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10.) Explain the concepts of Fibonacci in nature using three items as examples.

Fibonacci used his pattern (0, 1, 1, 2, 3, 5, 8, 13, 21...keep adding the previous two digits) to describe and count a honeybee's ancestors. There is one queen, many worker bees, and some drone bees. The females are produced when the queen mates with a male; therefore, they have two parents. Male bees, however, only have one parent, a female. Male drone bees have 1 parent, 2 grandparents, 3 great-grandparents, 5 great-great grandparents, and so on. Female bees have 2 parents, 3 grandparents, 5 great-grandparents, 8 great-great grandparents, and so on, continuing with the Fibonacci numbers.

Fibonacci also used his spiral pattern to describe the distribution of segments of the pinecones. One pattern is shown above in the yellow, the other in the green. There are 8 or 13 whirls on a pinecone, depending on which direction you follow. Each level, when viewed from the side, has a certain number of scales that matches a Fibonacci number.

You can also explain how plant leaves go up in spirals, so that, when viewed from the top, we can see that each leaf gets as much sunlight as possible. These spirals are in increments and proportions according to Fibonacci numbers.

Fibonacci picture found at http://www.math.gatech.edu/~morley/images/Fibonacci.jpeg

Information on honeybees found at http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html

Information and pictures of pinecones and plants found at http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fibonacci2.html