# fibonacci numbers

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Month 0 1 pair. Month 1 1 pair. Month 2 2 pairs. Month 3 3 pairs. Fibonacci numbers. 1 December, 2014 Jenny Gage University of Cambridge. Introductions and preliminary task Humphrey Davy – flowers Seven Kings – flowers John of Gaunt – pine cones or pineapples - PowerPoint PPT Presentation

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• Fibonacci numbers

*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 nature

• Fibonacci numbers in natureAn 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 youd never have thought of

• Complete the table of Fibonacci numbers

• 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?

• 1 1 = 12 1 = 23 2 = 1.55 3 = 1.6678 5 = 1.613 8 = 1.62521 13 = 1.61534 21 = 1.6191 1 = 11 2 = 0.52 3 = 0.6673 5 = 0.65 8 = 0.6258 13 = 0.61513 21 = 0.61921 34 = 0.617 1.618 0.618

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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.Find the sum of the first 1, 2, 3, 4, Fibonacci numbersAdd up F1, F1 + F3, F1 + F3 + F5, Add up F2, F2 + F4, F2 + F4 + F6, Divide each Fibonacci number by 11, ignoring any remainders.Report back at 13.45

E WJ GS KH D

• Are our bodies based on 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 back at 14.00

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

Compare your spirals with this nautilus shellDisplay of spirals at 14.25

• What have Fibonacci numbers got to do with:Pascals triangleCoin combinationsBrick wallsRabbits eating lettucesCombine all that you want to say into one report

Report back at 14.53