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

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 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

Complete the table of Fibonacci numbers 1 2 3 Complete the table of Fibonacci numbers

1 2 3 5 8 13 21

1 2 3 5 8 13 21 34 55 89 144

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

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 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

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

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

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

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