The Golden Ratio in Nature

    Growth in nature depends on a variety of factors.
    The following "rules" do not apply to all circumstances -- they are a "fascinatingly prevalent tendency"

    The ratio of the number of female drones to the number of male drones in any large beehive is close to the Golden Ratio

    		The ratio of each spiral's radius to the next approaches the golden ratio-- the shape does not change as the size increases. That is, if you zoom in on a small portion, or zoom out to look at the whole, it has the same shape.

    This is an example of a logarithmic spiral (a logarithmic spiral is any spiral where the radius increases by the same proportion by which the length of the spiral increases). There are many examples of logarithmic spirals in nature: sunflowers, whirlpools, hurricanes, spiral galaxies.

    		Looking down at the top of the pineapple, and picturing the "leaves" as being added in a tightly wound spiral as it develops, the "leaves" advance at roughly 137.5 degrees all around the circle. (Similarly for the sunflowr and the pinecone).

    What is the significance of 137.5 degrees? It is determined by the Golden Ratio -- the angle that divides a complete turn in a Golden Ratio is (roughly) 222.5 degrees:

    360/222.5 = 1.617977528 =1.618, to the third decimal place.

    Of course, 222.5 degrees is more than half way around the circle. We could get to the same place by going in the opposite direction around the circle. By how many degrees?

    360-222.5=137.5 degrees!

    The eye will pick out two spirals. The number of counter-clockwise and clockwise spirals tends to be consecutive Fibonacci numbers ... which are also related to the Golden Ratio.

    This growth angle of 137.5 degrees is called the Golden Angle. It seems to be the most efficient angle of growth. Mathematical algorithms have demonstrated that when a growth angle equal to the Golden Angle is used, structures that closely resemble sunfowers are obtained.


    		If you dissect a rose (petal by petal), you will discover the positions of if its tightly packed petals. The figure below the rose (from Mario Livio's book The Golden Ratio) shows a diagram with the positions of the petals being numbered. The angles defining the positions of the petals --in fractions of a full turn -- are the fractional parts of simple multiples of the Golden Ratio.

    For example,

    • Petal 1 is 0.618 of a turn from Petal 0.
      .618 is the fractional part of 1 x Phi
    • Petal 2 is .236 of a turn from Petal 1.
      .236 is the fractional part of 2 x Phi

    The Golden Ratio in Architecture

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