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Fibonacci numbers Month 0 1 pair Month 1 1 pair Month 2 2 pairs Month 3 3 pairs 10 February, 2014 Jenny Gage University of Cambridge

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

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Fibonacci numbers in art and nature

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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.

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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

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1123 Complete the table of Fibonacci numbers

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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?

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1 ÷ 1 = 1 2 ÷ 1 = 2 3 ÷ 2 = ÷ 3 = ÷ 5 = ÷ 8 = ÷ 13 = ÷ 21 = ÷ 1 = 1 1 ÷ 2 = ÷ 3 = ÷ 5 = ÷ 8 = ÷ 13 = ÷ 21 = ÷ 34 =

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Some mathematical properties of 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 F 1, F 1 + F 3, F 1 + F 3 + F 5, … 3. Add up F 2, F 2 + F 4, F 2 + F 4 + F 6, … 4. Divide each Fibonacci number by 11, ignoring any remainders. Report back at E W J G S K H D

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Are our bodies based on Fibonacci numbers? 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) What do you notice? Report back at 14.00

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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

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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

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