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What is the golden mean? How is it seen in nature?

The golden mean is a mathematical quantity represented by the symbol φ and defined by φ = 1 + 1/φ. Its value is (1 + √5)/2, or approximately 1.618.

φ is known by various names such as the golden ratio, golden propertion, golden section, etc.  The ratio of 1.618 to 1 has long been considered an aesthetically pleasing proportion and is often used in art and architecture. The ancients spotted so many occurrences of the ratio in nature that they believed it was favored by the gods, which is why ϕ is sometimes called the Divine Proportion.

φ does appear in nature, although some of the most famous examples, such as the spiral shell of the chambered nautilus, are incorrect. The ratio of each spiral's diameter to the next is not, despite popular belief, golden. Beautiful and elegant, yes, but not a golden spiral. If you want more examples, Google "golden ratio myths".

Genuine examples of φ can be found in the plant world. Look at the seed head of a sunflower (http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html). The seeds are arranged in opposing spirals, and the ratio of each rotation's diameter to the next is--φ. Why do the seeds arrange that way naturally? Note that the pattern allows contiguous, compact growth of the seed head. Maintaining the ratio φ allows all the spaces to be filled. Since φ is irrational, the spirals do not repeat in the same place and seeds don't overlap. It's efficient, and the seeds are less exposed.

There are many more examples; Google "golden ratio plants."

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#### Janet Yang

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I can answer questions in Algebra, Basic Math, Calculus, Differential Equations, Geometry, Number Theory, and Word Problems. I would not feel comfortable answering questions in Probability and Statistics or Topology because I have not studied these in depth.

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