Maravillas Matematicas - Fibonacci

Bunnies, Bees, and Pineapples - Oh, my! An exploration of the Fibonacci Sequence
Amanda Seitz Sam Houston State University
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Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

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Fibonacci Sequence
• This famous mathematical sequence was discovered by Leonardo Fibonacci in 1202 with his publication of Liber Abaci. • It was unofficially recognized before by the ancientEgyptians and their Greek students, but French mathematician Edouard Lucas later named the sequence after Leonardo Fibonacci.
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Fibonacci’s History
• Leonardo of Pisa
– Born in Pisa, Italy in 1175. – Traveled with his father to the north coast of Africa. – While there, he is believed to have studied from Muslim schoolmasters and learned the Hindu-Arabic numeral system.
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Fibonacci’s History– Wrote under the name Fibonacci which was derived from filius Bonacci. – Influenced the change from Roman numerals to Hindu-Arabic numerals in Europe with his work Liber Abaci.

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Rabbit Problem
• Fibonacci’s work discussed a “rabbit problem,” which asked, "How many pairs of rabbits will there be after a year if it is assumed that every month each pair produces one new pair, which beginsto bear young two months after its own birth?” • The solution to this problem made the sequence famous.
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Rabbit Problem
• Young adult rabbits • Baby rabbits

• Adult breeding rabbits
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Rabbit Problem – January 1

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Rabbit Problem – February 1

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Rabbit Problem – March 1

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Rabbit Problem – April 1

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Rabbit Problem – May 1

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Rabbit Problem – June 113

Rabbit Problem

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Fibonacci Sequence in Nature
• The Fibonacci sequence can also be found in nature.

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A Bee’s Genealogy
• A drone, or male bee – hatches from an unfertilized egg. • A female bee – hatches from an egg that has been fertilized. • When finding the ancestry of a male bee, we observe that the Fibonacci sequence is formed.
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A Male Bee’s Ancestry
• Let fdenote a female bee.
• Let m denote a male bee.

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A Male Bee’s Ancestry
• Let

f denote a female bee.

• Let m denote a male bee.

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A Bee’s Genealogy
Total number of female bees 0,1,1,2,3,5,8,13…

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A Male Bee’s Ancestry
• Let

f denote a female bee.

• Let m denote a male bee.

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A Bee’s Genealogy

• Total number of male bees

1,0,1,1,2,3,5,8…

21A Male Bee’s Ancestry
• Let

f denote a female bee.

• Let m denote a male bee.

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A Bee’s Genealogy

• Total number of bees

1,1,2,3,5,8,13,21…
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A Bee’s Genealogy
• Total number of male bees • Total number of female bees • Total number of bees 1,0,1,1,2,3,5,8…

0,1,1,2,3,5,8,13…

1,1,2,3,5,8,13,21…
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A Pineapple’s Scales

• In this example, the pineapplehas: - 5 spirals ascending slowly to the right. - 8 spirals ascending quickly to the left. - 13 spirals ascending quickly to the right.
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A Pineapple’s Scales
• 3, 5, 8, 13, and even 21 spirals can be found according to size. • Consecutive sets of spirals run opposite of each other. (i.e., 5 would run opposite of 8). • The higher the number of spirals, the steeper that spiral will be. Thus,no spirals moving opposite of each other will have the same spiral.
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A Coneflower’s Spiraling Seeds

• The seeds in the coneflower above form 34 spiraling to the left (green) and 55 spirals going to the right (blue).
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A Pinecone’s Spiraling Bracts

• The bracts on the pinecone above form 8 spirals going to the right (green) and 13 spiraling to the left (yellow).
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The MathBehind the Sequence
• As stated before, the Fibonacci sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21… • The next term of the sequence can be found by Sn = Sn-2 + Sn-1, where Sn represents the nth term in the sequence with the two initials S1 = 1 and S2 = 1.

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The Math Behind the Sequence
So we see, 1, 1, 1+1, 1+2, 2+3, 3+5, 5+8, 8+13, … 1, 1, 2, 3, 5, 8, 13, 21, …

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The Golden...