Fibonacci numbers

1
Fibonacci numbers

• 27 October, 2013
• Jenny Gage
• University of Cambridge

2

• Introductions and preliminary task
• Humphrey Davy flowers
• Seven Kings flowers
• John of Gaunt pine cones or pineapples
• Ellen Wilkinson pine cones or pineapples

3
Fibonacci numbers in art and nature
4
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.

5

• 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
  Hindu-Arabic numbers compared to Roman numerals
• Rabbit problem in the follow-up work
• About how maths is related to all kinds of things
  youd never have thought of

6
1 1 2 3



Complete the table of Fibonacci numbers
7
1 1 2 3 5
8 13 21


8
1 1 2 3 5
8 13 21 34 55
89 144

9
1 1 2 3 5
8 13 21 34 55
89 144 233 377 610
987
10
1 1 2 3 5
8 13 21 34 55
89 144 233 377 610
987 1597 2584 4181 6765
11

• 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?

12
2 1 2
5 3 1.667
13 8 1.625
34 21 1.619
? 1.618
21 13 1.615
8 5 1.6
3 2 1.5
1 1 1
1 1 1
2 3 0.667
5 8 0.625
? 0.618
13 21 0.619
21 34 0.617
3 5 0.6
8 13 0.615
1 2 0.5
13
Some mathematical properties of Fibonacci numbers
Report back at 13.45

• Try one or more of these.
• Try to find some general rule or pattern.
• Go high enough to see if your rules or patterns
  break down after a bit!
• Justify your answers if possible.

1. Find the sum of the first 1, 2, 3, 4, Fibonacci
   numbers
2. Add up F1, F1 F3, F1 F3 F5,
3. Add up F2, F2 F4, F2 F4 F6,
4. Divide each Fibonacci number by 11, ignoring any
   remainders.

E W
J G
S K
H D
14
Are our bodies based on Fibonacci numbers?
What do you notice?

• 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)

Report back at 14.00
15
Spirals

• Use the worksheet, and pencils, compasses and
  rulers, to create spirals based on Fibonacci
  numbers
• Compare your spirals with this nautilus shell

Display of spirals at 14.25
16

• What have Fibonacci numbers got to do with
• Pascals triangle
• Coin combinations
• Brick walls
• Rabbits eating lettuces
• Combine all that you want to say into one report

Report back at 14.53