View
255
Download
2
Category
Preview:
Citation preview
5.5 Fibonacci's Rabbits 2
“Fibonacci” (Leonardo de Pisa)
1170-1240
And this man’s claim to fame is …?
5.5 Fibonacci's Rabbits 4
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
5.5 Fibonacci's Rabbits 5
The Fibonacci Rabbit Problem
How many pairs of rabbits will
there be 12 months later?
5.5 Fibonacci's Rabbits 6
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
5.5 Fibonacci's Rabbits 7
Jan 1
Feb 1
Mar 12
0
1
0
5.5 Fibonacci's Rabbits 8
Apr 1
May 1
Mar 12
3
4
0
1
2 1
0
0
0
5.5 Fibonacci's Rabbits 9
Apr 1
May 1
June 1
3 1 0
4 2 1 0 0
5 3 2 1 1
5.5 Fibonacci's Rabbits 10
Pairs this month
Generalize
Pairs last month
Pairs of newborns= +
Pairs this month
Pairs last month
Pairs 2 months
ago
= +
5.5 Fibonacci's Rabbits 11
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
5.5 Fibonacci's Rabbits 12
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
5.5 Fibonacci's Rabbits 13
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)
5.5 Fibonacci's Rabbits 14
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
5.5 Fibonacci's Rabbits 15
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)
5.5 Fibonacci's Rabbits 16
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
5.5 Fibonacci's Rabbits 17
Exponential Growth
112358 1321345589
144 1 year
2 years
3 years
4 years
46,368
14,930,352
4,807,526,976
Examples
5.5 Fibonacci's Rabbits 20
Chromatic Scale
Fibonacci numbers?
5.5 Fibonacci's Rabbits 22
One-petaled ... white calla lily
5.5 Fibonacci's Rabbits 23
Two-petaled flowers… euphorbia
5.5 Fibonacci's Rabbits 24
Three petals… trillium
5.5 Fibonacci's Rabbits 26
Eight-petaled… bloodroot
5.5 Fibonacci's Rabbits 27
Thirteen... black-eyed susan
5.5 Fibonacci's Rabbits 28
Twenty-one… shasta daisy with 21 petals
5.5 Fibonacci's Rabbits 29
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
5.5 Fibonacci's Rabbits 30
“Fibs”
Six line, 20 syllable poem
OneSmall,
Precise,Poetic,
Spiraling mixture:Math plus poetry yields the Fib
5.5 Fibonacci's Rabbits 34
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
5.5 Fibonacci's Rabbits 38
Parthenon - Athens
5.5 Fibonacci's Rabbits 40
Spiral Generated by Golden Ratio
5.5 Fibonacci's Rabbits 44
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.
5.5 Fibonacci's Rabbits 48
(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.
5.5 Fibonacci's Rabbits 49
Spiral out Keep going Spiral out Keep going Spiral out Keep going Spiral out! Keep going
More lyrics
5.5 Fibonacci's Rabbits 50
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.”
5.5 Fibonacci's Rabbits 51
Fibonacci Joke
How much does a large order of Fibonaccos cost?.
The price of a medium
The price of a small
+
Recommended