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# Visual Patterns

## Describe non-arithmetic sequences by finding a rule.

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The Math Behind It All

Credit: Don Hankins
Source: http://www.flickr.com/photos/23905174@N00/2465852393/
License: CC BY-NC 3.0

Have you ever wondered if there was some sort of mathematical principle behind certain natural phenomena? Well, as it turns out, one such idea can be seen in the reproductive patterns of rabbits and bees, the petals of many flowers, the structure of pine cones, the shape of artichokesand much more! How can this be?

#### Why It Matters

In 1202, an Italian mathematician by the name of Leonardo Fibonacci wanted to investigate how quickly rabbit populations would grow under ideal circumstances. Under such conditions, each female would produce exactly two offspring, one male and one female, and each new pair could only reproduce after two months of maturing. So, starting with a single pair, there would be one pair in the first month. In the second month, there would still only be one pair, as they cannot yet breed. However, by the third month, the original pair would produce a pair of offspring, yielding two pairs total. In the fourth month, the original pair would produce another pair of offspring while the second pair of rabbits would still be maturing, so there would be three pairs total. By the fifth month, both the original and second pair of rabbits could reproduce, which would bring the total to five pairs. By this point, a pattern had begun to reveal itself: 1, 1, 2, 3, 5… It soon became clear that the next number in the sequence was determined by adding the previous two terms. This discovery would not be that surprising until people began to realize that the Fibonacci sequence of 1, 1, 2, 3, 5, 8, 13… appeared in nature in many other instances!

Credit: Jessica Crabtree
Source: http://www.flickr.com/photos/57403893@N02/5286398108/
License: CC BY-NC 3.0

This pattern of numbers applies to much more beyond Fibonacci’s rabbits! It just so happens that male bees only have a mother, while female bees have both a mother and a father. So, if you look at the ancestry of a male bee, it will have one parent (the mother), two grandparents, three great grandparents, five great-great grandparents, and so on. The pattern appears again. In addition, the flowering patterns of several plants, such as daisies and sunflowers, are related to the Fibonacci sequence.

See for yourself: http://www.youtube.com/watch?v=ahXIMUkSXX0

#### Explore More

For even more examples of how the Fibonacci sequence occurs in nature, check out the link below.

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