetic nor geometric however it can be represented by a recursive formula

Fn = Fn- + Fn- 1 2 The golden ratio concerns the ratio of the consecutive numbers in the Fibonacci sequence When one term is divided by the previous term the answer is always close to 1.618 As you move further down the Fibonacci sequence, the answer gets closer and closer to 1.618 1.618 is known as phi (this figure is rounded) Phi is an irrational number meaning it has an infinite number of decimal places and never repeats itself, it is also known as the Golden Ratio (a real number that cannot be written as a simple fraction) Another example of an irrational number is pi which is rounded as 3.14 It is called the golden ratio because it is said to be a divine proportion Many artists including Leonardo Davinci used the golden ratio and golden rectangle(i.e. Mona Lisa) The Great Pyramid of Giza, believed to be 4,600 years old has dimensions based on the Golden Ratio Golden rectangles were often used for beauty and balance in the design of architecture for the Greeks Used in Renaissance art Used in Egyptian pyramids Used in the Notre dame in Paris The ratio of length to width in the golden rectangle is approx. 1.618 (the golden ratio) The rectangle has been considered the most aesthetically pleasing to the eye. DID YOU KNOW?

If you cut off a square section with sides equal to the shorter side of the golden rectangle the piece remaining is also a golden rectangle The Fibonacci sequence is 1, 1, 3, 5, 8, 11, ..... The first square is 1 by 1, and then you add another square that’s also 1 by 1 . Then you follow by adding a square that’s 2 by 2 continue to add squares that are the length of the longer side of the rectangles being created You’ll notice the longer side is always the successive Fibonacci number The larger the rectangle the more it'll appear to be a golden rectangle Fibonacci squares are also known as whirling squares because as more rectangles are added, they grow in the clockwise direction of a spiral

Problem #1

Part 1: Creating the Arithmetic Sequence (An arithmetic sequence is when each term is calculated by adding the same constant to the previous term)

a (initial term) = 1531 & d (common difference) = 77

tn= a+(n - 1)d

=1531 + (n - 1) 77

= 77n + 1454

Therefore the formula is tn= 77n + 1454

Part 2: The year that corresponds with the 25th appearance (sub 25 for n)

t(25) = 77 (25) +1454

= 3379

Therefore it's 25th appearance will be in 3379 Fibonacci Sprial Fibonacci Spiral is a geometric spiral This spiral is created by drawing arcs connecting the opposite corners of the squares in the Fibonacci squares Problem #2

a) The next 3 years that will be the year of the dragon are 2012, 2024, 2036 (adding 12 each time)

b) Graph for Part a

c) The formula for the nth term for year of the Dragon

tn= a+( n - 1 )d

= 2000+(n-1)12

= 12n+ 1988

d)

Is 2146 the year of the dragon?

2146= 12n+1988

158= 12n

13.16= n

Therefore, not the year of dragon because their is a decimal answer

Is 2168 the year of the Dragon?

2168=12n+1988

180= 12n

15= n

Therefore, 2168 will be the year of dragon because the term number is a natural number. Fibonacci was a great man as evident by his many accomplishments. He created the most famous recursion formula of all time, the Fibonacci sequence. Fibonacci also focused on connecting the world around him to math. The problems he posed in his books were often based on real life scenarios. Two problems are about to be posed, both of which are also based on real life scenarios, and like Fibonacci we're going to create formulas to describe the sequences we encounter in order to solve the problems. The "Smart" Board Here's a little video to reinforce the concepts Term Number Year of The Dragon DID YOU KNOW?

If you look closely at Pascal's Triangle you can see Fibonacci's sequence DID YOU KNOW?

There is a statue of Fibonacci in the Camposanto gallery in Pisa DID YOU KNOW?

The golden ratio is often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron DID YOU KNOW?

Golden ratio can be achieved if you divide a line into two parts so that: the longer part divided by the smaller part is also equal to the whole length divided by the longer part. The golden rectangle and golden ratio can be found in nature, human body proportions and architecture!

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# Fibonacci

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