There is one thing more painful than learning from mistakes (our own or somebody else's): Not learning from them.  Barbara Johnson 
Fibonacci SequencePhi Golden Ratio or Golden Mean RatioThe Phi Golden Ratio or Golden Mean Ratio was first published by Fibonacci, apparently as a simple exercise without realizing its far reaching implications in nature. Fibonacci's example was: A pair of rabbits are put in a field and if rabbits take a month to become mature, then produce a new pair every month after that, how many pairs will there be in twelve months time? The answer to this riddle is the famous Fibonacci Sequence, which represents the number of rabbits each month. It looks like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. Starting with 1, 1, the next value is simply the addition of the two previous values. This series of values wasn't known as Fibonancci Series until a French matematician, Edouard Lucas (18421891), found many more applications for the series.
Fibonacci Sequence in NatureFibonacci's example problem has a wide variety of applications in nature since it illustrates the growth pattern of many species including trees, seeds, honeybee genetics, etc.
Optimal Packing with the Fibonacci SeriesOne of the places that the Fibonacci Series shows up routinely in nature is in the growth of seeds on pine cones and some flowers. The number of seeds in each row is increased by the Fibonacci series. This series allows for optimal packing of seeds in a spiral manner. New seeds on a sunflower are only formed at the very center. Yet, when you see a fully grown sunflower, you'll notice that it has hundreds of seeds that are all evenly spaced throughout the face. Look closely and you can see the spiral of seeds. You'll notice the same in the daisy. The magic of seed and plants growing like this comes from the magic in the Fibonacci series and Phi golden mean.
FlowersMany flowers and plants will have some Fibonacci number of pedals or leaves. For example:
Examples of the Fibonacci Sequence in the Real World


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