Transcript Document

Fibonacci
numbers
Month 0
1 pair
Month 1
1 pair
21 July, 2015
Month 2
2 pairs
Jenny Gage
University of Cambridge
Month 3
3 pairs
Introductions and preliminary task
Humphrey Davy – flowers
Seven Kings – flowers
John of Gaunt – pine cones or
pineapples
Ellen Wilkinson – pine cones or
pineapples
Fibonacci numbers in art and nature
Fibonacci numbers in nature
An example of efficiency in nature.
As each row of seeds in a sunflower or
pine cone, or petals on a flower grows,
it tries to put the maximum number in
the smallest space.
Fibonacci numbers are the whole
numbers which express the golden
ratio, which corresponds to the angle
which maximises number of items in
the smallest space.
Why are they called Fibonnaci numbers?
Leonardo of Pisa, c1175 – c1250
Liber Abaci, 1202, one of the first books to be
published by a European
One of the first people to introduce the decimal
number system into Europe
On his travels saw the advantage of the HinduArabic numbers compared to Roman numerals
Rabbit problem – in the follow-up work
About how maths is related to all kinds of things
you’d never have thought of
1
1
2
3
Complete the table of
Fibonacci numbers
1
1
2
8
13
21
3
5
1
1
2
3
5
8
13
21
34
55
89
144
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
Find the ratio of successive
Fibonacci numbers:
1 : 1, 2 : 1, 3 : 2, 5 : 3, 8 : 5, …
1 : 1, 1 : 2, 2 : 3, 3 : 5, 5 : 8, …
What do you notice?
2÷1=2
2
5 ÷ 3 = 1.667
13 ÷ 8 = 1.625
34 ÷ 
21 1.618
= 1.619
21 ÷ 13 = 1.615
8 ÷ 5 = 1.6
3 ÷ 2 = 1.5
1.75
1.5
1.25
1
0.75
1÷1=1
1÷1=1
2 ÷ 3 = 0.667
5 ÷ 8 = 0.625
0.5
1
1 ÷ 2 = 0.5
2
3
0.618
13 ÷ 21 = 
0.619
21 ÷ 34 = 0.617
3 ÷ 5 = 0.6 8 ÷ 13 = 0.615
4
5
6
7
8
9
10
Some mathematical properties
of Fibonacci numbers
Report
back at
13.45
1.
Try one or more of
these.
Find the sum of
the first 1, 2, 3, 4,
… Fibonacci
numbers
Try to find some
general rule or
pattern.
2.
Add up F1, F1 + F3,
F 1 + F 3 + F 5, …
Go high enough to
see if your rules or
patterns break down
after a bit!
3.
Add up F2, F2 + F4,
F 2 + F 4 + F 6, …
4.
Divide each
Fibonacci number
by 11, ignoring any
remainders.
Justify your answers
if possible.
EW
JG
SK
HD
Are our bodies based on
Fibonacci numbers?
Find the ratio of
Height (red) : Top
of head to
fingertips (blue)
Top of head to
fingertips (blue) :
Top of head to
elbows (green)
Length of forearm
(yellow) : length
of hand (purple)
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back at
14.00
What do
you
notice?
Spirals
Use the worksheet,
and pencils,
compasses and rulers,
to create spirals based
on Fibonacci numbers
Display of spirals
at 14.25
Compare your spirals
with this nautilus shell
What have Fibonacci numbers got
to do with:
Pascal’s triangle
Coin combinations
Brick walls
Rabbits eating lettuces
Report
back at
14.53
Combine all that you want to say
into one report