1. 27 March, 2017 Jenny Gage University of Cambridge
Fibonacci numbers
Month 0
1 pair
Month 1
Month 2
2 pairs
Month 3
3 pairs
27 March, 2017
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 you’d never have thought of

6. Complete the table of Fibonacci numbers1 2 3
Complete the table of Fibonacci numbers

7. 1 2 3 5 8 13 21

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

9. 1 2 3 5 8 13 21 34 55 89
144
233
377
610
987

10. 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
13 ÷ 21 = 0.619
 0.618
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.
Find the sum of the first 1, 2, 3, 4, … Fibonacci numbers
Add up F1, F1 + F3, F1 + F3 + F5, …
Add up F2, F2 + F4, F2 + F4 + F6, …
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

16. What have Fibonacci numbers got to do with:
Pascal’s triangle
Coin combinations
Brick walls
Rabbits eating lettuces
Combine all that you want to say into one report
Report back at 14.53