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Consider the reproductive cycle of bees. Each male bee has a mother but no father; each female bee has both a mother and a father. If we examine the generations we see the following family tree: We easily see that the number of ancestors in each generation is the sum of the two numbers before it. For example, our male bee has three great-grandparents, two grandparents, and one parent, and 3 = 2 + 1. The number of ancestors a bee has in generation n is defined by the Fibonacci sequence; we can also see this by applying the rule of sum. As a second example, consider light entering two adjacent planes of glass: At any meeting surface (between the two panes of glass, or between the glass and air), the light may either reflect or continue straight through (refract). For example, here is the light bouncing seven times before it leaves the glass. |