Remember the way the golden spiral was constructed. That same shape is found in many shells, particularly the nautilus. The golden spiral increased in size by a factor of Ø every 1/4 turn. The spiral in a nautilus shell increases in size by a factor of Ø every full turn. That means that given two points on the spiral a full turn apart, the one further out is f times as far away from the center as the first point. That's because the spiral is equiangular, or logarithmic.

      In plants, the relationship is a little less obvious, but still there. In 1979, a scientist named Vogel (which incidentally means "bird" in German) theorized that primordia, no matter what they developed into, could fill space most efficiently when the divergence angle was an irrational multiple of 360. Arranging leaves, branches, and seeds this way insures that as the plant grows rain will be channeled down to the roots, and everything will receive the most sunlight possible. If you'll recall, the most irrational of all numbers is Ø.

      At the tip of each branch, is a meristem, which is where all new cells are formed. It is continuously pushed out and rotated by the growth of older cells behind it. The angle at which it turns is about 137.5 degrees. If you take Ø, multiply it by 360, and then subtract the product from 360, the resulting difference is 137.5.

      This characteristic was first emphasized by the Bravais brothers in 1837, and later reexamined by Stephane Douady and Yves Couder. Douady and Couder devised computer models to expand upon this theory. The found that the primordia growing on the meristem diverged at an angle of 360(1-Ø), or approximately 137.5.

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