Fibonacci numbers Month 0 1 pair Month 1
Fibonacci numbers Month 0 1 pair Month 1 1 pair 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, c 1175 – c 1250 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
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 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÷ 1=1 2 ÷ 3 = 0. 667 5 ÷ 8 = 0. 625 1 ÷ 2 = 0. 5 0. 618 13 ÷ 21 = 0. 619 21 ÷ 34 = 0. 617 3 ÷ 5 = 0. 6 8 ÷ 13 = 0. 615
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 F 1, F 1 + F 3 + F 5, … Go high enough to see if your rules or patterns break down after a bit! 3. Add up F 2, 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 elbows (green) Length of forearm (yellow) : length of hand (purple) Report 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
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