Download Fibonacci Numbers

Based on the following PowerPoint's
Fibonacci Numbers
The Fibonacci Numbers and The Golden Section
M Golenberke 6/2012
Who Was Fibonacci?
~ Born in Pisa, Italy in 1175 AD
~ Full name was Leonardo Pisano
~ Grew up with a North African education under the Moors
~ Traveled extensively around the Mediterranean coast
~ Met with many merchants and learned their systems of arithmetic
~ Realized the advantages of the Hindu-Arabic system
The Fibonacci Numbers
~ Were introduced in The Book of Calculating
~ Series begins with 0 and 1
~ Next number is found by adding the last two numbers together
~ Number obtained is the next number in the series
~ Pattern is repeated over and over
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
Fn + 2 = Fn + 1 + Fn
The Fibonacci Numbers
}+
The number pattern that you have been using is known as
the Fibonacci sequence.
1 1 2
The Fibonacci Numbers
}+
The number pattern that you have been using is known as
the Fibonacci sequence.
1 1 2 3
The Fibonacci Numbers
}+
The number pattern that you have been using is known as
the Fibonacci sequence.
1 1 2 3 5 8 13 21 34
55
These numbers can be seen in many natural situations
Fibonacci’s Rabbits
Problem:
Suppose a newly-born pair of rabbits (one male, one female) are put in
a field. Rabbits are able to mate at the age of one month so that at the
end of its second month, a female can produce another pair of rabbits.
Suppose that the rabbits never die and that the female always produces
one new pair (one male, one female) every month from the second
month on. How many pairs will there be in one year?
Fibonacci’s Rabbits Continued
~ End of the first month = 1 pair
~ End of the second month = 2 pair
~ End of the third month = 3 pair
~ End of the fourth month = 5 pair
~ 5 pairs of rabbits produced in one year
1, 1, 2, 3, 5, 8, 13, 21, 34, …
The Fibonacci Numbers in Nature
~ Fibonacci spiral found in both snail and sea shells
The Fibonacci Numbers in
Nature Continued
~ Sneezewort (Achillea ptarmica) shows the Fibonacci numbers
Fibonacci’s sequence… in nature
One plant in particular shows
the Fibonacci numbers in the
number of "growing points"
that it has.
Suppose that when a plant
puts out a new shoot, that
shoot has to grow two
months before it is strong
enough to support branching.
If it branches every month
after that at the growing
point, we get the picture
shown here.
A plant that grows very much
like this is the "sneezewort“.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…
The Fibonacci Numbers in
Nature Continued
~ Plants show the Fibonacci numbers in the arrangements of their leaves
~ Three clockwise rotations, passing five leaves
~ Two counter-clockwise rotations
The Fibonacci Numbers in
Nature Continued
~ Pinecones clearly show the Fibonacci spiral
Fibonacci’s sequence… in nature
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…
The Fibonacci Numbers in
Nature Continued
Lilies and irises = 3 petals
Corn marigolds = 13 petals
Buttercups and wild roses = 5 petals
Black-eyed Susan’s = 21 petals
The Fibonacci Numbers in
Nature Continued
~ The Fibonacci numbers are found in the arrangement of seeds on flower heads
~ 55 spirals spiraling outwards and 34 spirals spiraling inwards
Fibonacci’s sequence… in nature
On many plants, the number of petals is a
Fibonacci number:
Buttercups have 5 petals; lilies and iris have 3
petals; some delphiniums have 8; corn marigolds
have 13 petals; some asters have 21 whereas
daisies can be found with 34, 55 or even 89 petals.
13 petals: ragwort, corn marigold, cineraria, some
daisies
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae
family.
Some species are very precise about the number
of petals they have - eg buttercups, but others
have petals that are very near those above, with
the average being a Fibonacci number.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…
The Fibonacci Numbers in Nature
Continued
~ Fibonacci spiral can be found in cauliflower
Fibonacci’s sequence… in nature
Take a look at a cauliflower.
Count the number of florets in the spirals.
The number in one direction and in the other
will be Fibonacci numbers.
If you look closer at a single floret, it is a mini
cauliflower with its own little florets all
arranged in spirals around a centre.
If you can, count the spirals in both
directions. How many are there?
Now look at the stem. Where the florets
are rather like a pinecone or pineapple.
The florets were arranged in spirals up
the stem. Counting them again shows
the Fibonacci numbers.
Try the same thing for broccoli.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…
The Fibonacci Numbers in
Nature Continued
~ The Fibonacci numbers can be found in pineapples and bananas
~ Bananas have 3 or 5 flat sides
~ Pineapple scales have Fibonacci spirals in sets of 8, 13, 21
The Fibonacci Numbers in Nature
Continued
~ The Fibonacci numbers can be found in the human hand and fingers
~ Person has 2 hands, which contain 5 fingers
~ Each finger has 3 parts separated by 2 knuckles
Fibonacci in Nature
The lengths of bones in a
hand are Fibonacci
numbers.
Class Activity
http://www.missmaggie.org/mission4_part
s/eng/teaching/pdfs/mathinnature.pdf
Patterns in Fibonacci
Complete the ‘Fibonacci Numbers’ sheet using column addition
1+1
1+2
2+3
3+5
5+8
8+13
13+21
=
=
=
=
=
=
=
2
3
5
8
13
21
34
Colour all the even numbers in blue and all the even numbers
in red.
What do you notice about the patterns in the colouring?
Patterns in Fibonacci
On another completed Fibonacci sheet,
use your knowledge of divisibility rules to
a) colour in blue the multiples of 5
b) colour in red the multiples of 3
c) underline the multiples of 6
What patterns can you see?
What if you extend the sequence?
Are there any patterns in the multiples of 10?
Square Spiral
29 little boxes down
1 little square
15 little
boxes
across
Box
Side
1
1
29 little boxes down
1 more little square
15 little
boxes
across
Box
Side
1
1
2
1
29 little boxes down
15 little
boxes
across
Box
Side
1
1
2
1
3
2
Box
Side
1
1
2
1
3
2
4
Box
Side
1
1
2
1
3
2
4
Now complete the pattern to fill
your page as much as possible.
Box
Side
1
1
2
1
3
2
4
Box
Side
1
1
2
1
3
2
4
Box
Side
1
1
2
1
3
2
4
Box
Side
1
1
2
1
3
2
4
Draw your own “Nautilus”
shell by drawing quarter
circles in each square and
joining to form the spiral
Fibonacci in Nature
Spirals, like the one you drew earlier are common in nature.