fibonacci numbers

FibonacFibonacci ci

numbernumberss

Month 01 pair

Month 11 pair

Month 22 pairs

Month 33 pairs

19 April 2023

Jenny GageUniversity of Cambridge

Introductions and preliminary taskHumphrey Davy – flowersSeven Kings – flowersJohn of Gaunt – pine cones or pineapplesEllen Wilkinson – pine cones or pineapples

Fibonacci numbers in art and Fibonacci numbers in art and naturenature

Fibonacci numbers in Fibonacci numbers in naturenature

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 – c1250Liber Abaci, 1202, one of the first books to be published by a EuropeanOne of the first people to introduce the decimal number system into EuropeOn his travels saw the advantage of the Hindu-Arabic numbers compared to Roman numeralsRabbit problem – in the follow-up workAbout 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 3 5

8 13 21

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?

0.5

0.75

1

1.25

1.5

1.75

2

1 2 3 4 5 6 7 8 9 10

1 ÷ 1 = 1

2 ÷ 1 = 2

3 ÷ 2 = 1.5

5 ÷ 3 = 1.667

8 ÷ 5 = 1.6

13 ÷ 8 = 1.625

21 ÷ 13 = 1.615

34 ÷ 21 = 1.619

1 ÷ 1 = 1

1 ÷ 2 = 0.5

2 ÷ 3 = 0.667

3 ÷ 5 = 0.6

5 ÷ 8 = 0.625

8 ÷ 13 = 0.61513 ÷ 21 = 0.619

21 ÷ 34 = 0.617

1.618

0.618

Some mathematical properties Some mathematical properties of Fibonacci numbersof Fibonacci numbers

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.

Report Report back at back at 13.4513.45 E

W

J G

S K

H D

Are our bodies based on Are our bodies based on Fibonacci numbers?Fibonacci numbers?

Find the ratio ofHeight (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)

What do you notice?

Report Report back at back at 14.0014.00

SpiralsSpiralsUse the worksheet, and pencils, compasses and rulers, to create spirals based on Fibonacci numbers

Compare your spirals with this nautilus shell

Display of Display of spirals at spirals at

14.2514.25

What have Fibonacci numbers got to do with:

Pascal’s triangleCoin combinationsBrick wallsRabbits eating lettuces

Combine all that you want to say into one report

Report Report back at back at 14.5314.53