5.5 Fibonacci's Rabbits 1 Section 5.5 Fibonacci’s Problem.

5.5 Fibonacci's Rabbits 1

Section 5.5

Fibonacci’s Problem


“Fibonacci” (Leonardo de Pisa)

1170-1240

And this man’s claim to fame is …?

http://images.google.com/imgres?imgurl=http://www.uni-flensburg.de/mathe/zero/mgalerie/fibonacci/fibonacci.jpg&imgrefurl=http://www.uni-flensburg.de/mathe/zero/veranst/arithalgebra/schreiber/glossar/fibonacci_folge.htm&h=207&w=174&sz=11&tbnid=2XhQHmVuXxSwHM:&tbnh=100&tbnw=84&hl=en&start=6&prev=/images%3Fq%3DFibonacci%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DGGLD,GGLD:2004-25,GGLD:en%26sa%3DN


Rabbit Rules

1. All pairs of rabbits consist of a male and female

2. One pair of newborn rabbits is placed in hutch on January 1

3. When this pair is 2 months old they produce a pair of baby rabbits

4. Every month afterwards they produce another pair

5. All rabbits produce pairs in the same manner

6. Rabbits don’t die


The Fibonacci Rabbit Problem

How many pairs of rabbits will

there be 12 months later?


How many pairs of rabbits will there be on June 1?

100%

0% 0%0%0%

1. 5

2. 7

3. 8

4. 11

5. 13


Jan 1

Feb 1

Mar 12

0

1

0

http://images.google.com/imgres?imgurl=http://www.rabbits-bunnies.com/pictures-images-photos/rabbits-bunnies-01.jpg&imgrefurl=http://www.rabbits-bunnies.com/&h=158&w=180&sz=12&hl=en&start=3&tbnid=0yPp36sy9-xlIM:&tbnh=89&tbnw=101&prev=/images%3Fq%3Drabbits%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DTSHA,TSHA:2005-13,TSHA:en%26sa%3DN



http://images.google.com/imgres?imgurl=http://www.designsbydorian.com/images/brown%2520rabbits.jpg&imgrefurl=http://www.designsbydorian.com/Dorian/Rabbits_Wedding_Cake_Tops.htm&h=288&w=288&sz=11&hl=en&start=12&tbnid=Ff2ZMY0hjsOMLM:&tbnh=115&tbnw=115&prev=/images%3Fq%3Drabbits%26ndsp%3D20%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DTSHA,TSHA:2005-13,TSHA:en%26sa%3DN


Apr 1

May 1

Mar 12

3

4

0

1

2 1

0

0

0


http://images.google.com/imgres?imgurl=http://home1.pacific.net.sg/~siowinc/big_grey_rabbit.jpg&imgrefurl=http://cgi.ebay.com.sg/VERY-RARE-11-HUGE-Sylvanian-Families-Rabbit-pair-NEW_W0QQitemZ270002405663QQihZ017QQcategoryZ2459QQcmdZViewItem&h=389&w=505&sz=54&hl=en&start=16&tbnid=U0ko8sCXT6b3iM:&tbnh=100&tbnw=130&prev=/images%3Fq%3Drabbits%2Bpair%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DTSHA,TSHA:2005-13,TSHA:en%26sa%3DN











Apr 1

May 1

June 1

3 1 0

4 2 1 0 0

5 3 2 1 1





















Pairs this month

Generalize

Pairs last month

Pairs of newborns= +

Pairs this month

Pairs last month

Pairs 2 months

ago

= +


Pattern of the Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, …

So the answer to the original problem is .

Joke

Rule: Every term is the of the two preceding terms


The 16th term of the Fibonacci sequence is 987 and the 17th term is 1597. What is the 19th term?

2584

4181

6765

0%

100%

0%

1. 2584

2. 4181

3. 6765


Change Fibonacci’s problem slightly so that each pair of adult rabbits produces 2 pairs per litter. Which recursion formula best describes the rabbit population?

33%

33%

33%

1. This month = Last month + (Two months ago)

2. This month = Last month + 2*(Two months ago)

3. This month = 2*Last month + (Two months ago)


As in the last problem assume that each pair produces 2 pairs per litter from the second month on.

How many pairs will there be in 5 months?

1. 8

2. 10

3. 11


New Problem: Assume each pair of adult rabbits produces one pair monthly from the 4th month on. Which recursive formula best describes the rabbit population?

1. This month = Last month + (Four months ago)

2. This month = Last month + 3*(Four months ago)

3. This month = Last month + 4*(Four months ago)


As in last problem, assume each pair of adult rabbits produces one pair monthly from the 4th month on. How many pairs of rabbits will there be in 7 months?

1. 5

2. 8

3. 13

4. 21


Exponential Growth

112358 1321345589

144 1 year

2 years

3 years

4 years

46,368

14,930,352

4,807,526,976

Examples


End of 5.5


Fibonacci Suite for retuned piano, seven hands

Music


Chromatic Scale

Fibonacci numbers?

http://images.google.com/imgres?imgurl=http://www.smu.edu/totw/keybrd2.gif&imgrefurl=http://www.smu.edu/totw/def.htm&h=177&w=306&sz=9&hl=en&start=1&tbnid=SmceQpK5b0iq1M:&tbnh=68&tbnw=117&prev=/images%3Fq%3Dchromatic%2Bscale%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DGGLD,GGLD:2005-49,GGLD:en%26sa%3DN


5 Spirals

Botany


One-petaled ... white calla lily

http://ccins.camosun.bc.ca/~jbritton/fibslide/fib01.jpg


Two-petaled flowers… euphorbia



Three petals… trillium



Five petals morning glory


Eight-petaled… bloodroot



Thirteen... black-eyed susan



Twenty-one… shasta daisy with 21 petals



I

am

sitting

quietly,

listening for the

quiet noises in the darkness,

ghostly images flying between the tall pine trees,

illusion created by the mind, made by shadows, the brain playing tricks on

itself.

It sits there, the raven, black as night, looking at me with its dark eyes in the dark night. Inspiration comes. Words

form in my head. Evermore.

Poetry

Jim T. Henriksen

http://allpoetry.com/Poem/1733415

http://allpoetry.com/Poem/1733415


“Fibs”

Six line, 20 syllable poem

OneSmall,

Precise,Poetic,

Spiraling mixture:Math plus poetry yields the Fib


Investments

Robert Fischer, leader of the Fibonacci approach to trading


Education


21 341313

Algebra

55 …3421

21* = * =441

442

853211


Fibonacci and the Greeks

8/5 = 1.613/8 = 1.62521/13 = 1.6153846…34/21 = 1.6190476…55/34 = 1.6176470…89/55 = 1.6181818…144/89 = 1.6179775…233/144 = 1.6180555…

2/1 = 23/2 = 1.5

5/3 = 1.666666…

1, 1, 2, 3, 5, 8, 13, 21, 34, …

1/1 = 1


Golden Number

1.61808…


Golden Rectangle

1

1.618


Mona Lisa

Classical Art


Parthenon - Athens

http://images.google.com/imgres?imgurl=http://mp082.k12.sd.us/parthenon.gif&imgrefurl=http://mp082.k12.sd.us/&h=435&w=580&sz=113&hl=en&start=7&um=1&tbnid=KUwpc5NuFG6EPM:&tbnh=101&tbnw=134&prev=/images%3Fq%3Dparthenon%2B%26svnum%3D10%26um%3D1%26hl%3Den%26rls%3DTSHA,TSHA:2005-13,TSHA:en


The Vitruvian Man


Spiral Generated by Golden Ratio

http://nlvm.usu.edu/en/nav/frames_asid_133_g_3_t_3.html




The Nautilus Shell


Modern Art

Piet Mondrian


Diets and Fitness


Mo and the Boys

There are 100 measures in the first movement. The first section, with the theme, has 32 measures, and the last section, with theme variations, has 68 measures. This is a perfect division, using natural numbers, with the golden section.

Although there is no physical evidence that Mozart used the Fibonacci sequence in his music, it is still very easy to see the use of perfect divisions.


Fibonacci Rap


Fibonacci Waltz


Lateralus

By Tool


(1)Black, (1) then,

(2) white are,(3) all I see,

(5) in my infancy,(8) red and yellow then

came to be, (5) reaching out to me,

(3) lets me see. (2) There is,

(1) so,

“Lateralus”

(1) much, (2) more that

(3) beckons me,(5) to look through to

these,(8) infinite possibilities. (13) As below so above and beyond I imagine,(8) drawn outside the

lines of reason.(5) Push the envelope.

(3) Watch it bend.


Spiral out Keep going Spiral out Keep going Spiral out Keep going Spiral out! Keep going

More lyrics


Facts

• Keenan begins singing at 1:37 into the song. 1 minute 37 seconds, or 97 seconds, is approximately 1.618 of a full minute.

• The time signatures of the chorus change from 9/8 to 8/8 to 7/8; as drummer Danny Carey says, "It was originally titled 9-8-7. For the time signatures. Then it turned out that 987 was the 17th number of the Fibonacci sequence. So that was cool.”


Fibonacci Joke

How much does a large order of Fibonaccos cost?.

The price of a medium

The price of a small

+


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