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The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. [8] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically . Natural patterns include symmetries , trees , spirals , meanders , waves , foams , tessellations , cracks and stripes. [ 1 ]
The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers .
The Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 C.E.) uses the golden ratio in the design including its plan, the prayer space, court, and minaret, [19] but the ratio does not appear in the original parts of the mosque. [20]
Leaves may also be whorled if several leaves arise, or appear to arise, from the same level (at the same node) on a stem. Veronicastrum virginicum has whorls of leaves separated by long internodes. With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the same node), on opposite sides of the stem. An opposite ...
Approximate and true golden spirals: the green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a golden spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.
The British actor’s eye, eyebrow, nose, lips, chin, jaw, and facial shape measurements were found to be 93.04% aligned with the Golden Ratio, an equation used by the ancient Greeks to measure ...
Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur in Plants is a book on the mathematics of plant structure, and in particular on phyllotaxis, the arrangement of leaves on plant stems. It was written by Irving Adler, and published in 2012 by World Scientific.