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Many works of art are claimed to have been designed using the golden ratio. However, many of these claims are disputed, or refuted by measurement. [1] The golden ratio, an irrational number, is approximately 1.618; it is often denoted by the Greek letter φ .
These golden, radiant streams, like Yggdrasill’s branches, create, sustain, and transform the world around us." Image credits: Nature Photographer of the Year (NPOTY) 2024 #15 Category Birds ...
The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. [92] Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".
Patterns in Nature. Little, Brown & Co. Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. Patterns from nature (as art) Edmaier, Bernard. Patterns of the Earth. Phaidon Press, 2007. Macnab, Maggie. Design by Nature: Using Universal Forms and Principles in Design. New Riders, 2012. Nakamura, Shigeki.
The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the golden ratio. In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the larger arc is the same as ...
where F n is the nth Fibonacci number. The ratio of numbers of kites to darts in any sufficiently large P2 Penrose tiling pattern therefore approximates to the golden ratio φ. [47] A similar result holds for the ratio of the number of thick rhombs to thin rhombs in the P3 Penrose tiling. [45]
Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences . The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. [ 1 ]
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