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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".
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 .
For example, the golden ratio, φ ≈ 1.618, is a real quadratic integer that is greater than 1, while the absolute value of its conjugate, −φ −1 ≈ −0.618, is less than 1. Therefore, φ is a Pisot number. Its minimal polynomial is x 2 − x − 1.
In geometry, a golden rectangle is a rectangle with side lengths in golden ratio +:, or :, with approximately equal to 1.618 or 89/55. Golden rectangles exhibit a special form of self-similarity : if a square is added to the long side, or removed from the short side, the result is a golden rectangle as well.
Several properties and common features of the Penrose tilings involve the golden ratio = + (approximately 1.618). [31] [32] This is the ratio of chord lengths to side lengths in a regular pentagon, and satisfies φ = 1 + 1/ φ.
The ratio of the progression of side lengths is , where = (+) / is the golden ratio, and the progression can be written: ::, or approximately 1 : 1.272 : 1.618. Squares on the edges of this triangle have areas in another geometric progression, 1 : φ : φ 2 {\displaystyle 1:\varphi :\varphi ^{2}} .
Dynamic symmetry is a proportioning system and natural design methodology described in Hambidge's books. The system uses dynamic rectangles, including root rectangles based on ratios such as √ 2, √ 3, √ 5, the golden ratio (φ = 1.618...), its square root (√ φ = 1.272...), and its square (φ 2 = 2.618....), and the silver ratio (=).
Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. [12] Both volumes have formulas involving the golden ratio but are taken to different powers. [1] Golden rectangle may also related to both regular icosahedron and regular dodecahedron. The regular ...