The seeds of a sunflower, the spines of a cactus, and the bracts of a pinecone all form whirling spiral patterns. Remarkable for their regularity and beauty, these natural structures also show some surprising mathematical properties.
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In more than 90 percent of the spiral formations made by plants, the angle between successive elements of the spiral—the leaves on a stem, for example—is approximately the golden angle. That geometrical quantity, which is about 137.5°, is closely related to the famous “golden ratio” and was first studied by the ancient Greeks. Furthermore, hidden within the spirals of many plants is a pattern involving the celebrated Fibonacci sequence of numbers, which is closely related to the golden ratio. Spiral patterns involving the golden angle and the Fibonacci sequence pop up throughout the natural world, in objects as disparate as galaxies and seashells.